Optimal. Leaf size=107 \[ \frac {i a d (c-i d)^2 \tan (e+f x)}{f}+\frac {i a (c+d \tan (e+f x))^3}{3 f}+\frac {a (d+i c) (c+d \tan (e+f x))^2}{2 f}+\frac {a (d+i c)^3 \log (\cos (e+f x))}{f}+a x (c-i d)^3 \]
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Rubi [A] time = 0.13, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {3528, 3525, 3475} \[ \frac {i a d (c-i d)^2 \tan (e+f x)}{f}+\frac {i a (c+d \tan (e+f x))^3}{3 f}+\frac {a (d+i c) (c+d \tan (e+f x))^2}{2 f}+\frac {a (d+i c)^3 \log (\cos (e+f x))}{f}+a x (c-i d)^3 \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3528
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^3 \, dx &=\frac {i a (c+d \tan (e+f x))^3}{3 f}+\int (c+d \tan (e+f x))^2 (a (c-i d)+a (i c+d) \tan (e+f x)) \, dx\\ &=\frac {a (i c+d) (c+d \tan (e+f x))^2}{2 f}+\frac {i a (c+d \tan (e+f x))^3}{3 f}+\int \left (a (c-i d)^2+i a (c-i d)^2 \tan (e+f x)\right ) (c+d \tan (e+f x)) \, dx\\ &=a (c-i d)^3 x+\frac {i a (c-i d)^2 d \tan (e+f x)}{f}+\frac {a (i c+d) (c+d \tan (e+f x))^2}{2 f}+\frac {i a (c+d \tan (e+f x))^3}{3 f}-\left (a (i c+d)^3\right ) \int \tan (e+f x) \, dx\\ &=a (c-i d)^3 x+\frac {a (i c+d)^3 \log (\cos (e+f x))}{f}+\frac {i a (c-i d)^2 d \tan (e+f x)}{f}+\frac {a (i c+d) (c+d \tan (e+f x))^2}{2 f}+\frac {i a (c+d \tan (e+f x))^3}{3 f}\\ \end {align*}
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Mathematica [B] time = 4.28, size = 219, normalized size = 2.05 \[ \frac {(\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x)) \left (-2 d \left (-9 c^2+9 i c d+4 d^2\right ) (\tan (e)+i) \sin (f x)+d^2 \cos (e) (\tan (e)+i) (9 c+2 d \tan (e)-3 i d) \sec (e+f x)+12 f x (c-i d)^3 (\cos (e)-i \sin (e)) \cos (e+f x)-3 i (c-i d)^3 (\cos (e)-i \sin (e)) \cos (e+f x) \log \left (\cos ^2(e+f x)\right )-6 (c-i d)^3 (\cos (e)-i \sin (e)) \cos (e+f x) \tan ^{-1}(\tan (2 e+f x))+2 d^3 (\tan (e)+i) \sin (f x) \sec ^2(e+f x)\right )}{6 f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 276, normalized size = 2.58 \[ -\frac {18 \, a c^{2} d - 18 i \, a c d^{2} - 8 \, a d^{3} + {\left (18 \, a c^{2} d - 36 i \, a c d^{2} - 18 \, a d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (36 \, a c^{2} d - 54 i \, a c d^{2} - 18 \, a d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )} - {\left (-3 i \, a c^{3} - 9 \, a c^{2} d + 9 i \, a c d^{2} + 3 \, a d^{3} + {\left (-3 i \, a c^{3} - 9 \, a c^{2} d + 9 i \, a c d^{2} + 3 \, a d^{3}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-9 i \, a c^{3} - 27 \, a c^{2} d + 27 i \, a c d^{2} + 9 \, a d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-9 i \, a c^{3} - 27 \, a c^{2} d + 27 i \, a c d^{2} + 9 \, a d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{3 \, {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.00, size = 597, normalized size = 5.58 \[ \frac {-3 i \, a c^{3} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 9 \, a c^{2} 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 ) + 9 i \, a c 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 ) + 3 \, a d^{3} e^{\left (6 i \, f x + 6 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 9 i \, a c^{3} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 27 \, a c^{2} 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 ) + 27 i \, a c 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 ) + 9 \, a d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 9 i \, a c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 27 \, a c^{2} 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 ) + 27 i \, a c 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 ) + 9 \, a d^{3} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 18 \, a c^{2} d e^{\left (4 i \, f x + 4 i \, e\right )} + 36 i \, a c d^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 18 \, a d^{3} e^{\left (4 i \, f x + 4 i \, e\right )} - 36 \, a c^{2} d e^{\left (2 i \, f x + 2 i \, e\right )} + 54 i \, a c d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 18 \, a d^{3} e^{\left (2 i \, f x + 2 i \, e\right )} - 3 i \, a c^{3} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 9 \, a c^{2} d \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 9 i \, a c d^{2} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) + 3 \, a d^{3} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right ) - 18 \, a c^{2} d + 18 i \, a c d^{2} + 8 \, a d^{3}}{3 \, {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, 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 = 256, normalized size = 2.39 \[ \frac {i a \,d^{3} \left (\tan ^{3}\left (f x +e \right )\right )}{3 f}+\frac {3 i a \left (\tan ^{2}\left (f x +e \right )\right ) c \,d^{2}}{2 f}+\frac {3 i a \,c^{2} d \tan \left (f x +e \right )}{f}-\frac {i a \,d^{3} \tan \left (f x +e \right )}{f}+\frac {a \left (\tan ^{2}\left (f x +e \right )\right ) d^{3}}{2 f}+\frac {3 a \tan \left (f x +e \right ) c \,d^{2}}{f}-\frac {3 i a \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c \,d^{2}}{2 f}-\frac {a \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) d^{3}}{2 f}+\frac {i a \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{3}}{2 f}+\frac {3 a \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{2} d}{2 f}-\frac {3 i a \arctan \left (\tan \left (f x +e \right )\right ) c^{2} d}{f}+\frac {i a \arctan \left (\tan \left (f x +e \right )\right ) d^{3}}{f}+\frac {a \arctan \left (\tan \left (f x +e \right )\right ) c^{3}}{f}-\frac {3 a \arctan \left (\tan \left (f x +e \right )\right ) c \,d^{2}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 145, normalized size = 1.36 \[ -\frac {-2 i \, a d^{3} \tan \left (f x + e\right )^{3} + 3 \, {\left (-3 i \, a c d^{2} - a d^{3}\right )} \tan \left (f x + e\right )^{2} - 6 \, {\left (a c^{3} - 3 i \, a c^{2} d - 3 \, a c d^{2} + i \, a d^{3}\right )} {\left (f x + e\right )} + 3 \, {\left (-i \, a c^{3} - 3 \, a c^{2} d + 3 i \, a c d^{2} + a d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right ) - {\left (18 i \, a c^{2} d + 18 \, a c d^{2} - 6 i \, a d^{3}\right )} \tan \left (f x + e\right )}{6 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.07, size = 122, normalized size = 1.14 \[ \frac {\mathrm {tan}\left (e+f\,x\right )\,\left (3{}\mathrm {i}\,a\,c^2\,d+3\,a\,c\,d^2-1{}\mathrm {i}\,a\,d^3\right )}{f}+\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (1{}\mathrm {i}\,a\,c^3+3\,a\,c^2\,d-3{}\mathrm {i}\,a\,c\,d^2-a\,d^3\right )}{f}+\frac {{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (\frac {a\,d^3}{2}+\frac {3{}\mathrm {i}\,a\,c\,d^2}{2}\right )}{f}+\frac {a\,d^3\,{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}}{3\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.98, size = 236, normalized size = 2.21 \[ - \frac {i a \left (c - i d\right )^{3} \log {\left (e^{2 i f x} + e^{- 2 i e} \right )}}{f} + \frac {18 i a c^{2} d + 18 a c d^{2} - 8 i a d^{3} + \left (36 i a c^{2} d e^{2 i e} + 54 a c d^{2} e^{2 i e} - 18 i a d^{3} e^{2 i e}\right ) e^{2 i f x} + \left (18 i a c^{2} d e^{4 i e} + 36 a c d^{2} e^{4 i e} - 18 i a d^{3} e^{4 i e}\right ) e^{4 i f x}}{- 3 i f e^{6 i e} e^{6 i f x} - 9 i f e^{4 i e} e^{4 i f x} - 9 i f e^{2 i e} e^{2 i f x} - 3 i f} \]
Verification of antiderivative is not currently implemented for this CAS.
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