Optimal. Leaf size=19 \[ B \log (\sin (x)+1)-\frac {A \cos (x)}{\sin (x)+1} \]
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Rubi [A] time = 0.07, antiderivative size = 19, normalized size of antiderivative = 1.00, 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 31
Rule 2648
Rule 2667
Rule 4401
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*}
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Mathematica [B] time = 0.05, 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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fricas [A] time = 0.45, size = 38, normalized size = 2.00 \[ -\frac {A \cos \relax (x) - {\left (B \cos \relax (x) + B \sin \relax (x) + B\right )} \log \left (\sin \relax (x) + 1\right ) - A \sin \relax (x) + A}{\cos \relax (x) + \sin \relax (x) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 43, normalized size = 2.26 \[ -B \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + 2 \, B \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) - \frac {2 \, {\left (B \tan \left (\frac {1}{2} \, x\right ) + A + B\right )}}{\tan \left (\frac {1}{2} \, x\right ) + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 35, normalized size = 1.84 \[ -B \ln \left (\tan ^{2}\left (\frac {x}{2}\right )+1\right )-\frac {2 A}{\tan \left (\frac {x}{2}\right )+1}+2 B \ln \left (\tan \left (\frac {x}{2}\right )+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 24, normalized size = 1.26 \[ B \log \left (\sin \relax (x) + 1\right ) - \frac {2 \, A}{\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 15.15, size = 34, normalized size = 1.79 \[ 2\,B\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )+1\right )-\frac {2\,A}{\mathrm {tan}\left (\frac {x}{2}\right )+1}-B\,\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.05, size = 94, normalized size = 4.95 \[ - \frac {2 A}{\tan {\left (\frac {x}{2} \right )} + 1} + \frac {2 B \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )} \tan {\left (\frac {x}{2} \right )}}{\tan {\left (\frac {x}{2} \right )} + 1} + \frac {2 B \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{\tan {\left (\frac {x}{2} \right )} + 1} - \frac {B \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \tan {\left (\frac {x}{2} \right )}}{\tan {\left (\frac {x}{2} \right )} + 1} - \frac {B \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )}}{\tan {\left (\frac {x}{2} \right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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