Optimal. Leaf size=97 \[ \frac {c^2 (A+3 i B)}{a^2 f (-\tan (e+f x)+i)}-\frac {c^2 (-B+i A)}{a^2 f (-\tan (e+f x)+i)^2}-\frac {B c^2 \log (\cos (e+f x))}{a^2 f}+\frac {i B c^2 x}{a^2} \]
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Rubi [A] time = 0.15, antiderivative size = 97, 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} \[ \frac {c^2 (A+3 i B)}{a^2 f (-\tan (e+f x)+i)}-\frac {c^2 (-B+i A)}{a^2 f (-\tan (e+f x)+i)^2}-\frac {B c^2 \log (\cos (e+f x))}{a^2 f}+\frac {i B c^2 x}{a^2} \]
Antiderivative was successfully verified.
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Rule 77
Rule 3588
Rubi steps
\begin {align*} \int \frac {(A+B \tan (e+f x)) (c-i c \tan (e+f x))^2}{(a+i a \tan (e+f x))^2} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(A+B x) (c-i c x)}{(a+i a x)^3} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \operatorname {Subst}\left (\int \left (\frac {2 i (A+i B) c}{a^3 (-i+x)^3}+\frac {(A+3 i B) c}{a^3 (-i+x)^2}+\frac {B c}{a^3 (-i+x)}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i B c^2 x}{a^2}-\frac {B c^2 \log (\cos (e+f x))}{a^2 f}-\frac {(i A-B) c^2}{a^2 f (i-\tan (e+f x))^2}+\frac {(A+3 i B) c^2}{a^2 f (i-\tan (e+f x))}\\ \end {align*}
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Mathematica [A] time = 2.81, size = 140, normalized size = 1.44 \[ \frac {c^2 \sec ^2(e+f x) \left (\cos (2 (e+f x)) \left (-i A+2 B \log \left (\cos ^2(e+f x)\right )+B\right )-A \sin (2 (e+f x))-i B \sin (2 (e+f x))+2 i B \sin (2 (e+f x)) \log \left (\cos ^2(e+f x)\right )+4 B \tan ^{-1}(\tan (f x)) (\sin (2 (e+f x))-i \cos (2 (e+f x)))-4 B\right )}{4 a^2 f (\tan (e+f x)-i)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.00, size = 88, normalized size = 0.91 \[ \frac {{\left (8 i \, B c^{2} f x e^{\left (4 i \, f x + 4 i \, e\right )} - 4 \, B c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 4 \, B c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, A - B\right )} c^{2}\right )} e^{\left (-4 i \, f x - 4 i \, e\right )}}{4 \, a^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.27, size = 201, normalized size = 2.07 \[ -\frac {\frac {6 \, B c^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{2}} - \frac {12 \, B c^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}{a^{2}} + \frac {6 \, B c^{2} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{a^{2}} + \frac {25 \, B c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 12 \, A c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 112 i \, B c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 198 \, B c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, A c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 112 i \, B c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 25 \, B c^{2}}{a^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - i\right )}^{4}}}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 116, normalized size = 1.20 \[ -\frac {i c^{2} A}{f \,a^{2} \left (\tan \left (f x +e \right )-i\right )^{2}}+\frac {c^{2} B}{f \,a^{2} \left (\tan \left (f x +e \right )-i\right )^{2}}-\frac {3 i c^{2} B}{f \,a^{2} \left (\tan \left (f x +e \right )-i\right )}-\frac {c^{2} A}{f \,a^{2} \left (\tan \left (f x +e \right )-i\right )}+\frac {c^{2} B \ln \left (\tan \left (f x +e \right )-i\right )}{f \,a^{2}} \]
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 = 8.51, size = 104, normalized size = 1.07 \[ \frac {c^2\,\left (2\,B+A\,\mathrm {tan}\left (e+f\,x\right )+B\,\mathrm {tan}\left (e+f\,x\right )\,3{}\mathrm {i}+B\,\ln \left (-1-\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )-B\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\ln \left (-1-\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )+B\,\mathrm {tan}\left (e+f\,x\right )\,\ln \left (-1-\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,2{}\mathrm {i}\right )}{a^2\,f\,{\left (1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.62, size = 207, normalized size = 2.13 \[ \frac {2 i B c^{2} x}{a^{2}} - \frac {B c^{2} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{a^{2} f} + \begin {cases} - \frac {\left (- 4 B a^{2} c^{2} f e^{4 i e} e^{- 2 i f x} + \left (- i A a^{2} c^{2} f e^{2 i e} + B a^{2} c^{2} f e^{2 i e}\right ) e^{- 4 i f x}\right ) e^{- 6 i e}}{4 a^{4} f^{2}} & \text {for}\: 4 a^{4} f^{2} e^{6 i e} \neq 0 \\x \left (- \frac {2 i B c^{2}}{a^{2}} + \frac {i \left (- i A c^{2} + 2 B c^{2} e^{4 i e} - 2 B c^{2} e^{2 i e} + B c^{2}\right ) e^{- 4 i e}}{a^{2}}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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