Optimal. Leaf size=105 \[ \frac {i \left (-\left (a^2 (B-i C)\right )+2 a A b-b^2 (B+i C)\right ) \log (a+i b \sin (x)+b \cos (x))}{2 a^2 b}+\frac {x (2 a A-b (B+i C))}{2 a^2}+\frac {(-C+i B) (\cos (x)-i \sin (x))}{2 a} \]
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Rubi [A] time = 0.07, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {3130} \[ \frac {i \left (a^2 (-(B-i C))+2 a A b-b^2 (B+i C)\right ) \log (a+i b \sin (x)+b \cos (x))}{2 a^2 b}+\frac {x (2 a A-b (B+i C))}{2 a^2}+\frac {(-C+i B) (\cos (x)-i \sin (x))}{2 a} \]
Antiderivative was successfully verified.
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Rule 3130
Rubi steps
\begin {align*} \int \frac {A+B \cos (x)+C \sin (x)}{a+b \cos (x)+i b \sin (x)} \, dx &=\frac {(2 a A-b (B+i C)) x}{2 a^2}+\frac {i \left (2 a A b-a^2 (B-i C)-b^2 (B+i C)\right ) \log (a+b \cos (x)+i b \sin (x))}{2 a^2 b}+\frac {(i B-C) (\cos (x)-i \sin (x))}{2 a}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 165, normalized size = 1.57 \[ \frac {x \left (a^2 (B-i C)+2 a A b-b^2 (B+i C)\right )+\left (a^2 (-C-i B)+2 i a A b+b^2 (C-i B)\right ) \log \left (a^2+2 a b \cos (x)+b^2\right )+2 \left (a^2 (B-i C)-2 a A b+b^2 (B+i C)\right ) \tan ^{-1}\left (\frac {(a+b) \cot \left (\frac {x}{2}\right )}{a-b}\right )+2 a b (B+i C) \sin (x)+2 i a b (B+i C) \cos (x)}{4 a^2 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.46, size = 89, normalized size = 0.85 \[ \frac {{\left ({\left (i \, B - C\right )} a b + {\left (2 \, A a b - {\left (B + i \, C\right )} b^{2}\right )} x e^{\left (i \, x\right )} + {\left ({\left (-i \, B - C\right )} a^{2} + 2 i \, A a b + {\left (-i \, B + C\right )} b^{2}\right )} e^{\left (i \, x\right )} \log \left (\frac {b e^{\left (i \, x\right )} + a}{b}\right )\right )} e^{\left (-i \, x\right )}}{2 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 203, normalized size = 1.93 \[ -\frac {{\left (-2 i \, A a + i \, B b - C b\right )} \log \left (-a \tan \left (\frac {1}{2} \, x\right )^{2} + b \tan \left (\frac {1}{2} \, x\right )^{2} - 2 i \, a \tan \left (\frac {1}{2} \, x\right ) + a + b\right )}{4 \, a^{2}} - \frac {{\left (2 i \, A a - i \, B b + C b\right )} \log \left (\tan \left (\frac {1}{2} \, x\right ) - i\right )}{2 \, a^{2}} + \frac {{\left (2 \, B a^{2} - 2 i \, C a^{2} - 2 \, A a b + B b^{2} + i \, C b^{2}\right )} {\left (x + 2 \, \arctan \left (\frac {-i \, a \cos \relax (x) - a \sin \relax (x) - i \, a}{a \cos \relax (x) - i \, a \sin \relax (x) - a + 2 \, b}\right )\right )}}{4 \, a^{2} b} - \frac {-2 i \, A a \tan \left (\frac {1}{2} \, x\right ) + i \, B b \tan \left (\frac {1}{2} \, x\right ) - C b \tan \left (\frac {1}{2} \, x\right ) - 2 \, A a - 2 \, B a - 2 i \, C a + B b + i \, C b}{2 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) - i\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.18, size = 257, normalized size = 2.45 \[ \frac {C \ln \left (\tan \left (\frac {x}{2}\right )+i\right )}{2 b}+\frac {i B \ln \left (\tan \left (\frac {x}{2}\right )+i\right )}{2 b}+\frac {i C}{a \left (\tan \left (\frac {x}{2}\right )-i\right )}+\frac {B}{a \left (\tan \left (\frac {x}{2}\right )-i\right )}-\frac {i \ln \left (\tan \left (\frac {x}{2}\right )-i\right ) A}{a}+\frac {i \ln \left (\tan \left (\frac {x}{2}\right )-i\right ) b B}{2 a^{2}}-\frac {\ln \left (\tan \left (\frac {x}{2}\right )-i\right ) b C}{2 a^{2}}-\frac {\ln \left (i a +i b +a \tan \left (\frac {x}{2}\right )-b \tan \left (\frac {x}{2}\right )\right ) C}{2 b}+\frac {b \ln \left (i a +i b +a \tan \left (\frac {x}{2}\right )-b \tan \left (\frac {x}{2}\right )\right ) C}{2 a^{2}}+\frac {i \ln \left (i a +i b +a \tan \left (\frac {x}{2}\right )-b \tan \left (\frac {x}{2}\right )\right ) A}{a}-\frac {i \ln \left (i a +i b +a \tan \left (\frac {x}{2}\right )-b \tan \left (\frac {x}{2}\right )\right ) B}{2 b}-\frac {i b \ln \left (i a +i b +a \tan \left (\frac {x}{2}\right )-b \tan \left (\frac {x}{2}\right )\right ) B}{2 a^{2}} \]
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 = 6.93, size = 132, normalized size = 1.26 \[ -\ln \left (a+b-a\,\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}+b\,\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}\right )\,\left (\frac {\frac {C}{2}+\frac {B\,1{}\mathrm {i}}{2}}{b}-\frac {\frac {C\,b^2}{2}-\frac {B\,b^2\,1{}\mathrm {i}}{2}+A\,a\,b\,1{}\mathrm {i}}{a^2\,b}\right )+\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )\,\left (C+B\,1{}\mathrm {i}\right )}{2\,b}+\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-\mathrm {i}\right )\,\left (B\,b-2\,A\,a+C\,b\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,a^2}+\frac {5\,B+C\,5{}\mathrm {i}}{5\,a\,\left (\mathrm {tan}\left (\frac {x}{2}\right )-\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.37, size = 138, normalized size = 1.31 \[ \begin {cases} - \frac {\left (- i B + C\right ) e^{- i x}}{2 a} & \text {for}\: 2 a \neq 0 \\x \left (- \frac {2 A a - B b - i C b}{2 a^{2}} - \frac {i \left (2 i A a + i B a - i B b - C a + C b\right )}{2 a^{2}}\right ) & \text {otherwise} \end {cases} - \frac {x \left (- 2 A a + B b + i C b\right )}{2 a^{2}} - \frac {i \left (- 2 A a b + B a^{2} + B b^{2} - i C a^{2} + i C b^{2}\right ) \log {\left (\frac {a}{b} + e^{i x} \right )}}{2 a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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