Optimal. Leaf size=90 \[ \frac {\left (i a^2 (B+i C)+b^2 (C+i B)\right ) \log (a-i b \sin (x)+b \cos (x))}{2 a^2 b}-\frac {b x (B-i C)}{2 a^2}-\frac {(C+i B) (\cos (x)+i \sin (x))}{2 a} \]
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Rubi [A] time = 0.08, antiderivative size = 85, normalized size of antiderivative = 0.94, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {3130} \[ -\frac {b x (B-i C)}{2 a^2}+\frac {1}{2} \left (\frac {b (C+i B)}{a^2}+\frac {i (B+i C)}{b}\right ) \log (a-i b \sin (x)+b \cos (x))-\frac {(C+i B) (\cos (x)+i \sin (x))}{2 a} \]
Antiderivative was successfully verified.
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Rule 3130
Rubi steps
\begin {align*} \int \frac {B \cos (x)+C \sin (x)}{a+b \cos (x)-i b \sin (x)} \, dx &=-\frac {b (B-i C) x}{2 a^2}+\frac {1}{2} \left (\frac {i (B+i C)}{b}+\frac {b (i B+C)}{a^2}\right ) \log (a+b \cos (x)-i b \sin (x))-\frac {(i B+C) (\cos (x)+i \sin (x))}{2 a}\\ \end {align*}
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Mathematica [B] time = 0.29, size = 195, normalized size = 2.17 \[ \frac {x \left (a^2 B+i a^2 C-b^2 B+i b^2 C\right )}{4 a^2 b}+\frac {i \left (a^2 B+i a^2 C+b^2 B-i b^2 C\right ) \log \left (a^2+2 a b \cos (x)+b^2\right )}{4 a^2 b}+\frac {\left (a^2 B+i a^2 C+b^2 B-i b^2 C\right ) \tan ^{-1}\left (\frac {(a+b) \cos \left (\frac {x}{2}\right )}{a \sin \left (\frac {x}{2}\right )-b \sin \left (\frac {x}{2}\right )}\right )}{2 a^2 b}+\frac {(B-i C) \sin (x)}{2 a}-\frac {i (B-i C) \cos (x)}{2 a} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 68, normalized size = 0.76 \[ \frac {{\left (B + i \, C\right )} a^{2} x + {\left (-i \, B - C\right )} a b e^{\left (i \, x\right )} + {\left ({\left (i \, B - C\right )} a^{2} + {\left (i \, B + C\right )} b^{2}\right )} \log \left (\frac {a e^{\left (i \, x\right )} + b}{a}\right )}{2 \, a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 178, normalized size = 1.98 \[ -\frac {{\left (-i \, B b - C b\right )} \log \left (-a \tan \left (\frac {1}{2} \, x\right )^{2} + b \tan \left (\frac {1}{2} \, x\right )^{2} + 2 i \, a \tan \left (\frac {1}{2} \, x\right ) + a + b\right )}{4 \, a^{2}} - \frac {{\left (i \, B b + C b\right )} \log \left (\tan \left (\frac {1}{2} \, x\right ) + i\right )}{2 \, a^{2}} + \frac {{\left (2 \, B a^{2} + 2 i \, C a^{2} + B b^{2} - i \, C b^{2}\right )} {\left (x + 2 \, \arctan \left (\frac {i \, a \cos \relax (x) - a \sin \relax (x) + i \, a}{a \cos \relax (x) + i \, a \sin \relax (x) - a + 2 \, b}\right )\right )}}{4 \, a^{2} b} - \frac {-i \, B b \tan \left (\frac {1}{2} \, x\right ) - C b \tan \left (\frac {1}{2} \, x\right ) - 2 \, B a + 2 i \, C a + B b - i \, C b}{2 \, a^{2} {\left (\tan \left (\frac {1}{2} \, x\right ) + i\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.19, size = 388, normalized size = 4.31 \[ -\frac {i C}{a \left (\tan \left (\frac {x}{2}\right )+i\right )}+\frac {B}{a \left (\tan \left (\frac {x}{2}\right )+i\right )}-\frac {i \ln \left (\tan \left (\frac {x}{2}\right )+i\right ) b B}{2 a^{2}}-\frac {\ln \left (\tan \left (\frac {x}{2}\right )+i\right ) b C}{2 a^{2}}+\frac {C \ln \left (\tan \left (\frac {x}{2}\right )-i\right )}{2 b}-\frac {i B \ln \left (\tan \left (\frac {x}{2}\right )-i\right )}{2 b}+\frac {a \ln \left (i a +i b -a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )\right ) C}{2 b \left (-a +b \right )}-\frac {\ln \left (i a +i b -a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )\right ) C}{2 \left (-a +b \right )}-\frac {b \ln \left (i a +i b -a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )\right ) C}{2 a \left (-a +b \right )}+\frac {b^{2} \ln \left (i a +i b -a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )\right ) C}{2 a^{2} \left (-a +b \right )}-\frac {i a \ln \left (i a +i b -a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )\right ) B}{2 b \left (-a +b \right )}+\frac {i \ln \left (i a +i b -a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )\right ) B}{-2 a +2 b}-\frac {i b \ln \left (i a +i b -a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )\right ) B}{2 a \left (-a +b \right )}+\frac {i b^{2} \ln \left (i a +i b -a \tan \left (\frac {x}{2}\right )+b \tan \left (\frac {x}{2}\right )\right ) B}{2 a^{2} \left (-a +b \right )} \]
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 = 4.51, size = 118, normalized size = 1.31 \[ \ln \left (a+b+a\,\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}-b\,\mathrm {tan}\left (\frac {x}{2}\right )\,1{}\mathrm {i}\right )\,\left (\frac {-\frac {C}{2}+\frac {B\,1{}\mathrm {i}}{2}}{b}+\frac {\frac {C\,b^2}{2}+\frac {B\,b^2\,1{}\mathrm {i}}{2}}{a^2\,b}\right )+\frac {B-C\,1{}\mathrm {i}}{a\,\left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1{}\mathrm {i}\right )\,\left (C\,b+B\,b\,1{}\mathrm {i}\right )}{2\,a^2}-\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.81, size = 102, normalized size = 1.13 \[ \begin {cases} - \frac {\left (i B + C\right ) e^{i x}}{2 a} & \text {for}\: 2 a \neq 0 \\x \left (- \frac {B + i C}{2 b} + \frac {B a + B b + i C a - i C b}{2 a b}\right ) & \text {otherwise} \end {cases} - \frac {x \left (- B - i C\right )}{2 b} + \frac {i \left (B a^{2} + B b^{2} + i C a^{2} - i C b^{2}\right ) \log {\left (e^{i x} + \frac {b}{a} \right )}}{2 a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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