Optimal. Leaf size=24 \[ \frac {\log (a+b \cos (x))}{b}+\frac {x \sin (x)}{a+b \cos (x)} \]
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Rubi [A] time = 0.06, antiderivative size = 24, 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 = {4593, 2668, 31} \[ \frac {\log (a+b \cos (x))}{b}+\frac {x \sin (x)}{a+b \cos (x)} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2668
Rule 4593
Rubi steps
\begin {align*} \int \frac {x (b+a \cos (x))}{(a+b \cos (x))^2} \, dx &=\frac {x \sin (x)}{a+b \cos (x)}-\int \frac {\sin (x)}{a+b \cos (x)} \, dx\\ &=\frac {x \sin (x)}{a+b \cos (x)}+\frac {\operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \cos (x)\right )}{b}\\ &=\frac {\log (a+b \cos (x))}{b}+\frac {x \sin (x)}{a+b \cos (x)}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 24, normalized size = 1.00 \[ \frac {\log (a+b \cos (x))}{b}+\frac {x \sin (x)}{a+b \cos (x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 36, normalized size = 1.50 \[ \frac {b x \sin \relax (x) + {\left (b \cos \relax (x) + a\right )} \log \left (-b \cos \relax (x) - a\right )}{b^{2} \cos \relax (x) + a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.32, size = 397, normalized size = 16.54 \[ \frac {a \log \left (\frac {4 \, {\left (a^{2} \tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, a b \tan \left (\frac {1}{2} \, x\right )^{4} + b^{2} \tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + a^{2} + 2 \, a b + b^{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} - b \log \left (\frac {4 \, {\left (a^{2} \tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, a b \tan \left (\frac {1}{2} \, x\right )^{4} + b^{2} \tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + a^{2} + 2 \, a b + b^{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} + 8 \, b x \tan \left (\frac {1}{2} \, x\right ) + a \log \left (\frac {4 \, {\left (a^{2} \tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, a b \tan \left (\frac {1}{2} \, x\right )^{4} + b^{2} \tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) + b \log \left (\frac {4 \, {\left (a^{2} \tan \left (\frac {1}{2} \, x\right )^{4} - 2 \, a b \tan \left (\frac {1}{2} \, x\right )^{4} + b^{2} \tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, a^{2} \tan \left (\frac {1}{2} \, x\right )^{2} - 2 \, b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + a^{2} + 2 \, a b + b^{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} - b^{2} \tan \left (\frac {1}{2} \, x\right )^{2} + a b + b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.41, size = 91, normalized size = 3.79 \[ \frac {2 x \tan \left (\frac {x}{2}\right )+2 x \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+a +b \right )}+\frac {\ln \left (a \left (\tan ^{2}\left (\frac {x}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {x}{2}\right )\right )+a +b \right )}{b}-\frac {\ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )}{b} \]
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 [B] time = 2.74, size = 68, normalized size = 2.83 \[ \frac {\ln \left (b+2\,a\,{\mathrm {e}}^{x\,1{}\mathrm {i}}+b\,{\mathrm {e}}^{x\,2{}\mathrm {i}}\right )}{b}-\frac {x\,2{}\mathrm {i}}{b}+\frac {x\,2{}\mathrm {i}+\frac {a\,x\,{\mathrm {e}}^{x\,1{}\mathrm {i}}\,2{}\mathrm {i}}{b}}{b+2\,a\,{\mathrm {e}}^{x\,1{}\mathrm {i}}+b\,{\mathrm {e}}^{x\,2{}\mathrm {i}}} \]
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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