Optimal. Leaf size=40 \[ a x (B+i A)+\frac {a A \log (\sin (c+d x))}{d}-\frac {i a B \log (\cos (c+d x))}{d} \]
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Rubi [A] time = 0.07, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3589, 3475, 3531} \[ a x (B+i A)+\frac {a A \log (\sin (c+d x))}{d}-\frac {i a B \log (\cos (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3531
Rule 3589
Rubi steps
\begin {align*} \int \cot (c+d x) (a+i a \tan (c+d x)) (A+B \tan (c+d x)) \, dx &=(i a B) \int \tan (c+d x) \, dx+\int \cot (c+d x) (a A+a (i A+B) \tan (c+d x)) \, dx\\ &=a (i A+B) x-\frac {i a B \log (\cos (c+d x))}{d}+(a A) \int \cot (c+d x) \, dx\\ &=a (i A+B) x-\frac {i a B \log (\cos (c+d x))}{d}+\frac {a A \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 49, normalized size = 1.22 \[ \frac {a A (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+i a A x-\frac {i a B \log (\cos (c+d x))}{d}+a B x \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 36, normalized size = 0.90 \[ \frac {-i \, B a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + A a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.41, size = 74, normalized size = 1.85 \[ -\frac {i \, B a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right ) + i \, B a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right ) - A a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 2 \, {\left (A a - i \, B a\right )} \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 56, normalized size = 1.40 \[ i A a x +\frac {i A a c}{d}-\frac {i a B \ln \left (\cos \left (d x +c \right )\right )}{d}+a B x +\frac {a A \ln \left (\sin \left (d x +c \right )\right )}{d}+\frac {B a c}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 49, normalized size = 1.22 \[ \frac {2 \, {\left (d x + c\right )} {\left (i \, A + B\right )} a - {\left (A - i \, B\right )} a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 2 \, A a \log \left (\tan \left (d x + c\right )\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.23, size = 36, normalized size = 0.90 \[ \frac {A\,a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )}{d}-\frac {a\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.94, size = 97, normalized size = 2.42 \[ \frac {A a \log {\left (\frac {A a + i B a}{- A a e^{2 i c} - i B a e^{2 i c}} + e^{2 i d x} \right )}}{d} - \frac {i B a \log {\left (\frac {- A a - i B a}{- A a e^{2 i c} - i B a e^{2 i c}} + e^{2 i d x} \right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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