Optimal. Leaf size=84 \[ -\frac{i \left (a^2 (-B)+2 a A b-b^2 B\right ) \log (a-i b \sin (x)+b \cos (x))}{2 a^2 b}+\frac{x (2 a A-b B)}{2 a^2}+\frac{B \sin (x)}{2 a}-\frac{i B \cos (x)}{2 a} \]
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Rubi [A] time = 0.0424567, antiderivative size = 84, normalized size of antiderivative = 1., 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 = {3132} \[ -\frac{i \left (a^2 (-B)+2 a A b-b^2 B\right ) \log (a-i b \sin (x)+b \cos (x))}{2 a^2 b}+\frac{x (2 a A-b B)}{2 a^2}+\frac{B \sin (x)}{2 a}-\frac{i B \cos (x)}{2 a} \]
Antiderivative was successfully verified.
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Rule 3132
Rubi steps
\begin{align*} \int \frac{A+B \cos (x)}{a+b \cos (x)-i b \sin (x)} \, dx &=\frac{(2 a A-b B) x}{2 a^2}-\frac{i B \cos (x)}{2 a}-\frac{i \left (2 a A b-a^2 B-b^2 B\right ) \log (a+b \cos (x)-i b \sin (x))}{2 a^2 b}+\frac{B \sin (x)}{2 a}\\ \end{align*}
Mathematica [A] time = 0.182585, size = 147, normalized size = 1.75 \[ \frac{2 \left (a^2 B-2 a A b+b^2 B\right ) \tan ^{-1}\left (\frac{(a+b) \cot \left (\frac{x}{2}\right )}{a-b}\right )-2 i a A b \log \left (a^2+2 a b \cos (x)+b^2\right )+i a^2 B \log \left (a^2+2 a b \cos (x)+b^2\right )+i b^2 B \log \left (a^2+2 a b \cos (x)+b^2\right )+a^2 B x+2 a A b x+2 a b B \sin (x)-2 i a b B \cos (x)-b^2 B x}{4 a^2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.08, size = 284, normalized size = 3.4 \begin{align*}{\frac{iA}{a}\ln \left ( \tan \left ({\frac{x}{2}} \right ) +i \right ) }-{\frac{{\frac{i}{2}}bB}{{a}^{2}}\ln \left ( \tan \left ({\frac{x}{2}} \right ) +i \right ) }+{\frac{B}{a} \left ( \tan \left ({\frac{x}{2}} \right ) +i \right ) ^{-1}}+{\frac{iA}{-a+b}\ln \left ( ia+ib-a\tan \left ({\frac{x}{2}} \right ) +b\tan \left ({\frac{x}{2}} \right ) \right ) }-{\frac{iAb}{a \left ( -a+b \right ) }\ln \left ( ia+ib-a\tan \left ({\frac{x}{2}} \right ) +b\tan \left ({\frac{x}{2}} \right ) \right ) }-{\frac{{\frac{i}{2}}aB}{b \left ( -a+b \right ) }\ln \left ( ia+ib-a\tan \left ({\frac{x}{2}} \right ) +b\tan \left ({\frac{x}{2}} \right ) \right ) }+{\frac{{\frac{i}{2}}B}{-a+b}\ln \left ( ia+ib-a\tan \left ({\frac{x}{2}} \right ) +b\tan \left ({\frac{x}{2}} \right ) \right ) }-{\frac{{\frac{i}{2}}bB}{a \left ( -a+b \right ) }\ln \left ( ia+ib-a\tan \left ({\frac{x}{2}} \right ) +b\tan \left ({\frac{x}{2}} \right ) \right ) }+{\frac{{\frac{i}{2}}{b}^{2}B}{{a}^{2} \left ( -a+b \right ) }\ln \left ( ia+ib-a\tan \left ({\frac{x}{2}} \right ) +b\tan \left ({\frac{x}{2}} \right ) \right ) }-{\frac{{\frac{i}{2}}B}{b}\ln \left ( \tan \left ({\frac{x}{2}} \right ) -i \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89394, size = 132, normalized size = 1.57 \begin{align*} \frac{B a^{2} x - i \, B a b e^{\left (i \, x\right )} +{\left (i \, B a^{2} - 2 i \, A a b + i \, B b^{2}\right )} \log \left (\frac{a e^{\left (i \, x\right )} + b}{a}\right )}{2 \, a^{2} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.884295, size = 51, normalized size = 0.61 \begin{align*} \left (- \frac{i A}{a} + \frac{i B}{2 b} + \frac{i B b}{2 a^{2}}\right ) \log{\left (e^{i x} + \frac{b}{a} \right )} + \frac{B a x - i B b e^{i x}}{2 a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15591, size = 209, normalized size = 2.49 \begin{align*} -\frac{2 \,{\left (B a^{3} - 2 \, A a^{2} b - B a^{2} b + 2 \, A a b^{2} + B a b^{2} - B b^{3}\right )} \log \left (-a \tan \left (\frac{1}{2} \, x\right ) + b \tan \left (\frac{1}{2} \, x\right ) + i \, a + i \, b\right )}{4 i \, a^{3} b - 4 i \, a^{2} b^{2}} - \frac{i \, B \log \left (\tan \left (\frac{1}{2} \, x\right ) - i\right )}{2 \, b} - \frac{{\left (-2 i \, A a + i \, B b\right )} \log \left (\tan \left (\frac{1}{2} \, x\right ) + i\right )}{2 \, a^{2}} - \frac{2 i \, A a \tan \left (\frac{1}{2} \, x\right ) - i \, B b \tan \left (\frac{1}{2} \, x\right ) - 2 \, A a - 2 \, B a + B b}{2 \, a^{2}{\left (\tan \left (\frac{1}{2} \, x\right ) + i\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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