Optimal. Leaf size=114 \[ -\frac{i (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \left (\frac{c+d \tan (e+f x)}{c+i d}\right )^{-n} F_1\left (m;-n,1;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d},\frac{1}{2} (i \tan (e+f x)+1)\right )}{2 f m} \]
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Rubi [A] time = 0.169104, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107, Rules used = {3564, 137, 136} \[ -\frac{i (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \left (\frac{c+d \tan (e+f x)}{c+i d}\right )^{-n} F_1\left (m;-n,1;m+1;-\frac{d (i \tan (e+f x)+1)}{i c-d},\frac{1}{2} (i \tan (e+f x)+1)\right )}{2 f m} \]
Antiderivative was successfully verified.
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Rule 3564
Rule 137
Rule 136
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \, dx &=\frac{\left (i a^2\right ) \operatorname{Subst}\left (\int \frac{(a+x)^{-1+m} \left (c-\frac{i d x}{a}\right )^n}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac{\left (i a^2 (c+d \tan (e+f x))^n \left (\frac{c+d \tan (e+f x)}{c+i d}\right )^{-n}\right ) \operatorname{Subst}\left (\int \frac{(a+x)^{-1+m} \left (\frac{c}{c+i d}-\frac{i d x}{a (c+i d)}\right )^n}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=-\frac{i F_1\left (m;-n,1;1+m;-\frac{d (1+i \tan (e+f x))}{i c-d},\frac{1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \left (\frac{c+d \tan (e+f x)}{c+i d}\right )^{-n}}{2 f m}\\ \end{align*}
Mathematica [F] time = 6.78242, size = 0, normalized size = 0. \[ \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^n \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.875, size = 0, normalized size = 0. \begin{align*} \int \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{m} \left ( c+d\tan \left ( fx+e \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}{\left (d \tan \left (f x + e\right ) + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\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} \left (\frac{{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}{\left (d \tan \left (f x + e\right ) + c\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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