Optimal. Leaf size=44 \[ \frac {a (B+i A) \log (\sin (c+d x))}{d}-a x (A-i B)-\frac {a A \cot (c+d x)}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {3591, 3531, 3475} \[ \frac {a (B+i A) \log (\sin (c+d x))}{d}-a x (A-i B)-\frac {a A \cot (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3531
Rule 3591
Rubi steps
\begin {align*} \int \cot ^2(c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx &=-\frac {a A \cot (c+d x)}{d}+\int \cot (c+d x) (a (i A+B)-a (A-i B) \tan (c+d x)) \, dx\\ &=-a (A-i B) x-\frac {a A \cot (c+d x)}{d}+(a (i A+B)) \int \cot (c+d x) \, dx\\ &=-a (A-i B) x-\frac {a A \cot (c+d x)}{d}+\frac {a (i A+B) \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [C] time = 0.19, size = 84, normalized size = 1.91 \[ -\frac {a A \cot (c+d x) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};-\tan ^2(c+d x)\right )}{d}+\frac {i a A (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+\frac {a B (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+i a B x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.57, size = 62, normalized size = 1.41 \[ \frac {-2 i \, A a + {\left ({\left (i \, A + B\right )} a e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (-i \, A - B\right )} a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d e^{\left (2 i \, d x + 2 i \, c\right )} - d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.65, size = 104, normalized size = 2.36 \[ \frac {A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 4 \, {\left (-i \, A a - B a\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) + 2 \, {\left (i \, A a + B a\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + \frac {-2 i \, A a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - A a}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 71, normalized size = 1.61 \[ i B a x +\frac {i A a \ln \left (\sin \left (d x +c \right )\right )}{d}-a A x +\frac {i B a c}{d}-\frac {a A \cot \left (d x +c \right )}{d}-\frac {A a c}{d}+\frac {a B \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.94, size = 64, normalized size = 1.45 \[ -\frac {2 \, {\left (d x + c\right )} {\left (A - i \, B\right )} a + {\left (i \, A + B\right )} a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, {\left (i \, A + B\right )} a \log \left (\tan \left (d x + c\right )\right ) + \frac {2 \, A a}{\tan \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.21, size = 39, normalized size = 0.89 \[ -\frac {A\,a\,\mathrm {cot}\left (c+d\,x\right )}{d}+\frac {a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.46, size = 53, normalized size = 1.20 \[ \frac {2 i A a}{- d e^{2 i c} e^{2 i d x} + d} + \frac {i a \left (A - i B\right ) \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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