Optimal. Leaf size=99 \[ \frac{\tan (e+f x) (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m F_1(n p+1;1-m,1;n p+2;-i \tan (e+f x),i \tan (e+f x)) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.187287, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3578, 3564, 135, 133} \[ \frac{\tan (e+f x) (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m F_1(n p+1;1-m,1;n p+2;-i \tan (e+f x),i \tan (e+f x)) \left (c (d \tan (e+f x))^p\right )^n}{f (n p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3578
Rule 3564
Rule 135
Rule 133
Rubi steps
\begin{align*} \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^m \, dx &=\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+i a \tan (e+f x))^m \, dx\\ &=\frac{\left (i a^2 (d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{i d x}{a}\right )^{n p} (a+x)^{-1+m}}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac{\left (i a (1+i \tan (e+f x))^{-m} (d \tan (e+f x))^{-n p} \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^m\right ) \operatorname{Subst}\left (\int \frac{\left (-\frac{i d x}{a}\right )^{n p} \left (1+\frac{x}{a}\right )^{-1+m}}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac{F_1(1+n p;1-m,1;2+n p;-i \tan (e+f x),i \tan (e+f x)) (1+i \tan (e+f x))^{-m} \tan (e+f x) \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^m}{f (1+n p)}\\ \end{align*}
Mathematica [F] time = 5.65535, size = 0, normalized size = 0. \[ \int \left (c (d \tan (e+f x))^p\right )^n (a+i a \tan (e+f x))^m \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.506, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( d\tan \left ( fx+e \right ) \right ) ^{p} \right ) ^{n} \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\left (d \tan \left (f x + e\right )\right )^{p} c\right )^{n}{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\left (c \left (\frac{-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{p}\right )^{n} \left (\frac{2 \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (\left (d \tan \left (f x + e\right )\right )^{p} c\right )^{n}{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]