Optimal. Leaf size=25 \[ \frac {\log (a+b \sin (x))}{b}-\frac {x \cos (x)}{a+b \sin (x)} \]
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Rubi [A] time = 0.05, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4592, 2668, 31} \[ \frac {\log (a+b \sin (x))}{b}-\frac {x \cos (x)}{a+b \sin (x)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2668
Rule 4592
Rubi steps
\begin {align*} \int \frac {x (b+a \sin (x))}{(a+b \sin (x))^2} \, dx &=-\frac {x \cos (x)}{a+b \sin (x)}+\int \frac {\cos (x)}{a+b \sin (x)} \, dx\\ &=-\frac {x \cos (x)}{a+b \sin (x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \sin (x)\right )}{b}\\ &=\frac {\log (a+b \sin (x))}{b}-\frac {x \cos (x)}{a+b \sin (x)}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 25, normalized size = 1.00 \[ \frac {\log (a+b \sin (x))}{b}-\frac {x \cos (x)}{a+b \sin (x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.71, size = 35, normalized size = 1.40 \[ -\frac {b x \cos \relax (x) - {\left (b \sin \relax (x) + a\right )} \log \left (b \sin \relax (x) + a\right )}{b^{2} \sin \relax (x) + a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 283, normalized size = 11.32 \[ \frac {4 \, b x \tan \left (\frac {1}{2} \, x\right )^{2} + a \log \left (\frac {4 \, {\left (a^{2} \tan \left (\frac {1}{2} \, x\right )^{4} + 4 \, a b \tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 4 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 4 \, a b \tan \left (\frac {1}{2} \, x\right ) + a^{2}\right )}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b \log \left (\frac {4 \, {\left (a^{2} \tan \left (\frac {1}{2} \, x\right )^{4} + 4 \, a b \tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 4 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 4 \, a b \tan \left (\frac {1}{2} \, x\right ) + a^{2}\right )}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right ) - 4 \, b x + a \log \left (\frac {4 \, {\left (a^{2} \tan \left (\frac {1}{2} \, x\right )^{4} + 4 \, a b \tan \left (\frac {1}{2} \, x\right )^{3} + 2 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 4 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + 4 \, a b \tan \left (\frac {1}{2} \, x\right ) + a^{2}\right )}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right )}{2 \, {\left (a b \tan \left (\frac {1}{2} \, x\right )^{2} + 2 \, b^{2} \tan \left (\frac {1}{2} \, x\right ) + a b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.70, size = 80, normalized size = 3.20 \[ \frac {x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )-x}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a \right )}+\frac {\ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+2 b \tan \left (\frac {x}{2}\right )+a \right )}{b}-\frac {\ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}{b} \]
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 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {x\,\left (b+a\,\sin \relax (x)\right )}{{\left (a+b\,\sin \relax (x)\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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