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 = 0.09, 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 69
Rule 70
Rule 3523
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*}
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Mathematica [B] time = 14.43, 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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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\frac {2 \, c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n} e^{\left (2 i \, f m x + 2 i \, e m + m \log \left (\frac {a}{c}\right ) + m \log \left (\frac {2 \, c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.03, size = 0, normalized size = 0.00 \[ \int \left (a +i a \tan \left (f x +e \right )\right )^{m} \left (c -i c \tan \left (f x +e \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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