Optimal. Leaf size=50 \[ -\frac {a \cot ^2(c+d x)}{2 d}-\frac {i a \cot (c+d x)}{d}-\frac {a \log (\sin (c+d x))}{d}-i a x \]
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Rubi [A] time = 0.07, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3529, 3531, 3475} \[ -\frac {a \cot ^2(c+d x)}{2 d}-\frac {i a \cot (c+d x)}{d}-\frac {a \log (\sin (c+d x))}{d}-i a x \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3529
Rule 3531
Rubi steps
\begin {align*} \int \cot ^3(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac {a \cot ^2(c+d x)}{2 d}+\int \cot ^2(c+d x) (i a-a \tan (c+d x)) \, dx\\ &=-\frac {i a \cot (c+d x)}{d}-\frac {a \cot ^2(c+d x)}{2 d}+\int \cot (c+d x) (-a-i a \tan (c+d x)) \, dx\\ &=-i a x-\frac {i a \cot (c+d x)}{d}-\frac {a \cot ^2(c+d x)}{2 d}-a \int \cot (c+d x) \, dx\\ &=-i a x-\frac {i a \cot (c+d x)}{d}-\frac {a \cot ^2(c+d x)}{2 d}-\frac {a \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] time = 0.16, size = 68, normalized size = 1.36 \[ -\frac {a \left (\cot ^2(c+d x)+2 \log (\tan (c+d x))+2 \log (\cos (c+d x))\right )}{2 d}-\frac {i a \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 83, normalized size = 1.66 \[ \frac {4 \, a e^{\left (2 i \, d x + 2 i \, c\right )} - {\left (a e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 2 \, a}{d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.61, size = 102, normalized size = 2.04 \[ -\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 16 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 8 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) - 4 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \frac {12 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4 i \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 55, normalized size = 1.10 \[ -i a x -\frac {i a \cot \left (d x +c \right )}{d}-\frac {i a c}{d}-\frac {a \left (\cot ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.74, size = 58, normalized size = 1.16 \[ -\frac {2 i \, {\left (d x + c\right )} a - a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, a \log \left (\tan \left (d x + c\right )\right ) + \frac {2 i \, a \tan \left (d x + c\right ) + a}{\tan \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.79, size = 47, normalized size = 0.94 \[ -\frac {a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d}-\frac {\frac {a}{2}+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}{d\,{\mathrm {tan}\left (c+d\,x\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.34, size = 88, normalized size = 1.76 \[ - \frac {a \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} + \frac {4 i a e^{2 i c} e^{2 i d x} - 2 i a}{i d e^{4 i c} e^{4 i d x} - 2 i d e^{2 i c} e^{2 i d x} + i d} \]
Verification of antiderivative is not currently implemented for this CAS.
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