Optimal. Leaf size=58 \[ \frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right ),2\right )}{b^2}-\frac{2 x \sqrt{\csc (a+b x)}}{b} \]
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Rubi [A] time = 0.0320025, antiderivative size = 58, 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 = {4213, 3771, 2641} \[ \frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{b^2}-\frac{2 x \sqrt{\csc (a+b x)}}{b} \]
Antiderivative was successfully verified.
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Rule 4213
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int x \cos (a+b x) \csc ^{\frac{3}{2}}(a+b x) \, dx &=-\frac{2 x \sqrt{\csc (a+b x)}}{b}+\frac{2 \int \sqrt{\csc (a+b x)} \, dx}{b}\\ &=-\frac{2 x \sqrt{\csc (a+b x)}}{b}+\frac{\left (2 \sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx}{b}\\ &=-\frac{2 x \sqrt{\csc (a+b x)}}{b}+\frac{4 \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{b^2}\\ \end{align*}
Mathematica [A] time = 0.148038, size = 46, normalized size = 0.79 \[ -\frac{2 \sqrt{\csc (a+b x)} \left (2 \sqrt{\sin (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )+b x\right )}{b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.109, size = 0, normalized size = 0. \begin{align*} \int x\cos \left ( bx+a \right ) \left ( \csc \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 ) \csc \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 ) \csc \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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