Optimal. Leaf size=91 \[ -\frac {i a^2 B x}{c^2}-\frac {a^2 B \log (\cos (e+f x))}{c^2 f}+\frac {a^2 (i A+B)}{c^2 f (i+\tan (e+f x))^2}-\frac {a^2 (A-3 i B)}{c^2 f (i+\tan (e+f x))} \]
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Rubi [A]
time = 0.10, 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 = {3669, 78}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 3669
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) \text {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) \text {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] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(184\) vs. \(2(91)=182\).
time = 1.35, size = 184, normalized size = 2.02 \begin {gather*} \frac {a^2 \left (4 B-i \cos (2 (e+f x)) \left (A-i B+8 B f x-2 i B \log \left (\cos ^2(e+f x)\right )\right )+A \sin (2 (e+f x))-i B \sin (2 (e+f x))-8 B f x \sin (2 (e+f x))+2 i B \log \left (\cos ^2(e+f x)\right ) \sin (2 (e+f x))+4 B \text {ArcTan}(\tan (3 e+f x)) (i \cos (2 (e+f x))+\sin (2 (e+f x)))\right ) (\cos (2 (e+2 f x))+i \sin (2 (e+2 f x)))}{4 c^2 f (\cos (f x)+i \sin (f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 64, normalized size = 0.70
method | result | size |
derivativedivides | \(\frac {a^{2} \left (-\frac {-2 i A -2 B}{2 \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {-3 i B +A}{i+\tan \left (f x +e \right )}+B \ln \left (i+\tan \left (f x +e \right )\right )\right )}{f \,c^{2}}\) | \(64\) |
default | \(\frac {a^{2} \left (-\frac {-2 i A -2 B}{2 \left (i+\tan \left (f x +e \right )\right )^{2}}-\frac {-3 i B +A}{i+\tan \left (f x +e \right )}+B \ln \left (i+\tan \left (f x +e \right )\right )\right )}{f \,c^{2}}\) | \(64\) |
risch | \(-\frac {{\mathrm e}^{4 i \left (f x +e \right )} B \,a^{2}}{4 c^{2} f}-\frac {i {\mathrm e}^{4 i \left (f x +e \right )} a^{2} A}{4 c^{2} f}+\frac {a^{2} B \,{\mathrm e}^{2 i \left (f x +e \right )}}{c^{2} f}+\frac {2 i a^{2} B e}{c^{2} f}-\frac {a^{2} B \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}{c^{2} f}\) | \(103\) |
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 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.52, size = 65, normalized size = 0.71 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.32, size = 160, normalized size = 1.76 \begin {gather*} - \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}\: 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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 201 vs. \(2 (84) = 168\).
time = 0.75, size = 201, normalized size = 2.21 \begin {gather*} -\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} \end {gather*}
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 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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