Optimal. Leaf size=19 \[ B \log (\sin (x)+1)-\frac{A \cos (x)}{\sin (x)+1} \]
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Rubi [A] time = 0.0683109, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {4401, 2648, 2667, 31} \[ B \log (\sin (x)+1)-\frac{A \cos (x)}{\sin (x)+1} \]
Antiderivative was successfully verified.
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Rule 4401
Rule 2648
Rule 2667
Rule 31
Rubi steps
\begin{align*} \int \frac{A+B \cos (x)}{1+\sin (x)} \, dx &=\int \left (\frac{A}{1+\sin (x)}+\frac{B \cos (x)}{1+\sin (x)}\right ) \, dx\\ &=A \int \frac{1}{1+\sin (x)} \, dx+B \int \frac{\cos (x)}{1+\sin (x)} \, dx\\ &=-\frac{A \cos (x)}{1+\sin (x)}+B \operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\sin (x)\right )\\ &=B \log (1+\sin (x))-\frac{A \cos (x)}{1+\sin (x)}\\ \end{align*}
Mathematica [B] time = 0.0493335, size = 42, normalized size = 2.21 \[ \frac{2 A \sin \left (\frac{x}{2}\right )}{\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )}+2 B \log \left (\sin \left (\frac{x}{2}\right )+\cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 35, normalized size = 1.8 \begin{align*} -B\ln \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) -2\,{\frac{A}{\tan \left ( x/2 \right ) +1}}+2\,B\ln \left ( \tan \left ( x/2 \right ) +1 \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.995644, size = 32, normalized size = 1.68 \begin{align*} B \log \left (\sin \left (x\right ) + 1\right ) - \frac{2 \, A}{\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.99319, size = 127, normalized size = 6.68 \begin{align*} -\frac{A \cos \left (x\right ) -{\left (B \cos \left (x\right ) + B \sin \left (x\right ) + B\right )} \log \left (\sin \left (x\right ) + 1\right ) - A \sin \left (x\right ) + A}{\cos \left (x\right ) + \sin \left (x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.41307, size = 94, normalized size = 4.95 \begin{align*} - \frac{2 A}{\tan{\left (\frac{x}{2} \right )} + 1} + \frac{2 B \log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )} \tan{\left (\frac{x}{2} \right )}}{\tan{\left (\frac{x}{2} \right )} + 1} + \frac{2 B \log{\left (\tan{\left (\frac{x}{2} \right )} + 1 \right )}}{\tan{\left (\frac{x}{2} \right )} + 1} - \frac{B \log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} + 1 \right )} \tan{\left (\frac{x}{2} \right )}}{\tan{\left (\frac{x}{2} \right )} + 1} - \frac{B \log{\left (\tan ^{2}{\left (\frac{x}{2} \right )} + 1 \right )}}{\tan{\left (\frac{x}{2} \right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.11003, size = 58, normalized size = 3.05 \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 ) + 1 \right |}\right ) - \frac{2 \,{\left (B \tan \left (\frac{1}{2} \, x\right ) + A + B\right )}}{\tan \left (\frac{1}{2} \, x\right ) + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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