What is the integral of tan^4(x)sec^3(x) ?

The integral of tan^4(x)sec^3(x) is

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integral of tan^4(x)sec^3(x)

\int tan^{4} (x) sec^{3} (x) d x

\frac{1}{16} sec (x) tan (x) + \frac{1}{16} ln ∣ tan (x) + sec (x) ∣ + \frac{1}{6} sec^{6} (x) sin (x) + \frac{5}{24} sec^{4} (x) sin (x) - \frac{1}{2} sec^{4} (x) sin (x) + C

Solution steps

\int tan^{4} (x) sec^{3} (x) d x

Rewrite using trig identities

= \int (- 1 + sec^{2} (x))^{2} sec^{3} (x) d x

Expand

(- 1 + sec^{2} (x))^{2} sec^{3} (x) : sec^{3} (x) - 2 sec^{5} (x) + sec^{7} (x)

= \int sec^{3} (x) - 2 sec^{5} (x) + sec^{7} (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int sec^{3} (x) d x - \int 2 sec^{5} (x) d x + \int sec^{7} (x) d x

\int sec^{3} (x) d x = \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

\int 2 sec^{5} (x) d x = 2 (\frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣))

\int sec^{7} (x) d x = \frac{1}{6} sec^{6} (x) sin (x) + \frac{5}{48} (2 sec^{4} (x) sin (x) + 3 (sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣))

= \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ - 2 (\frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)) + \frac{1}{6} sec^{6} (x) sin (x) + \frac{5}{48} (2 sec^{4} (x) sin (x) + 3 (sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣))

Simplify

\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ - 2 (\frac{sec ^{4} ( x ) sin ( x )}{4} + \frac{3}{4} (\frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)) + \frac{1}{6} sec^{6} (x) sin (x) + \frac{5}{48} (2 sec^{4} (x) sin (x) + 3 (sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣)) : \frac{1}{16} sec (x) tan (x) + \frac{1}{16} ln ∣ tan (x) + sec (x) ∣ + \frac{1}{6} sec^{6} (x) sin (x) + \frac{5}{24} sec^{4} (x) sin (x) - \frac{1}{2} sec^{4} (x) sin (x)

= \frac{1}{16} sec (x) tan (x) + \frac{1}{16} ln ∣ tan (x) + sec (x) ∣ + \frac{1}{6} sec^{6} (x) sin (x) + \frac{5}{24} sec^{4} (x) sin (x) - \frac{1}{2} sec^{4} (x) sin (x)

Add a constant to the solution

= \frac{1}{16} sec (x) tan (x) + \frac{1}{16} ln ∣ tan (x) + sec (x) ∣ + \frac{1}{6} sec^{6} (x) sin (x) + \frac{5}{24} sec^{4} (x) sin (x) - \frac{1}{2} sec^{4} (x) sin (x) + C