Optimal. Leaf size=85 \[ -\frac {\left (a^2 C+2 i a A b-b^2 C\right ) \log (a-i b \sin (x)+b \cos (x))}{2 a^2 b}+\frac {x (2 a A+i b C)}{2 a^2}-\frac {i C \sin (x)}{2 a}-\frac {C \cos (x)}{2 a} \]
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Rubi [A] time = 0.05, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {3131} \[ -\frac {\left (a^2 C+2 i a A b-b^2 C\right ) \log (a-i b \sin (x)+b \cos (x))}{2 a^2 b}+\frac {x (2 a A+i b C)}{2 a^2}-\frac {i C \sin (x)}{2 a}-\frac {C \cos (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 3131
Rubi steps
\begin {align*} \int \frac {A+C \sin (x)}{a+b \cos (x)-i b \sin (x)} \, dx &=\frac {(2 a A+i b C) x}{2 a^2}-\frac {C \cos (x)}{2 a}-\frac {\left (2 i a A b+a^2 C-b^2 C\right ) \log (a+b \cos (x)-i b \sin (x))}{2 a^2 b}-\frac {i C \sin (x)}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 152, normalized size = 1.79 \[ \frac {2 i \left (a^2 C+2 i a A b-b^2 C\right ) \tan ^{-1}\left (\frac {(a+b) \cot \left (\frac {x}{2}\right )}{a-b}\right )-2 i a A b \log \left (a^2+2 a b \cos (x)+b^2\right )-a^2 C \log \left (a^2+2 a b \cos (x)+b^2\right )+b^2 C \log \left (a^2+2 a b \cos (x)+b^2\right )+i a^2 C x+2 a A b x-2 i a b C \sin (x)-2 a b C \cos (x)+i b^2 C x}{4 a^2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 57, normalized size = 0.67 \[ \frac {i \, C a^{2} x - C a b e^{\left (i \, x\right )} - {\left (C a^{2} + 2 i \, A a b - C b^{2}\right )} \log \left (\frac {a e^{\left (i \, x\right )} + b}{a}\right )}{2 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 169, normalized size = 1.99 \[ -\frac {{\left (2 i \, A a - C b\right )} \log \left (-a \tan \left (\frac {1}{2} \, x\right )^{2} + b \tan \left (\frac {1}{2} \, x\right )^{2} + 2 i \, a \tan \left (\frac {1}{2} \, x\right ) + a + b\right )}{4 \, a^{2}} - \frac {{\left (-2 i \, A a + C b\right )} \log \left (\tan \left (\frac {1}{2} \, x\right ) + i\right )}{2 \, a^{2}} - \frac {{\left (-2 i \, C a^{2} + 2 \, A a b + i \, C b^{2}\right )} {\left (x + 2 \, \arctan \left (\frac {i \, a \cos \relax (x) - a \sin \relax (x) + i \, a}{a \cos \relax (x) + i \, a \sin \relax (x) - a + 2 \, b}\right )\right )}}{4 \, a^{2} b} - \frac {2 i \, A a \tan \left (\frac {1}{2} \, x\right ) - C b \tan \left (\frac {1}{2} \, x\right ) - 2 \, A a + 2 i \, C a - i \, C b}{2 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + i\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 280, normalized size = 3.29 \[ -\frac {i C}{a \left (\tan \left (\frac {x}{2}\right )+i\right )}+\frac {i \ln \left (\tan \left (\frac {x}{2}\right )+i\right ) A}{a}-\frac {\ln \left (\tan \left (\frac {x}{2}\right )+i\right ) b C}{2 a^{2}}+\frac {C \ln \left (\tan \left (\frac {x}{2}\right )-i\right )}{2 b}+\frac {a \ln \left (i a +i b -a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )\right ) C}{2 b \left (-a +b \right )}-\frac {\ln \left (i a +i b -a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )\right ) C}{2 \left (-a +b \right )}-\frac {b \ln \left (i a +i b -a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )\right ) C}{2 a \left (-a +b \right )}+\frac {b^{2} \ln \left (i a +i b -a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )\right ) C}{2 a^{2} \left (-a +b \right )}+\frac {i \ln \left (i a +i b -a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )\right ) A}{-a +b}-\frac {i b \ln \left (i a +i b -a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )\right ) A}{a \left (-a +b \right )} \]
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 = 4.34, size = 96, normalized size = 1.13 \[ -\ln \left (a+b+a\,\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}-b\,\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}\right )\,\left (\frac {C}{2\,b}-\frac {C\,b}{2\,a^2}+\frac {A\,1{}\mathrm {i}}{a}\right )-\frac {C\,1{}\mathrm {i}}{a\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )}+\frac {C\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-\mathrm {i}\right )}{2\,b}+\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )\,\left (-\frac {C\,b}{2}+A\,a\,1{}\mathrm {i}\right )}{a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.74, size = 80, normalized size = 0.94 \[ \frac {i C x}{2 b} + \begin {cases} - \frac {C e^{i x}}{2 a} & \text {for}\: 2 a \neq 0 \\x \left (- \frac {i C}{2 b} + \frac {i C a - i C b}{2 a b}\right ) & \text {otherwise} \end {cases} - \frac {\left (2 i A a b + C a^{2} - C b^{2}\right ) \log {\left (e^{i x} + \frac {b}{a} \right )}}{2 a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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