Integrand size = 23, antiderivative size = 99 \[ \int (d \cos (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\frac {(d \cos (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (1-m+2 p)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1+2 p),\frac {1}{2} (1-m+2 p),\frac {1}{2} (3+2 p),\sin ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1+2 p)} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3739, 2683, 2697} \[ \int (d \cos (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\frac {\tan (e+f x) \left (b \tan ^2(e+f x)\right )^p (d \cos (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (-m+2 p+1)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (2 p+1),\frac {1}{2} (-m+2 p+1),\frac {1}{2} (2 p+3),\sin ^2(e+f x)\right )}{f (2 p+1)} \]
[In]
[Out]
Rule 2683
Rule 2697
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \left (\tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \int (d \cos (e+f x))^m \tan ^{2 p}(e+f x) \, dx \\ & = \left ((d \cos (e+f x))^m \left (\frac {\sec (e+f x)}{d}\right )^m \tan ^{-2 p}(e+f x) \left (b \tan ^2(e+f x)\right )^p\right ) \int \left (\frac {\sec (e+f x)}{d}\right )^{-m} \tan ^{2 p}(e+f x) \, dx \\ & = \frac {(d \cos (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (1-m+2 p)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2} (1+2 p),\frac {1}{2} (1-m+2 p),\frac {1}{2} (3+2 p),\sin ^2(e+f x)\right ) \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1+2 p)} \\ \end{align*}
Time = 0.94 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.82 \[ \int (d \cos (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\frac {(d \cos (e+f x))^m \operatorname {Hypergeometric2F1}\left (1+\frac {m}{2},\frac {1}{2}+p,\frac {3}{2}+p,-\tan ^2(e+f x)\right ) \sec ^2(e+f x)^{m/2} \tan (e+f x) \left (b \tan ^2(e+f x)\right )^p}{f (1+2 p)} \]
[In]
[Out]
\[\int \left (d \cos \left (f x +e \right )\right )^{m} \left (b \tan \left (f x +e \right )^{2}\right )^{p}d x\]
[In]
[Out]
\[ \int (d \cos (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cos \left (f x + e\right )\right )^{m} \,d x } \]
[In]
[Out]
\[ \int (d \cos (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int \left (b \tan ^{2}{\left (e + f x \right )}\right )^{p} \left (d \cos {\left (e + f x \right )}\right )^{m}\, dx \]
[In]
[Out]
\[ \int (d \cos (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cos \left (f x + e\right )\right )^{m} \,d x } \]
[In]
[Out]
\[ \int (d \cos (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int { \left (b \tan \left (f x + e\right )^{2}\right )^{p} \left (d \cos \left (f x + e\right )\right )^{m} \,d x } \]
[In]
[Out]
Timed out. \[ \int (d \cos (e+f x))^m \left (b \tan ^2(e+f x)\right )^p \, dx=\int {\left (d\,\cos \left (e+f\,x\right )\right )}^m\,{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}^p \,d x \]
[In]
[Out]