Optimal. Leaf size=142 \[ -\frac {a^3 (-d+i c) (c-3 i d) \log (c \cos (e+f x)+d \sin (e+f x))}{d^2 f (c-i d)^2}+\frac {(c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{d f (c-i d) (c+d \tan (e+f x))}+\frac {4 a^3 x}{(c-i d)^2}+\frac {i a^3 \log (\cos (e+f x))}{d^2 f} \]
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Rubi [A] time = 0.37, antiderivative size = 142, 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 = {3553, 3589, 3475, 3531, 3530} \[ -\frac {a^3 (-d+i c) (c-3 i d) \log (c \cos (e+f x)+d \sin (e+f x))}{d^2 f (c-i d)^2}+\frac {(c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{d f (c-i d) (c+d \tan (e+f x))}+\frac {4 a^3 x}{(c-i d)^2}+\frac {i a^3 \log (\cos (e+f x))}{d^2 f} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3530
Rule 3531
Rule 3553
Rule 3589
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^3}{(c+d \tan (e+f x))^2} \, dx &=\frac {(c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f (c+d \tan (e+f x))}-\frac {\int \frac {(a+i a \tan (e+f x)) \left (-a^2 (c+3 i d)+a^2 (i c+d) \tan (e+f x)\right )}{c+d \tan (e+f x)} \, dx}{d (i c+d)}\\ &=\frac {(c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f (c+d \tan (e+f x))}-\frac {\left (i a^3\right ) \int \tan (e+f x) \, dx}{d^2}-\frac {\int \frac {-a^3 (c+3 i d) d+a^3 \left (c^2-i c d+4 d^2\right ) \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{d^2 (i c+d)}\\ &=-\frac {4 a^3 x}{(i c+d)^2}+\frac {i a^3 \log (\cos (e+f x))}{d^2 f}+\frac {(c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f (c+d \tan (e+f x))}+\frac {\left (a^3 (c+i d) (c-3 i d)\right ) \int \frac {d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{(c-i d) d^2 (i c+d)}\\ &=-\frac {4 a^3 x}{(i c+d)^2}+\frac {i a^3 \log (\cos (e+f x))}{d^2 f}-\frac {a^3 (i c-d) (c-3 i d) \log (c \cos (e+f x)+d \sin (e+f x))}{(c-i d)^2 d^2 f}+\frac {(c+i d) \left (a^3+i a^3 \tan (e+f x)\right )}{(c-i d) d f (c+d \tan (e+f x))}\\ \end {align*}
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Mathematica [B] time = 9.06, size = 1936, normalized size = 13.63 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.64, size = 298, normalized size = 2.10 \[ \frac {2 i \, a^{3} c^{2} d - 4 \, a^{3} c d^{2} - 2 i \, a^{3} d^{3} - {\left (a^{3} c^{3} - i \, a^{3} c^{2} d + 5 \, a^{3} c d^{2} + 3 i \, a^{3} d^{3} + {\left (a^{3} c^{3} - 3 i \, a^{3} c^{2} d + a^{3} c d^{2} - 3 i \, a^{3} d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (\frac {{\left (i \, c + d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c - d}{i \, c + d}\right ) + {\left (a^{3} c^{3} - i \, a^{3} c^{2} d + a^{3} c d^{2} - i \, a^{3} d^{3} + {\left (a^{3} c^{3} - 3 i \, a^{3} c^{2} d - 3 \, a^{3} c d^{2} + i \, a^{3} d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \log \left (e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )}{{\left (-i \, c^{3} d^{2} - 3 \, c^{2} d^{3} + 3 i \, c d^{4} + d^{5}\right )} f e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, c^{3} d^{2} - c^{2} d^{3} - i \, c d^{4} - d^{5}\right )} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.79, size = 392, normalized size = 2.76 \[ \frac {\frac {8 \, a^{3} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i\right )}{-i \, c^{2} - 2 \, c d + i \, d^{2}} + \frac {i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{d^{2}} + \frac {i \, a^{3} \log \left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}{d^{2}} + \frac {2 \, {\left (-i \, a^{3} c^{2} - 2 \, a^{3} c d - 3 i \, a^{3} d^{2}\right )} \log \left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - c\right )}{2 \, c^{2} d^{2} - 4 i \, c d^{3} - 2 \, d^{4}} - \frac {2 \, {\left (-i \, a^{3} c^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, a^{3} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 i \, a^{3} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 4 i \, a^{3} c^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a^{3} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 8 i \, a^{3} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + i \, a^{3} c^{4} + 2 \, a^{3} c^{3} d + 3 i \, a^{3} c^{2} d^{2}\right )}}{{\left (2 \, c^{3} d^{2} - 4 i \, c^{2} d^{3} - 2 \, c d^{4}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - c\right )}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.30, size = 438, normalized size = 3.08 \[ -\frac {i a^{3} \ln \left (c +d \tan \left (f x +e \right )\right ) c^{4}}{f \left (c^{2}+d^{2}\right )^{2} d^{2}}-\frac {6 i a^{3} \ln \left (c +d \tan \left (f x +e \right )\right ) c^{2}}{f \left (c^{2}+d^{2}\right )^{2}}+\frac {3 i a^{3} d^{2} \ln \left (c +d \tan \left (f x +e \right )\right )}{f \left (c^{2}+d^{2}\right )^{2}}+\frac {8 a^{3} d \ln \left (c +d \tan \left (f x +e \right )\right ) c}{f \left (c^{2}+d^{2}\right )^{2}}-\frac {i a^{3} c^{3}}{f \,d^{2} \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {3 i a^{3} c}{f \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {3 a^{3} c^{2}}{f d \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}-\frac {a^{3} d}{f \left (c^{2}+d^{2}\right ) \left (c +d \tan \left (f x +e \right )\right )}+\frac {2 i a^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c^{2}}{f \left (c^{2}+d^{2}\right )^{2}}-\frac {2 i a^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) d^{2}}{f \left (c^{2}+d^{2}\right )^{2}}+\frac {8 i a^{3} \arctan \left (\tan \left (f x +e \right )\right ) c d}{f \left (c^{2}+d^{2}\right )^{2}}-\frac {4 a^{3} \ln \left (1+\tan ^{2}\left (f x +e \right )\right ) c d}{f \left (c^{2}+d^{2}\right )^{2}}+\frac {4 a^{3} \arctan \left (\tan \left (f x +e \right )\right ) c^{2}}{f \left (c^{2}+d^{2}\right )^{2}}-\frac {4 a^{3} \arctan \left (\tan \left (f x +e \right )\right ) d^{2}}{f \left (c^{2}+d^{2}\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 248, normalized size = 1.75 \[ \frac {\frac {2 \, {\left (4 \, a^{3} c^{2} + 8 i \, a^{3} c d - 4 \, a^{3} d^{2}\right )} {\left (f x + e\right )}}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} + \frac {2 \, {\left (-i \, a^{3} c^{4} - 6 i \, a^{3} c^{2} d^{2} + 8 \, a^{3} c d^{3} + 3 i \, a^{3} d^{4}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{4} d^{2} + 2 \, c^{2} d^{4} + d^{6}} + \frac {{\left (4 i \, a^{3} c^{2} - 8 \, a^{3} c d - 4 i \, a^{3} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} + \frac {2 \, {\left (-i \, a^{3} c^{3} + 3 \, a^{3} c^{2} d + 3 i \, a^{3} c d^{2} - a^{3} d^{3}\right )}}{c^{3} d^{2} + c d^{4} + {\left (c^{2} d^{3} + d^{5}\right )} \tan \left (f x + e\right )}}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.82, size = 133, normalized size = 0.94 \[ -\frac {4\,a^3\,\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )}{f\,\left (c^2\,1{}\mathrm {i}+2\,c\,d-d^2\,1{}\mathrm {i}\right )}+\frac {a^3\,\left (-c^2\,1{}\mathrm {i}+2\,c\,d+d^2\,1{}\mathrm {i}\right )}{d^3\,f\,\left (\mathrm {tan}\left (e+f\,x\right )+\frac {c}{d}\right )\,\left (c-d\,1{}\mathrm {i}\right )}+\frac {a^3\,\ln \left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (c^2\,1{}\mathrm {i}+2\,c\,d+d^2\,3{}\mathrm {i}\right )}{d^2\,f\,{\left (d+c\,1{}\mathrm {i}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 33.04, size = 382, normalized size = 2.69 \[ - \frac {i a^{3} \left (c - 3 i d\right ) \left (c + i d\right ) \log {\left (e^{2 i f x} + \frac {i a^{3} c^{2} - \frac {a^{3} c d \left (c - 3 i d\right ) \left (c + i d\right )}{\left (c - i d\right )^{2}} + a^{3} c d + \frac {i a^{3} d^{2} \left (c - 3 i d\right ) \left (c + i d\right )}{\left (c - i d\right )^{2}} + 2 i a^{3} d^{2}}{i a^{3} c^{2} e^{2 i e} + 2 a^{3} c d e^{2 i e} + i a^{3} d^{2} e^{2 i e}} \right )}}{d^{2} f \left (c - i d\right )^{2}} + \frac {i a^{3} \log {\left (\frac {i a^{3} c^{2} + 2 a^{3} c d + i a^{3} d^{2}}{i a^{3} c^{2} e^{2 i e} + 2 a^{3} c d e^{2 i e} + i a^{3} d^{2} e^{2 i e}} + e^{2 i f x} \right )}}{d^{2} f} + \frac {- 2 i a^{3} c^{2} + 4 a^{3} c d + 2 i a^{3} d^{2}}{i c^{3} d f + c^{2} d^{2} f + i c d^{3} f + d^{4} f + \left (i c^{3} d f e^{2 i e} + 3 c^{2} d^{2} f e^{2 i e} - 3 i c d^{3} f e^{2 i e} - d^{4} f e^{2 i e}\right ) e^{2 i f x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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