Optimal. Leaf size=108 \[ -\frac{20 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right ),2\right )}{147 b^2}+\frac{4 \cos (a+b x)}{49 b^2 \csc ^{\frac{5}{2}}(a+b x)}+\frac{20 \cos (a+b x)}{147 b^2 \sqrt{\csc (a+b x)}}+\frac{2 x}{7 b \csc ^{\frac{7}{2}}(a+b x)} \]
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Rubi [A] time = 0.0596252, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {4213, 3769, 3771, 2641} \[ \frac{4 \cos (a+b x)}{49 b^2 \csc ^{\frac{5}{2}}(a+b x)}+\frac{20 \cos (a+b x)}{147 b^2 \sqrt{\csc (a+b x)}}-\frac{20 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{147 b^2}+\frac{2 x}{7 b \csc ^{\frac{7}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Rule 4213
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{x \cos (a+b x)}{\csc ^{\frac{5}{2}}(a+b x)} \, dx &=\frac{2 x}{7 b \csc ^{\frac{7}{2}}(a+b x)}-\frac{2 \int \frac{1}{\csc ^{\frac{7}{2}}(a+b x)} \, dx}{7 b}\\ &=\frac{2 x}{7 b \csc ^{\frac{7}{2}}(a+b x)}+\frac{4 \cos (a+b x)}{49 b^2 \csc ^{\frac{5}{2}}(a+b x)}-\frac{10 \int \frac{1}{\csc ^{\frac{3}{2}}(a+b x)} \, dx}{49 b}\\ &=\frac{2 x}{7 b \csc ^{\frac{7}{2}}(a+b x)}+\frac{4 \cos (a+b x)}{49 b^2 \csc ^{\frac{5}{2}}(a+b x)}+\frac{20 \cos (a+b x)}{147 b^2 \sqrt{\csc (a+b x)}}-\frac{10 \int \sqrt{\csc (a+b x)} \, dx}{147 b}\\ &=\frac{2 x}{7 b \csc ^{\frac{7}{2}}(a+b x)}+\frac{4 \cos (a+b x)}{49 b^2 \csc ^{\frac{5}{2}}(a+b x)}+\frac{20 \cos (a+b x)}{147 b^2 \sqrt{\csc (a+b x)}}-\frac{\left (10 \sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx}{147 b}\\ &=\frac{2 x}{7 b \csc ^{\frac{7}{2}}(a+b x)}+\frac{4 \cos (a+b x)}{49 b^2 \csc ^{\frac{5}{2}}(a+b x)}+\frac{20 \cos (a+b x)}{147 b^2 \sqrt{\csc (a+b x)}}-\frac{20 \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)}}{147 b^2}\\ \end{align*}
Mathematica [A] time = 0.420339, size = 93, normalized size = 0.86 \[ \frac{\sqrt{\csc (a+b x)} \left (80 \sqrt{\sin (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )+52 \sin (2 (a+b x))-6 \sin (4 (a+b x))-84 b x \cos (2 (a+b x))+21 b x \cos (4 (a+b x))+63 b x\right )}{588 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.099, 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 \frac{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 \frac{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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