Optimal. Leaf size=65 \[ -\frac{12 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{25 b^2}+\frac{4 \sin ^{\frac{3}{2}}(a+b x) \cos (a+b x)}{25 b^2}+\frac{2 x \sin ^{\frac{5}{2}}(a+b x)}{5 b} \]
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Rubi [A] time = 0.0323581, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3443, 2635, 2639} \[ -\frac{12 E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{25 b^2}+\frac{4 \sin ^{\frac{3}{2}}(a+b x) \cos (a+b x)}{25 b^2}+\frac{2 x \sin ^{\frac{5}{2}}(a+b x)}{5 b} \]
Antiderivative was successfully verified.
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Rule 3443
Rule 2635
Rule 2639
Rubi steps
\begin{align*} \int x \cos (a+b x) \sin ^{\frac{3}{2}}(a+b x) \, dx &=\frac{2 x \sin ^{\frac{5}{2}}(a+b x)}{5 b}-\frac{2 \int \sin ^{\frac{5}{2}}(a+b x) \, dx}{5 b}\\ &=\frac{4 \cos (a+b x) \sin ^{\frac{3}{2}}(a+b x)}{25 b^2}+\frac{2 x \sin ^{\frac{5}{2}}(a+b x)}{5 b}-\frac{6 \int \sqrt{\sin (a+b x)} \, dx}{25 b}\\ &=-\frac{12 E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{25 b^2}+\frac{4 \cos (a+b x) \sin ^{\frac{3}{2}}(a+b x)}{25 b^2}+\frac{2 x \sin ^{\frac{5}{2}}(a+b x)}{5 b}\\ \end{align*}
Mathematica [C] time = 0.925856, size = 108, normalized size = 1.66 \[ \frac{\sqrt{\sin (a+b x)} \left (4 \tan \left (\frac{1}{2} (a+b x)\right ) \sqrt{\sec ^2\left (\frac{1}{2} (a+b x)\right )} \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{7}{4},-\tan ^2\left (\frac{1}{2} (a+b x)\right )\right )+2 \sin (2 (a+b x))-5 b x \cos (2 (a+b x))-12 \tan \left (\frac{1}{2} (a+b x)\right )+5 b x\right )}{25 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.092, size = 0, normalized size = 0. \begin{align*} \int x\cos \left ( bx+a \right ) \left ( \sin \left ( bx+a \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cos \left (b x + a\right ) \sin \left (b x + a\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \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 x \cos \left (b x + a\right ) \sin \left (b x + a\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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