Optimal. Leaf size=150 \[ \frac{(B+i A) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 f n}-\frac{2^{n-1} (B (m-n)+i A (m+n)) (1-i \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n \text{Hypergeometric2F1}\left (m,-n,m+1,\frac{1}{2} (1+i \tan (e+f x))\right )}{f m n} \]
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Rubi [A] time = 0.227203, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {3588, 79, 70, 69} \[ \frac{(B+i A) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 f n}-\frac{2^{n-1} (B (m-n)+i A (m+n)) (1-i \tan (e+f x))^{-n} (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n \, _2F_1\left (m,-n;m+1;\frac{1}{2} (i \tan (e+f x)+1)\right )}{f m n} \]
Antiderivative was successfully verified.
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Rule 3588
Rule 79
Rule 70
Rule 69
Rubi steps
\begin{align*} \int (a+i a \tan (e+f x))^m (A+B \tan (e+f x)) (c-i c \tan (e+f x))^n \, dx &=\frac{(a c) \operatorname{Subst}\left (\int (a+i a x)^{-1+m} (A+B x) (c-i c x)^{-1+n} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{(i A+B) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 f n}-\frac{(a (i B (m-n)-A (m+n))) \operatorname{Subst}\left (\int (a+i a x)^{-1+m} (c-i c x)^n \, dx,x,\tan (e+f x)\right )}{2 f n}\\ &=\frac{(i A+B) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 f n}-\frac{\left (2^{-1+n} a (i B (m-n)-A (m+n)) (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 )^n (a+i a x)^{-1+m} \, dx,x,\tan (e+f x)\right )}{f n}\\ &=\frac{(i A+B) (a+i a \tan (e+f x))^m (c-i c \tan (e+f x))^n}{2 f n}-\frac{2^{-1+n} (B (m-n)+i A (m+n)) \, _2F_1\left (m,-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 n}\\ \end{align*}
Mathematica [A] time = 33.303, size = 197, normalized size = 1.31 \[ \frac{2^{m+n-1} \left (e^{i f x}\right )^m \left (\frac{e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \left (\frac{c}{1+e^{2 i (e+f x)}}\right )^n \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m \left ((m+1) (B-i A) \text{Hypergeometric2F1}\left (1,-n,m+1,-e^{2 i (e+f x)}\right )-i m (A-i B) e^{2 i (e+f x)} \text{Hypergeometric2F1}\left (1,1-n,m+2,-e^{2 i (e+f x)}\right )\right )}{f m (m+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 1.095, size = 0, normalized size = 0. \begin{align*} \int \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{m} \left ( A+B\tan \left ( fx+e \right ) \right ) \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 (B \tan \left (f x + e\right ) + A\right )}{\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 (\frac{{\left ({\left (A - i \, B\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + A + i \, B\right )} \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}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}, 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 (B \tan \left (f x + e\right ) + A\right )}{\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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