Optimal. Leaf size=38 \[ \frac{4 \text{EllipticF}\left (\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right ),2\right )}{b^2}-\frac{2 x}{b \sqrt{\sin (a+b x)}} \]
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Rubi [A] time = 0.0229804, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3443, 2641} \[ \frac{4 F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b^2}-\frac{2 x}{b \sqrt{\sin (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 3443
Rule 2641
Rubi steps
\begin{align*} \int \frac{x \cos (a+b x)}{\sin ^{\frac{3}{2}}(a+b x)} \, dx &=-\frac{2 x}{b \sqrt{\sin (a+b x)}}+\frac{2 \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx}{b}\\ &=\frac{4 F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right )}{b^2}-\frac{2 x}{b \sqrt{\sin (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.181046, size = 37, normalized size = 0.97 \[ \frac{2 \left (-2 \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )-\frac{b x}{\sqrt{\sin (a+b x)}}\right )}{b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.098, 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 \frac{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \cos{\left (a + b x \right )}}{\sin ^{\frac{3}{2}}{\left (a + b x \right )}}\, dx \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 \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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