Integrand size = 19, antiderivative size = 68 \[ \int \sec ^4(a+b x) (c \sin (a+b x))^m \, dx=\frac {\sqrt {\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) \sec (a+b x) (c \sin (a+b x))^{1+m}}{b c (1+m)} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2657} \[ \int \sec ^4(a+b x) (c \sin (a+b x))^m \, dx=\frac {\sqrt {\cos ^2(a+b x)} \sec (a+b x) (c \sin (a+b x))^{m+1} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {m+1}{2},\frac {m+3}{2},\sin ^2(a+b x)\right )}{b c (m+1)} \]
[In]
[Out]
Rule 2657
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) \sec (a+b x) (c \sin (a+b x))^{1+m}}{b c (1+m)} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.93 \[ \int \sec ^4(a+b x) (c \sin (a+b x))^m \, dx=\frac {\sqrt {\cos ^2(a+b x)} \operatorname {Hypergeometric2F1}\left (\frac {5}{2},\frac {1+m}{2},\frac {3+m}{2},\sin ^2(a+b x)\right ) (c \sin (a+b x))^m \tan (a+b x)}{b (1+m)} \]
[In]
[Out]
\[\int \left (\sec ^{4}\left (b x +a \right )\right ) \left (c \sin \left (b x +a \right )\right )^{m}d x\]
[In]
[Out]
\[ \int \sec ^4(a+b x) (c \sin (a+b x))^m \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right )^{4} \,d x } \]
[In]
[Out]
\[ \int \sec ^4(a+b x) (c \sin (a+b x))^m \, dx=\int \left (c \sin {\left (a + b x \right )}\right )^{m} \sec ^{4}{\left (a + b x \right )}\, dx \]
[In]
[Out]
\[ \int \sec ^4(a+b x) (c \sin (a+b x))^m \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right )^{4} \,d x } \]
[In]
[Out]
\[ \int \sec ^4(a+b x) (c \sin (a+b x))^m \, dx=\int { \left (c \sin \left (b x + a\right )\right )^{m} \sec \left (b x + a\right )^{4} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sec ^4(a+b x) (c \sin (a+b x))^m \, dx=\int \frac {{\left (c\,\sin \left (a+b\,x\right )\right )}^m}{{\cos \left (a+b\,x\right )}^4} \,d x \]
[In]
[Out]