Integrand size = 27, antiderivative size = 201 \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \, dx=\frac {\operatorname {AppellF1}\left (1+n p,-m,1,2+n p,-\frac {b \tan (e+f x)}{a},-i \tan (e+f x)\right ) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \left (1+\frac {b \tan (e+f x)}{a}\right )^{-m}}{2 f (1+n p)}+\frac {\operatorname {AppellF1}\left (1+n p,-m,1,2+n p,-\frac {b \tan (e+f x)}{a},i \tan (e+f x)\right ) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \left (1+\frac {b \tan (e+f x)}{a}\right )^{-m}}{2 f (1+n p)} \]
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Time = 0.32 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3659, 3656, 926, 140, 138} \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \, dx=\frac {\tan (e+f x) (a+b \tan (e+f x))^m \left (\frac {b \tan (e+f x)}{a}+1\right )^{-m} \left (c (d \tan (e+f x))^p\right )^n \operatorname {AppellF1}\left (n p+1,-m,1,n p+2,-\frac {b \tan (e+f x)}{a},-i \tan (e+f x)\right )}{2 f (n p+1)}+\frac {\tan (e+f x) (a+b \tan (e+f x))^m \left (\frac {b \tan (e+f x)}{a}+1\right )^{-m} \left (c (d \tan (e+f x))^p\right )^n \operatorname {AppellF1}\left (n p+1,-m,1,n p+2,-\frac {b \tan (e+f x)}{a},i \tan (e+f x)\right )}{2 f (n p+1)} \]
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Rule 138
Rule 140
Rule 926
Rule 3656
Rule 3659
Rubi steps \begin{align*} \text {integral}& = \left ((d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n\right ) \int (d \tan (e+f x))^{n p} (a+b \tan (e+f x))^m \, dx \\ & = \frac {\left ((d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \frac {(d x)^{n p} (a+b x)^m}{1+x^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\left ((d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \left (\frac {i (d x)^{n p} (a+b x)^m}{2 (i-x)}+\frac {i (d x)^{n p} (a+b x)^m}{2 (i+x)}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {\left (i (d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \frac {(d x)^{n p} (a+b x)^m}{i-x} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac {\left (i (d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n\right ) \text {Subst}\left (\int \frac {(d x)^{n p} (a+b x)^m}{i+x} \, dx,x,\tan (e+f x)\right )}{2 f} \\ & = \frac {\left (i (d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \left (1+\frac {b \tan (e+f x)}{a}\right )^{-m}\right ) \text {Subst}\left (\int \frac {(d x)^{n p} \left (1+\frac {b x}{a}\right )^m}{i-x} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac {\left (i (d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \left (1+\frac {b \tan (e+f x)}{a}\right )^{-m}\right ) \text {Subst}\left (\int \frac {(d x)^{n p} \left (1+\frac {b x}{a}\right )^m}{i+x} \, dx,x,\tan (e+f x)\right )}{2 f} \\ & = \frac {\operatorname {AppellF1}\left (1+n p,-m,1,2+n p,-\frac {b \tan (e+f x)}{a},-i \tan (e+f x)\right ) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \left (1+\frac {b \tan (e+f x)}{a}\right )^{-m}}{2 f (1+n p)}+\frac {\operatorname {AppellF1}\left (1+n p,-m,1,2+n p,-\frac {b \tan (e+f x)}{a},i \tan (e+f x)\right ) \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \left (1+\frac {b \tan (e+f x)}{a}\right )^{-m}}{2 f (1+n p)} \\ \end{align*}
\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \, dx=\int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \, dx \]
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\[\int \left (c \left (d \tan \left (f x +e \right )\right )^{p}\right )^{n} \left (a +b \tan \left (f x +e \right )\right )^{m}d x\]
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\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \, dx=\int { \left (\left (d \tan \left (f x + e\right )\right )^{p} c\right )^{n} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \, dx=\int \left (c \left (d \tan {\left (e + f x \right )}\right )^{p}\right )^{n} \left (a + b \tan {\left (e + f x \right )}\right )^{m}\, dx \]
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\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \, dx=\int { \left (\left (d \tan \left (f x + e\right )\right )^{p} c\right )^{n} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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\[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \, dx=\int { \left (\left (d \tan \left (f x + e\right )\right )^{p} c\right )^{n} {\left (b \tan \left (f x + e\right ) + a\right )}^{m} \,d x } \]
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Timed out. \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+b \tan (e+f x))^m \, dx=\int {\left (c\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^p\right )}^n\,{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m \,d x \]
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