Optimal. Leaf size=88 \[ \frac{\sin (x)}{2 \left (a^2-b^2\right ) (a+b \cos (x))}+\frac{x}{2 b (a+b \cos (x))^2}-\frac{a \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{b (a-b)^{3/2} (a+b)^{3/2}} \]
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Rubi [A] time = 0.103267, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4423, 2664, 12, 2659, 205} \[ \frac{\sin (x)}{2 \left (a^2-b^2\right ) (a+b \cos (x))}+\frac{x}{2 b (a+b \cos (x))^2}-\frac{a \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{b (a-b)^{3/2} (a+b)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4423
Rule 2664
Rule 12
Rule 2659
Rule 205
Rubi steps
\begin{align*} \int \frac{x \sin (x)}{(a+b \cos (x))^3} \, dx &=\frac{x}{2 b (a+b \cos (x))^2}-\frac{\int \frac{1}{(a+b \cos (x))^2} \, dx}{2 b}\\ &=\frac{x}{2 b (a+b \cos (x))^2}+\frac{\sin (x)}{2 \left (a^2-b^2\right ) (a+b \cos (x))}-\frac{\int \frac{a}{a+b \cos (x)} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac{x}{2 b (a+b \cos (x))^2}+\frac{\sin (x)}{2 \left (a^2-b^2\right ) (a+b \cos (x))}-\frac{a \int \frac{1}{a+b \cos (x)} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac{x}{2 b (a+b \cos (x))^2}+\frac{\sin (x)}{2 \left (a^2-b^2\right ) (a+b \cos (x))}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )}{b \left (a^2-b^2\right )}\\ &=-\frac{a \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{(a-b)^{3/2} b (a+b)^{3/2}}+\frac{x}{2 b (a+b \cos (x))^2}+\frac{\sin (x)}{2 \left (a^2-b^2\right ) (a+b \cos (x))}\\ \end{align*}
Mathematica [A] time = 0.315517, size = 85, normalized size = 0.97 \[ \frac{\frac{\sin (x) (a+b \cos (x))}{(a-b) (a+b)}+\frac{x}{b}}{2 (a+b \cos (x))^2}-\frac{a \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )}{b \left (b^2-a^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.231, size = 250, normalized size = 2.8 \begin{align*}{\frac{i \left ( -2\,i{a}^{2}x{{\rm e}^{2\,ix}}+2\,i{b}^{2}x{{\rm e}^{2\,ix}}+ba{{\rm e}^{3\,ix}}+2\,{a}^{2}{{\rm e}^{2\,ix}}+{b}^{2}{{\rm e}^{2\,ix}}+3\,ab{{\rm e}^{ix}}+{b}^{2} \right ) }{b \left ( b{{\rm e}^{2\,ix}}+2\,a{{\rm e}^{ix}}+b \right ) ^{2} \left ({a}^{2}-{b}^{2} \right ) }}-{\frac{{\frac{i}{2}}a}{ \left ( a+b \right ) \left ( a-b \right ) b}\ln \left ({{\rm e}^{ix}}+{\frac{1}{b} \left ( a\sqrt{{a}^{2}-{b}^{2}}+{a}^{2}-{b}^{2} \right ){\frac{1}{\sqrt{{a}^{2}-{b}^{2}}}}} \right ){\frac{1}{\sqrt{{a}^{2}-{b}^{2}}}}}+{\frac{{\frac{i}{2}}a}{ \left ( a+b \right ) \left ( a-b \right ) b}\ln \left ({{\rm e}^{ix}}+{\frac{1}{b} \left ( a\sqrt{{a}^{2}-{b}^{2}}-{a}^{2}+{b}^{2} \right ){\frac{1}{\sqrt{{a}^{2}-{b}^{2}}}}} \right ){\frac{1}{\sqrt{{a}^{2}-{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.68733, size = 944, normalized size = 10.73 \begin{align*} \left [\frac{{\left (a b^{2} \cos \left (x\right )^{2} + 2 \, a^{2} b \cos \left (x\right ) + a^{3}\right )} \sqrt{-a^{2} + b^{2}} \log \left (\frac{2 \, a b \cos \left (x\right ) +{\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + 2 \, \sqrt{-a^{2} + b^{2}}{\left (a \cos \left (x\right ) + b\right )} \sin \left (x\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (x\right )^{2} + 2 \, a b \cos \left (x\right ) + a^{2}}\right ) + 2 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x + 2 \,{\left (a^{3} b - a b^{3} +{\left (a^{2} b^{2} - b^{4}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{4 \,{\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5} +{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cos \left (x\right )^{2} + 2 \,{\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right )\right )}}, -\frac{{\left (a b^{2} \cos \left (x\right )^{2} + 2 \, a^{2} b \cos \left (x\right ) + a^{3}\right )} \sqrt{a^{2} - b^{2}} \arctan \left (-\frac{a \cos \left (x\right ) + b}{\sqrt{a^{2} - b^{2}} \sin \left (x\right )}\right ) -{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} x -{\left (a^{3} b - a b^{3} +{\left (a^{2} b^{2} - b^{4}\right )} \cos \left (x\right )\right )} \sin \left (x\right )}{2 \,{\left (a^{6} b - 2 \, a^{4} b^{3} + a^{2} b^{5} +{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} \cos \left (x\right )^{2} + 2 \,{\left (a^{5} b^{2} - 2 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (x\right )\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \sin \left (x\right )}{{\left (b \cos \left (x\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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