Optimal. Leaf size=98 \[ \frac {7}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {\log (\sin (c+d x))}{a^3 d}-\frac {7 i x}{8 a^3}+\frac {3}{8 a d (a+i a \tan (c+d x))^2}+\frac {1}{6 d (a+i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.23, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3559, 3596, 3531, 3475} \[ \frac {7}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {\log (\sin (c+d x))}{a^3 d}-\frac {7 i x}{8 a^3}+\frac {3}{8 a d (a+i a \tan (c+d x))^2}+\frac {1}{6 d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3531
Rule 3559
Rule 3596
Rubi steps
\begin {align*} \int \frac {\cot (c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=\frac {1}{6 d (a+i a \tan (c+d x))^3}+\frac {\int \frac {\cot (c+d x) (6 a-3 i a \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx}{6 a^2}\\ &=\frac {1}{6 d (a+i a \tan (c+d x))^3}+\frac {3}{8 a d (a+i a \tan (c+d x))^2}+\frac {\int \frac {\cot (c+d x) \left (24 a^2-18 i a^2 \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{24 a^4}\\ &=\frac {1}{6 d (a+i a \tan (c+d x))^3}+\frac {3}{8 a d (a+i a \tan (c+d x))^2}+\frac {7}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {\int \cot (c+d x) \left (48 a^3-42 i a^3 \tan (c+d x)\right ) \, dx}{48 a^6}\\ &=-\frac {7 i x}{8 a^3}+\frac {1}{6 d (a+i a \tan (c+d x))^3}+\frac {3}{8 a d (a+i a \tan (c+d x))^2}+\frac {7}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac {\int \cot (c+d x) \, dx}{a^3}\\ &=-\frac {7 i x}{8 a^3}+\frac {\log (\sin (c+d x))}{a^3 d}+\frac {1}{6 d (a+i a \tan (c+d x))^3}+\frac {3}{8 a d (a+i a \tan (c+d x))^2}+\frac {7}{8 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.42, size = 118, normalized size = 1.20 \[ \frac {\sec ^3(c+d x) (-51 \sin (c+d x)+84 i d x \sin (3 (c+d x))+2 \sin (3 (c+d x))+81 i \cos (c+d x)-96 \sin (3 (c+d x)) \log (\sin (c+d x))+\cos (3 (c+d x)) (96 i \log (\sin (c+d x))+84 d x+2 i))}{96 a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 77, normalized size = 0.79 \[ \frac {{\left (-180 i \, d x e^{\left (6 i \, d x + 6 i \, c\right )} + 96 \, e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) + 66 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 15 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{96 \, a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.63, size = 93, normalized size = 0.95 \[ -\frac {\frac {90 \, \log \left (\tan \left (d x + c\right ) - i\right )}{a^{3}} + \frac {6 \, \log \left (i \, \tan \left (d x + c\right ) - 1\right )}{a^{3}} - \frac {96 \, \log \left (\tan \left (d x + c\right )\right )}{a^{3}} - \frac {165 \, \tan \left (d x + c\right )^{3} - 579 i \, \tan \left (d x + c\right )^{2} - 699 \, \tan \left (d x + c\right ) + 301 i}{a^{3} {\left (\tan \left (d x + c\right ) - i\right )}^{3}}}{96 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 111, normalized size = 1.13 \[ -\frac {\ln \left (\tan \left (d x +c \right )+i\right )}{16 d \,a^{3}}+\frac {\ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}+\frac {i}{6 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{3}}-\frac {7 i}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )}-\frac {3}{8 d \,a^{3} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {15 \ln \left (\tan \left (d x +c \right )-i\right )}{16 d \,a^{3}} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.87, size = 120, normalized size = 1.22 \[ \frac {\frac {17}{12\,a^3}-\frac {7\,{\mathrm {tan}\left (c+d\,x\right )}^2}{8\,a^3}+\frac {\mathrm {tan}\left (c+d\,x\right )\,17{}\mathrm {i}}{8\,a^3}}{d\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^3\,1{}\mathrm {i}-3\,{\mathrm {tan}\left (c+d\,x\right )}^2+\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}+1\right )}-\frac {15\,\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )}{16\,a^3\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )}{16\,a^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{a^3\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.61, size = 184, normalized size = 1.88 \[ \begin {cases} \frac {\left (16896 a^{6} d^{2} e^{10 i c} e^{- 2 i d x} + 3840 a^{6} d^{2} e^{8 i c} e^{- 4 i d x} + 512 a^{6} d^{2} e^{6 i c} e^{- 6 i d x}\right ) e^{- 12 i c}}{24576 a^{9} d^{3}} & \text {for}\: 24576 a^{9} d^{3} e^{12 i c} \neq 0 \\x \left (- \frac {i \left (15 e^{6 i c} + 11 e^{4 i c} + 5 e^{2 i c} + 1\right ) e^{- 6 i c}}{8 a^{3}} + \frac {15 i}{8 a^{3}}\right ) & \text {otherwise} \end {cases} - \frac {15 i x}{8 a^{3}} + \frac {\log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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