Optimal. Leaf size=23 \[ B \log (1-\cos (x))-\frac{A \sin (x)}{1-\cos (x)} \]
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Rubi [A] time = 0.0783222, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {4401, 2648, 2667, 31} \[ B \log (1-\cos (x))-\frac{A \sin (x)}{1-\cos (x)} \]
Antiderivative was successfully verified.
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Rule 4401
Rule 2648
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{A+B \sin (x)}{1-\cos (x)} \, dx &=\int \left (-\frac{A}{-1+\cos (x)}-\frac{B \sin (x)}{-1+\cos (x)}\right ) \, dx\\ &=-\left (A \int \frac{1}{-1+\cos (x)} \, dx\right )-B \int \frac{\sin (x)}{-1+\cos (x)} \, dx\\ &=-\frac{A \sin (x)}{1-\cos (x)}+B \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,\cos (x)\right )\\ &=B \log (1-\cos (x))-\frac{A \sin (x)}{1-\cos (x)}\\ \end{align*}
Mathematica [A] time = 0.0465049, size = 20, normalized size = 0.87 \[ 2 B \log \left (\sin \left (\frac{x}{2}\right )\right )-A \cot \left (\frac{x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 31, normalized size = 1.4 \begin{align*} -B\ln \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) -{A \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+2\,B\ln \left ( \tan \left ( x/2 \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.95204, size = 26, normalized size = 1.13 \begin{align*} B \log \left (\cos \left (x\right ) - 1\right ) - \frac{A{\left (\cos \left (x\right ) + 1\right )}}{\sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.11904, size = 77, normalized size = 3.35 \begin{align*} \frac{B \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) \sin \left (x\right ) - A \cos \left (x\right ) - A}{\sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.680834, size = 27, normalized size = 1.17 \begin{align*} - \frac{A}{\tan{\left (\frac{x}{2} \right )}} - B \log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} + 1 \right )} + 2 B \log{\left (\tan{\left (\frac{x}{2} \right )} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17906, size = 53, normalized size = 2.3 \begin{align*} -B \log \left (\tan \left (\frac{1}{2} \, x\right )^{2} + 1\right ) + 2 \, B \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right ) - \frac{2 \, B \tan \left (\frac{1}{2} \, x\right ) + A}{\tan \left (\frac{1}{2} \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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