Optimal. Leaf size=60 \[ \frac{4 \text{EllipticF}\left (\frac{1}{2} (a+b x),2\right )}{9 b^2}+\frac{4 \sin (a+b x) \sqrt{\cos (a+b x)}}{9 b^2}-\frac{2 x \cos ^{\frac{3}{2}}(a+b x)}{3 b} \]
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Rubi [A] time = 0.0388699, antiderivative size = 60, 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 = {3444, 2635, 2641} \[ \frac{4 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{9 b^2}+\frac{4 \sin (a+b x) \sqrt{\cos (a+b x)}}{9 b^2}-\frac{2 x \cos ^{\frac{3}{2}}(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 3444
Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int x \sqrt{\cos (a+b x)} \sin (a+b x) \, dx &=-\frac{2 x \cos ^{\frac{3}{2}}(a+b x)}{3 b}+\frac{2 \int \cos ^{\frac{3}{2}}(a+b x) \, dx}{3 b}\\ &=-\frac{2 x \cos ^{\frac{3}{2}}(a+b x)}{3 b}+\frac{4 \sqrt{\cos (a+b x)} \sin (a+b x)}{9 b^2}+\frac{2 \int \frac{1}{\sqrt{\cos (a+b x)}} \, dx}{9 b}\\ &=-\frac{2 x \cos ^{\frac{3}{2}}(a+b x)}{3 b}+\frac{4 F\left (\left .\frac{1}{2} (a+b x)\right |2\right )}{9 b^2}+\frac{4 \sqrt{\cos (a+b x)} \sin (a+b x)}{9 b^2}\\ \end{align*}
Mathematica [A] time = 0.152188, size = 52, normalized size = 0.87 \[ \frac{4 \text{EllipticF}\left (\frac{1}{2} (a+b x),2\right )+2 \sqrt{\cos (a+b x)} (2 \sin (a+b x)-3 b x \cos (a+b x))}{9 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.092, size = 0, normalized size = 0. \begin{align*} \int x\sin \left ( bx+a \right ) \sqrt{\cos \left ( bx+a \right ) }\, 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 \sqrt{\cos \left (b x + a\right )} \sin \left (b x + a\right )\,{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 \sqrt{\cos \left (b x + a\right )} \sin \left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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