Optimal. Leaf size=123 \[ \frac {a^3 (A-5 i B) x}{c^2}-\frac {a^3 (i A+5 B) \log (\cos (e+f x))}{c^2 f}+\frac {i a^3 B \tan (e+f x)}{c^2 f}+\frac {2 a^3 (i A+B)}{c^2 f (i+\tan (e+f x))^2}-\frac {4 a^3 (A-2 i B)}{c^2 f (i+\tan (e+f x))} \]
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Rubi [A]
time = 0.12, antiderivative size = 123, 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 {4 a^3 (A-2 i B)}{c^2 f (\tan (e+f x)+i)}+\frac {2 a^3 (B+i A)}{c^2 f (\tan (e+f x)+i)^2}-\frac {a^3 (5 B+i A) \log (\cos (e+f x))}{c^2 f}+\frac {a^3 x (A-5 i B)}{c^2}+\frac {i a^3 B \tan (e+f x)}{c^2 f} \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))^3 (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)^2 (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 {i a^2 B}{c^3}-\frac {4 i a^2 (A-i B)}{c^3 (i+x)^3}+\frac {4 a^2 (A-2 i B)}{c^3 (i+x)^2}+\frac {a^2 (i A+5 B)}{c^3 (i+x)}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {a^3 (A-5 i B) x}{c^2}-\frac {a^3 (i A+5 B) \log (\cos (e+f x))}{c^2 f}+\frac {i a^3 B \tan (e+f x)}{c^2 f}+\frac {2 a^3 (i A+B)}{c^2 f (i+\tan (e+f x))^2}-\frac {4 a^3 (A-2 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. \(285\) vs. \(2(123)=246\).
time = 5.34, size = 285, normalized size = 2.32 \begin {gather*} \frac {a^3 \left (2 (i A+3 B) \cos (2 f x) \cos (e+f x) (\cos (e)-i \sin (e))+(A-i B) \cos (4 f x) \cos (e+f x) (-i \cos (e)+\sin (e))+2 (A-5 i B) f x \cos (e+f x) (\cos (3 e)-i \sin (3 e))-i (A-5 i B) \cos (e+f x) \log \left (\cos ^2(e+f x)\right ) (\cos (3 e)-i \sin (3 e))+2 B \sec (e) (i \cos (3 e)+\sin (3 e)) \sin (f x)+(A-3 i B) \cos (e+f x) (-2 \cos (e)+2 i \sin (e)) \sin (2 f x)+(A-i B) \cos (e+f x) (\cos (e)+i \sin (e)) \sin (4 f x)\right ) (\cos (e+f x)+i \sin (e+f x))^3 (A+B \tan (e+f x))}{2 c^2 f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 83, normalized size = 0.67
method | result | size |
derivativedivides | \(\frac {a^{3} \left (i B \tan \left (f x +e \right )-\frac {-8 i B +4 A}{i+\tan \left (f x +e \right )}+\left (i A +5 B \right ) \ln \left (i+\tan \left (f x +e \right )\right )-\frac {-4 i A -4 B}{2 \left (i+\tan \left (f x +e \right )\right )^{2}}\right )}{f \,c^{2}}\) | \(83\) |
default | \(\frac {a^{3} \left (i B \tan \left (f x +e \right )-\frac {-8 i B +4 A}{i+\tan \left (f x +e \right )}+\left (i A +5 B \right ) \ln \left (i+\tan \left (f x +e \right )\right )-\frac {-4 i A -4 B}{2 \left (i+\tan \left (f x +e \right )\right )^{2}}\right )}{f \,c^{2}}\) | \(83\) |
risch | \(-\frac {{\mathrm e}^{4 i \left (f x +e \right )} B \,a^{3}}{2 c^{2} f}-\frac {i {\mathrm e}^{4 i \left (f x +e \right )} A \,a^{3}}{2 c^{2} f}+\frac {3 \,{\mathrm e}^{2 i \left (f x +e \right )} B \,a^{3}}{c^{2} f}+\frac {i {\mathrm e}^{2 i \left (f x +e \right )} A \,a^{3}}{c^{2} f}+\frac {10 i a^{3} B e}{c^{2} f}-\frac {2 a^{3} A e}{c^{2} f}-\frac {2 B \,a^{3}}{f \,c^{2} \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )}-\frac {5 a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) B}{c^{2} f}-\frac {i a^{3} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right ) A}{c^{2} f}\) | \(189\) |
norman | \(\frac {\frac {\left (-5 i B \,a^{3}+A \,a^{3}\right ) x}{c}+\frac {2 i A \,a^{3}+6 B \,a^{3}}{c f}+\frac {\left (-5 i B \,a^{3}+A \,a^{3}\right ) x \left (\tan ^{4}\left (f x +e \right )\right )}{c}+\frac {i B \,a^{3} \left (\tan ^{5}\left (f x +e \right )\right )}{c f}+\frac {2 \left (-5 i B \,a^{3}+A \,a^{3}\right ) x \left (\tan ^{2}\left (f x +e \right )\right )}{c}-\frac {2 \left (-5 i B \,a^{3}+2 A \,a^{3}\right ) \left (\tan ^{3}\left (f x +e \right )\right )}{c f}+\frac {2 \left (3 i A \,a^{3}+5 B \,a^{3}\right ) \left (\tan ^{2}\left (f x +e \right )\right )}{c f}+\frac {5 i B \,a^{3} \tan \left (f x +e \right )}{c f}}{c \left (1+\tan ^{2}\left (f x +e \right )\right )^{2}}+\frac {\left (i A \,a^{3}+5 B \,a^{3}\right ) \ln \left (1+\tan ^{2}\left (f x +e \right )\right )}{2 c^{2} f}\) | \(244\) |
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 = 4.18, size = 144, normalized size = 1.17 \begin {gather*} \frac {{\left (-i \, A - B\right )} a^{3} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (i \, A + 5 \, B\right )} a^{3} e^{\left (4 i \, f x + 4 i \, e\right )} - 2 \, {\left (-i \, A - 3 \, B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 4 \, B a^{3} - 2 \, {\left ({\left (i \, A + 5 \, B\right )} a^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, A + 5 \, B\right )} a^{3}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{2 \, {\left (c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} + c^{2} f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.54, size = 236, normalized size = 1.92 \begin {gather*} - \frac {2 B a^{3}}{c^{2} f e^{2 i e} e^{2 i f x} + c^{2} f} - \frac {i a^{3} \left (A - 5 i B\right ) \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{c^{2} f} + \begin {cases} \frac {\left (2 i A a^{3} c^{2} f e^{2 i e} + 6 B a^{3} c^{2} f e^{2 i e}\right ) e^{2 i f x} + \left (- i A a^{3} c^{2} f e^{4 i e} - B a^{3} c^{2} f e^{4 i e}\right ) e^{4 i f x}}{2 c^{4} f^{2}} & \text {for}\: c^{4} f^{2} \neq 0 \\\frac {x \left (2 A a^{3} e^{4 i e} - 2 A a^{3} e^{2 i e} - 2 i B a^{3} e^{4 i e} + 6 i B a^{3} 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. 357 vs. \(2 (113) = 226\).
time = 0.82, size = 357, normalized size = 2.90 \begin {gather*} \frac {\frac {6 \, {\left (-i \, A a^{3} - 5 \, B a^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{c^{2}} + \frac {12 \, {\left (i \, A a^{3} + 5 \, B a^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}{c^{2}} - \frac {6 \, {\left (i \, A a^{3} + 5 \, B a^{3}\right )} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{c^{2}} - \frac {6 \, {\left (-i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 5 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i \, A a^{3} + 5 \, B a^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )} c^{2}} - \frac {25 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 125 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 100 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 548 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 198 i \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 894 \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 100 \, A a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 548 i \, B a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 25 i \, A a^{3} + 125 \, B a^{3}}{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 = 9.16, size = 182, normalized size = 1.48 \begin {gather*} -\frac {a^3\,\left (7\,B\,\mathrm {tan}\left (e+f\,x\right )+B\,6{}\mathrm {i}+A\,\mathrm {tan}\left (e+f\,x\right )\,4{}\mathrm {i}-2\,A+B\,{\mathrm {tan}\left (e+f\,x\right )}^2\,2{}\mathrm {i}+B\,{\mathrm {tan}\left (e+f\,x\right )}^3-A\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )+B\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,5{}\mathrm {i}+A\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\mathrm {tan}\left (e+f\,x\right )\,2{}\mathrm {i}+10\,B\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\mathrm {tan}\left (e+f\,x\right )+A\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^2-B\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,{\mathrm {tan}\left (e+f\,x\right )}^2\,5{}\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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