Optimal. Leaf size=153 \[ -\frac {2 a^3 (c-i d)^2 \tan (e+f x)}{f}-\frac {4 i a^3 (c-i d)^2 \log (\cos (e+f x))}{f}+4 a^3 x (c-i d)^2+\frac {2 c d (a+i a \tan (e+f x))^3}{3 f}+\frac {i a (c-i d)^2 (a+i a \tan (e+f x))^2}{2 f}-\frac {i d^2 (a+i a \tan (e+f x))^4}{4 a f} \]
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Rubi [A] time = 0.20, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3543, 3527, 3478, 3477, 3475} \[ -\frac {2 a^3 (c-i d)^2 \tan (e+f x)}{f}-\frac {4 i a^3 (c-i d)^2 \log (\cos (e+f x))}{f}+4 a^3 x (c-i d)^2+\frac {2 c d (a+i a \tan (e+f x))^3}{3 f}+\frac {i a (c-i d)^2 (a+i a \tan (e+f x))^2}{2 f}-\frac {i d^2 (a+i a \tan (e+f x))^4}{4 a f} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3477
Rule 3478
Rule 3527
Rule 3543
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^2 \, dx &=-\frac {i d^2 (a+i a \tan (e+f x))^4}{4 a f}+\int (a+i a \tan (e+f x))^3 \left (c^2-d^2+2 c d \tan (e+f x)\right ) \, dx\\ &=\frac {2 c d (a+i a \tan (e+f x))^3}{3 f}-\frac {i d^2 (a+i a \tan (e+f x))^4}{4 a f}+(c-i d)^2 \int (a+i a \tan (e+f x))^3 \, dx\\ &=\frac {i a (c-i d)^2 (a+i a \tan (e+f x))^2}{2 f}+\frac {2 c d (a+i a \tan (e+f x))^3}{3 f}-\frac {i d^2 (a+i a \tan (e+f x))^4}{4 a f}+\left (2 a (c-i d)^2\right ) \int (a+i a \tan (e+f x))^2 \, dx\\ &=4 a^3 (c-i d)^2 x-\frac {2 a^3 (c-i d)^2 \tan (e+f x)}{f}+\frac {i a (c-i d)^2 (a+i a \tan (e+f x))^2}{2 f}+\frac {2 c d (a+i a \tan (e+f x))^3}{3 f}-\frac {i d^2 (a+i a \tan (e+f x))^4}{4 a f}+\left (4 i a^3 (c-i d)^2\right ) \int \tan (e+f x) \, dx\\ &=4 a^3 (c-i d)^2 x-\frac {4 i a^3 (c-i d)^2 \log (\cos (e+f x))}{f}-\frac {2 a^3 (c-i d)^2 \tan (e+f x)}{f}+\frac {i a (c-i d)^2 (a+i a \tan (e+f x))^2}{2 f}+\frac {2 c d (a+i a \tan (e+f x))^3}{3 f}-\frac {i d^2 (a+i a \tan (e+f x))^4}{4 a f}\\ \end {align*}
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Mathematica [B] time = 9.23, size = 948, normalized size = 6.20 \[ \frac {\left (\frac {1}{3} \cos (3 e)-\frac {1}{3} i \sin (3 e)\right ) \left (-3 \sin (f x) d^2-2 i c \sin (f x) d\right ) (i \tan (e+f x) a+a)^3}{f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) (\cos (f x)+i \sin (f x))^3}+\frac {\cos ^2(e+f x) \left (\frac {1}{3} \cos (3 e)-\frac {1}{3} i \sin (3 e)\right ) \left (-9 \sin (f x) c^2+26 i d \sin (f x) c+15 d^2 \sin (f x)\right ) (i \tan (e+f x) a+a)^3}{f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) (\cos (f x)+i \sin (f x))^3}+\frac {x \cos ^3(e+f x) \left (2 c^2 \cos ^3(e)-2 d^2 \cos ^3(e)-4 i c d \cos ^3(e)-8 i c^2 \sin (e) \cos ^2(e)+8 i d^2 \sin (e) \cos ^2(e)-16 c d \sin (e) \cos ^2(e)-2 c^2 \cos (e)+2 d^2 \cos (e)-12 c^2 \sin ^2(e) \cos (e)+12 d^2 \sin ^2(e) \cos (e)+24 i c d \sin ^2(e) \cos (e)+4 i c d \cos (e)+8 i c^2 \sin ^3(e)-8 i d^2 \sin ^3(e)+16 c d \sin ^3(e)+4 i c^2 \sin (e)-4 i d^2 \sin (e)+8 c d \sin (e)+2 c^2 \sin ^3(e) \tan (e)-2 d^2 \sin ^3(e) \tan (e)-4 i c d \sin ^3(e) \tan (e)+2 c^2 \sin (e) \tan (e)-2 d^2 \sin (e) \tan (e)-4 i c d \sin (e) \tan (e)+i (c-i d)^2 (4 \cos (3 e)-4 i \sin (3 e)) \tan (e)\right ) (i \tan (e+f x) a+a)^3}{(\cos (f x)+i \sin (f x))^3}+\frac {\cos ^3(e+f x) \left (\cos \left (\frac {3 e}{2}\right ) c^2-i \sin \left (\frac {3 e}{2}\right ) c^2-2 i d \cos \left (\frac {3 e}{2}\right ) c-2 d \sin \left (\frac {3 e}{2}\right ) c-d^2 \cos \left (\frac {3 e}{2}\right )+i d^2 \sin \left (\frac {3 e}{2}\right )\right ) \left (-2 i \cos \left (\frac {3 e}{2}\right ) \log \left (\cos ^2(e+f x)\right )-2 \sin \left (\frac {3 e}{2}\right ) \log \left (\cos ^2(e+f x)\right )\right ) (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3}+\frac {\cos (e+f x) \left (3 \cos (e) c^2-18 i d \cos (e) c+4 d \sin (e) c-15 d^2 \cos (e)-6 i d^2 \sin (e)\right ) \left (-\frac {1}{6} i \cos (3 e)-\frac {1}{6} \sin (3 e)\right ) (i \tan (e+f x) a+a)^3}{f \left (\cos \left (\frac {e}{2}\right )-\sin \left (\frac {e}{2}\right )\right ) \left (\cos \left (\frac {e}{2}\right )+\sin \left (\frac {e}{2}\right )\right ) (\cos (f x)+i \sin (f x))^3}+\frac {\sec (e+f x) \left (-\frac {1}{4} i \cos (3 e) d^2-\frac {1}{4} \sin (3 e) d^2\right ) (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3}+\frac {(c-i d)^2 \cos ^3(e+f x) (4 f x \cos (3 e)-4 i f x \sin (3 e)) (i \tan (e+f x) a+a)^3}{f (\cos (f x)+i \sin (f x))^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 352, normalized size = 2.30 \[ \frac {-18 i \, a^{3} c^{2} - 52 \, a^{3} c d + 30 i \, a^{3} d^{2} + {\left (-24 i \, a^{3} c^{2} - 96 \, a^{3} c d + 72 i \, a^{3} d^{2}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-66 i \, a^{3} c^{2} - 228 \, a^{3} c d + 138 i \, a^{3} d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-60 i \, a^{3} c^{2} - 184 \, a^{3} c d + 108 i \, a^{3} d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-12 i \, a^{3} c^{2} - 24 \, a^{3} c d + 12 i \, a^{3} d^{2} + {\left (-12 i \, a^{3} c^{2} - 24 \, a^{3} c d + 12 i \, a^{3} d^{2}\right )} e^{\left (8 i \, f x + 8 i \, e\right )} + {\left (-48 i \, a^{3} c^{2} - 96 \, a^{3} c d + 48 i \, a^{3} d^{2}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-72 i \, a^{3} c^{2} - 144 \, a^{3} c d + 72 i \, a^{3} d^{2}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-48 i \, a^{3} c^{2} - 96 \, a^{3} c d + 48 i \, a^{3} d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{3 \, {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.55, size = 670, normalized size = 4.38 \[ \frac {-12 i \, a^{3} c^{2} e^{\left (8 i \, f x + 8 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 24 \, a^{3} c d e^{\left (8 i \, f x + 8 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 12 i \, a^{3} d^{2} e^{\left (8 i \, f x + 8 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 48 i \, a^{3} c^{2} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 96 \, a^{3} c d e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 48 i \, a^{3} d^{2} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 72 i \, a^{3} 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 ) - 144 \, a^{3} c d e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 72 i \, a^{3} d^{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 ) - 48 i \, a^{3} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 96 \, a^{3} c d e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 48 i \, a^{3} d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 24 i \, a^{3} c^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - 96 \, a^{3} c d e^{\left (6 i \, f x + 6 i \, e\right )} + 72 i \, a^{3} d^{2} e^{\left (6 i \, f x + 6 i \, e\right )} - 66 i \, a^{3} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 228 \, a^{3} c d e^{\left (4 i \, f x + 4 i \, e\right )} + 138 i \, a^{3} d^{2} e^{\left (4 i \, f x + 4 i \, e\right )} - 60 i \, a^{3} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 184 \, a^{3} c d e^{\left (2 i \, f x + 2 i \, e\right )} + 108 i \, a^{3} d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - 12 i \, a^{3} c^{2} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 24 \, a^{3} c d \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 12 i \, a^{3} d^{2} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 18 i \, a^{3} c^{2} - 52 \, a^{3} c d + 30 i \, a^{3} d^{2}}{3 \, {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.02, size = 290, normalized size = 1.90 \[ -\frac {i a^{3} d^{2} \left (\tan ^{4}\left (f x +e \right )\right )}{4 f}-\frac {2 i a^{3} \left (\tan ^{3}\left (f x +e \right )\right ) c d}{3 f}-\frac {i a^{3} \left (\tan ^{2}\left (f x +e \right )\right ) c^{2}}{2 f}+\frac {2 i a^{3} \left (\tan ^{2}\left (f x +e \right )\right ) d^{2}}{f}-\frac {a^{3} d^{2} \left (\tan ^{3}\left (f x +e \right )\right )}{f}+\frac {8 i a^{3} c d \tan \left (f x +e \right )}{f}-\frac {3 a^{3} \left (\tan ^{2}\left (f x +e \right )\right ) c d}{f}-\frac {3 a^{3} c^{2} \tan \left (f x +e \right )}{f}+\frac {4 a^{3} \tan \left (f x +e \right ) d^{2}}{f}+\frac {2 i a^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{2}}{f}-\frac {2 i a^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) d^{2}}{f}+\frac {4 a^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c d}{f}-\frac {8 i a^{3} \arctan \left (\tan \left (f x +e \right )\right ) c d}{f}+\frac {4 a^{3} \arctan \left (\tan \left (f x +e \right )\right ) c^{2}}{f}-\frac {4 a^{3} \arctan \left (\tan \left (f x +e \right )\right ) d^{2}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 181, normalized size = 1.18 \[ -\frac {3 i \, a^{3} d^{2} \tan \left (f x + e\right )^{4} + 4 \, {\left (2 i \, a^{3} c d + 3 \, a^{3} d^{2}\right )} \tan \left (f x + e\right )^{3} - {\left (-6 i \, a^{3} c^{2} - 36 \, a^{3} c d + 24 i \, a^{3} d^{2}\right )} \tan \left (f x + e\right )^{2} - 12 \, {\left (4 \, a^{3} c^{2} - 8 i \, a^{3} c d - 4 \, a^{3} d^{2}\right )} {\left (f x + e\right )} - 12 \, {\left (2 i \, a^{3} c^{2} + 4 \, a^{3} c d - 2 i \, a^{3} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) + {\left (36 \, a^{3} c^{2} - 96 i \, a^{3} c d - 48 \, a^{3} d^{2}\right )} \tan \left (f x + e\right )}{12 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.92, size = 217, normalized size = 1.42 \[ \frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {a^3\,d^2\,1{}\mathrm {i}}{2}-\frac {a^3\,\left (c^2\,1{}\mathrm {i}+4\,c\,d-d^2\,1{}\mathrm {i}\right )}{2}+a^3\,d\,\left (d+c\,1{}\mathrm {i}\right )\,1{}\mathrm {i}\right )}{f}-\frac {{\mathrm {tan}\left (e+f\,x\right )}^3\,\left (\frac {a^3\,d^2}{3}+\frac {2\,a^3\,d\,\left (d+c\,1{}\mathrm {i}\right )}{3}\right )}{f}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (a^3\,c^2\,4{}\mathrm {i}+8\,a^3\,c\,d-a^3\,d^2\,4{}\mathrm {i}\right )}{f}+\frac {\mathrm {tan}\left (e+f\,x\right )\,\left (a^3\,d^2+a^3\,\left (c^2\,1{}\mathrm {i}+4\,c\,d-d^2\,1{}\mathrm {i}\right )\,1{}\mathrm {i}-2\,a^3\,c\,\left (c-d\,1{}\mathrm {i}\right )+2\,a^3\,d\,\left (d+c\,1{}\mathrm {i}\right )\right )}{f}-\frac {a^3\,d^2\,{\mathrm {tan}\left (e+f\,x\right )}^4\,1{}\mathrm {i}}{4\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.13, size = 314, normalized size = 2.05 \[ - \frac {4 i a^{3} \left (c - i d\right )^{2} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{f} + \frac {18 i a^{3} c^{2} + 52 a^{3} c d - 30 i a^{3} d^{2} + \left (60 i a^{3} c^{2} e^{2 i e} + 184 a^{3} c d e^{2 i e} - 108 i a^{3} d^{2} e^{2 i e}\right ) e^{2 i f x} + \left (66 i a^{3} c^{2} e^{4 i e} + 228 a^{3} c d e^{4 i e} - 138 i a^{3} d^{2} e^{4 i e}\right ) e^{4 i f x} + \left (24 i a^{3} c^{2} e^{6 i e} + 96 a^{3} c d e^{6 i e} - 72 i a^{3} d^{2} e^{6 i e}\right ) e^{6 i f x}}{- 3 f e^{8 i e} e^{8 i f x} - 12 f e^{6 i e} e^{6 i f x} - 18 f e^{4 i e} e^{4 i f x} - 12 f e^{2 i e} e^{2 i f x} - 3 f} \]
Verification of antiderivative is not currently implemented for this CAS.
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