Integrand size = 25, antiderivative size = 80 \[ \int (d \cot (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\frac {(d \cot (e+f x))^m \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+n p),\frac {1}{2} (3-m+n p),-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1-m+n p)} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3740, 2684, 3557, 371} \[ \int (d \cot (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\frac {\tan (e+f x) (d \cot (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (-m+n p+1),\frac {1}{2} (-m+n p+3),-\tan ^2(e+f x)\right )}{f (-m+n p+1)} \]
[In]
[Out]
Rule 371
Rule 2684
Rule 3557
Rule 3740
Rubi steps \begin{align*} \text {integral}& = \left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int (d \cot (e+f x))^m (c \tan (e+f x))^{n p} \, dx \\ & = \left ((d \cot (e+f x))^m (c \tan (e+f x))^{m-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int (c \tan (e+f x))^{-m+n p} \, dx \\ & = \frac {\left (c (d \cot (e+f x))^m (c \tan (e+f x))^{m-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \text {Subst}\left (\int \frac {x^{-m+n p}}{c^2+x^2} \, dx,x,c \tan (e+f x)\right )}{f} \\ & = \frac {(d \cot (e+f x))^m \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+n p),\frac {1}{2} (3-m+n p),-\tan ^2(e+f x)\right ) \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1-m+n p)} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.96 \[ \int (d \cot (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\frac {d (d \cot (e+f x))^{-1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} (1-m+n p),\frac {1}{2} (3-m+n p),-\tan ^2(e+f x)\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (1-m+n p)} \]
[In]
[Out]
\[\int \left (d \cot \left (f x +e \right )\right )^{m} \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}d x\]
[In]
[Out]
\[ \int (d \cot (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \cot \left (f x + e\right )\right )^{m} \,d x } \]
[In]
[Out]
\[ \int (d \cot (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \left (d \cot {\left (e + f x \right )}\right )^{m}\, dx \]
[In]
[Out]
\[ \int (d \cot (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \cot \left (f x + e\right )\right )^{m} \,d x } \]
[In]
[Out]
\[ \int (d \cot (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int { \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \cot \left (f x + e\right )\right )^{m} \,d x } \]
[In]
[Out]
Timed out. \[ \int (d \cot (e+f x))^m \left (b (c \tan (e+f x))^n\right )^p \, dx=\int {\left (d\,\mathrm {cot}\left (e+f\,x\right )\right )}^m\,{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \]
[In]
[Out]