Optimal. Leaf size=66 \[ \frac{i (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n \, _2F_1\left (1,m+n;n+1;\frac{1}{2} (1-i \tan (e+f x))\right )}{2 f n} \]
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Rubi [A] time = 0.0940459, antiderivative size = 87, normalized size of antiderivative = 1.32, number of steps used = 3, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {3523, 70, 69} \[ -\frac{i 2^{n-1} (1-i \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n \, _2F_1\left (m,1-n;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right )}{f m} \]
Antiderivative was successfully verified.
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Rule 3523
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n \, dx &=\frac{(a c) \operatorname{Subst}\left (\int (a+i a x)^{-1+m} (c-i c x)^{-1+n} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\left (2^{-1+n} a (c-i c \tan (e+f x))^n \left (\frac{c-i c \tan (e+f x)}{c}\right )^{-n}\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2}-\frac{i x}{2}\right )^{-1+n} (a+i a x)^{-1+m} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{i 2^{-1+n} \, _2F_1\left (m,1-n;1+m;\frac{1}{2} (1+i \tan (e+f x))\right ) (1-i \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{f m}\\ \end{align*}
Mathematica [B] time = 14.1926, size = 142, normalized size = 2.15 \[ -\frac{i c 2^{m+n-1} \left (e^{i f x}\right )^m \left (\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \left (\frac{c}{1+e^{2 i (e+f x)}}\right )^{n-1} \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m \, _2F_1\left (1,1-n;m+1;-e^{2 i (e+f x)}\right )}{f m} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.653, size = 0, normalized size = 0. \begin{align*} \int \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{m} \left ( c-ic\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 (-i \, c \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{2 \, c}{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 (-i \, c \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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