Optimal. Leaf size=91 \[ -\frac {a^2 (A-3 i B)}{c^2 f (\tan (e+f x)+i)}+\frac {a^2 (B+i A)}{c^2 f (\tan (e+f x)+i)^2}-\frac {a^2 B \log (\cos (e+f x))}{c^2 f}-\frac {i a^2 B x}{c^2} \]
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Rubi [A] time = 0.15, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3588, 77} \[ -\frac {a^2 (A-3 i B)}{c^2 f (\tan (e+f x)+i)}+\frac {a^2 (B+i A)}{c^2 f (\tan (e+f x)+i)^2}-\frac {a^2 B \log (\cos (e+f x))}{c^2 f}-\frac {i a^2 B x}{c^2} \]
Antiderivative was successfully verified.
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Rule 77
Rule 3588
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^2 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^2} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(a+i a x) (A+B x)}{(c-i c x)^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left (-\frac {2 i a (A-i B)}{c^3 (i+x)^3}+\frac {a (A-3 i B)}{c^3 (i+x)^2}+\frac {a B}{c^3 (i+x)}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {i a^2 B x}{c^2}-\frac {a^2 B \log (\cos (e+f x))}{c^2 f}+\frac {a^2 (i A+B)}{c^2 f (i+\tan (e+f x))^2}-\frac {a^2 (A-3 i B)}{c^2 f (i+\tan (e+f x))}\\ \end {align*}
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Mathematica [B] time = 3.61, size = 184, normalized size = 2.02 \[ \frac {a^2 (\cos (2 (e+2 f x))+i \sin (2 (e+2 f x))) \left (-i \cos (2 (e+f x)) \left (A-2 i B \log \left (\cos ^2(e+f x)\right )+8 B f x-i B\right )+A \sin (2 (e+f x))-8 B f x \sin (2 (e+f x))-i B \sin (2 (e+f x))+2 i B \sin (2 (e+f x)) \log \left (\cos ^2(e+f x)\right )+4 B \tan ^{-1}(\tan (3 e+f x)) (\sin (2 (e+f x))+i \cos (2 (e+f x)))+4 B\right )}{4 c^2 f (\cos (f x)+i \sin (f x))^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.69, size = 62, normalized size = 0.68 \[ \frac {{\left (-i \, A - B\right )} a^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, B a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 4 \, B a^{2} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{4 \, c^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.91, size = 201, normalized size = 2.21 \[ -\frac {\frac {6 \, B a^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{c^{2}} - \frac {12 \, B a^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}{c^{2}} + \frac {6 \, B a^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{c^{2}} + \frac {25 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 12 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 112 i \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 198 \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, A a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 112 i \, B a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 25 \, B a^{2}}{c^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}^{4}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 116, normalized size = 1.27 \[ \frac {3 i a^{2} B}{f \,c^{2} \left (\tan \left (f x +e \right )+i\right )}-\frac {a^{2} A}{f \,c^{2} \left (\tan \left (f x +e \right )+i\right )}+\frac {a^{2} B \ln \left (\tan \left (f x +e \right )+i\right )}{f \,c^{2}}+\frac {i a^{2} A}{f \,c^{2} \left (\tan \left (f x +e \right )+i\right )^{2}}+\frac {a^{2} B}{f \,c^{2} \left (\tan \left (f x +e \right )+i\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.71, size = 104, normalized size = 1.14 \[ -\frac {a^2\,\left (B\,2{}\mathrm {i}+A\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}+3\,B\,\mathrm {tan}\left (e+f\,x\right )+B\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,1{}\mathrm {i}+2\,B\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\mathrm {tan}\left (e+f\,x\right )-B\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^2\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{c^2\,f\,{\left (-1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.63, size = 163, normalized size = 1.79 \[ - \frac {B a^{2} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{c^{2} f} + \begin {cases} - \frac {- 4 B a^{2} c^{2} f e^{2 i e} e^{2 i f x} + \left (i A a^{2} c^{2} f e^{4 i e} + B a^{2} c^{2} f e^{4 i e}\right ) e^{4 i f x}}{4 c^{4} f^{2}} & \text {for}\: 4 c^{4} f^{2} \neq 0 \\- \frac {x \left (- A a^{2} e^{4 i e} + i B a^{2} e^{4 i e} - 2 i B a^{2} e^{2 i e}\right )}{c^{2}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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