Optimal. Leaf size=85 \[ -\frac{4 \cos (a+b x) \sqrt{\csc (a+b x)}}{3 b^2}-\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{3 b^2}-\frac{2 x \csc ^{\frac{3}{2}}(a+b x)}{3 b} \]
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Rubi [A] time = 0.0401547, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4213, 3768, 3771, 2639} \[ -\frac{4 \cos (a+b x) \sqrt{\csc (a+b x)}}{3 b^2}-\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{3 b^2}-\frac{2 x \csc ^{\frac{3}{2}}(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 4213
Rule 3768
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int x \cos (a+b x) \csc ^{\frac{5}{2}}(a+b x) \, dx &=-\frac{2 x \csc ^{\frac{3}{2}}(a+b x)}{3 b}+\frac{2 \int \csc ^{\frac{3}{2}}(a+b x) \, dx}{3 b}\\ &=-\frac{4 \cos (a+b x) \sqrt{\csc (a+b x)}}{3 b^2}-\frac{2 x \csc ^{\frac{3}{2}}(a+b x)}{3 b}-\frac{2 \int \frac{1}{\sqrt{\csc (a+b x)}} \, dx}{3 b}\\ &=-\frac{4 \cos (a+b x) \sqrt{\csc (a+b x)}}{3 b^2}-\frac{2 x \csc ^{\frac{3}{2}}(a+b x)}{3 b}-\frac{\left (2 \sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \sqrt{\sin (a+b x)} \, dx}{3 b}\\ &=-\frac{4 \cos (a+b x) \sqrt{\csc (a+b x)}}{3 b^2}-\frac{2 x \csc ^{\frac{3}{2}}(a+b x)}{3 b}-\frac{4 \sqrt{\csc (a+b x)} E\left (\left .\frac{1}{2} \left (a-\frac{\pi }{2}+b x\right )\right |2\right ) \sqrt{\sin (a+b x)}}{3 b^2}\\ \end{align*}
Mathematica [A] time = 0.184442, size = 56, normalized size = 0.66 \[ -\frac{2 \csc ^{\frac{3}{2}}(a+b x) \left (\sin (2 (a+b x))-2 \sin ^{\frac{3}{2}}(a+b x) E\left (\left .\frac{1}{4} (-2 a-2 b x+\pi )\right |2\right )+b x\right )}{3 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.104, size = 0, normalized size = 0. \begin{align*} \int x\cos \left ( bx+a \right ) \left ( \csc \left ( bx+a \right ) \right ) ^{{\frac{5}{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{5}{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{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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