Optimal. Leaf size=85 \[ -\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right ),2\right )}{9 b^2}+\frac{4 \cos (a+b x)}{9 b^2 \sqrt{\csc (a+b x)}}+\frac{2 x}{3 b \csc ^{\frac{3}{2}}(a+b x)} \]
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Rubi [A] time = 0.0442773, 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, 3769, 3771, 2641} \[ \frac{4 \cos (a+b x)}{9 b^2 \sqrt{\csc (a+b x)}}-\frac{4 \sqrt{\sin (a+b x)} \sqrt{\csc (a+b x)} F\left (\left .\frac{1}{2} \left (a+b x-\frac{\pi }{2}\right )\right |2\right )}{9 b^2}+\frac{2 x}{3 b \csc ^{\frac{3}{2}}(a+b x)} \]
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)}{\sqrt{\csc (a+b x)}} \, dx &=\frac{2 x}{3 b \csc ^{\frac{3}{2}}(a+b x)}-\frac{2 \int \frac{1}{\csc ^{\frac{3}{2}}(a+b x)} \, dx}{3 b}\\ &=\frac{2 x}{3 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{4 \cos (a+b x)}{9 b^2 \sqrt{\csc (a+b x)}}-\frac{2 \int \sqrt{\csc (a+b x)} \, dx}{9 b}\\ &=\frac{2 x}{3 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{4 \cos (a+b x)}{9 b^2 \sqrt{\csc (a+b x)}}-\frac{\left (2 \sqrt{\csc (a+b x)} \sqrt{\sin (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (a+b x)}} \, dx}{9 b}\\ &=\frac{2 x}{3 b \csc ^{\frac{3}{2}}(a+b x)}+\frac{4 \cos (a+b x)}{9 b^2 \sqrt{\csc (a+b x)}}-\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)}}{9 b^2}\\ \end{align*}
Mathematica [A] time = 0.244726, size = 65, normalized size = 0.76 \[ \frac{2 \sqrt{\csc (a+b x)} \left (2 \sqrt{\sin (a+b x)} \text{EllipticF}\left (\frac{1}{4} (-2 a-2 b x+\pi ),2\right )+3 b x \sin ^2(a+b x)+\sin (2 (a+b x))\right )}{9 b^2} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{x\cos \left ( bx+a \right ){\frac{1}{\sqrt{\csc \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 \frac{x \cos \left (b x + a\right )}{\sqrt{\csc \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \cos{\left (a + b x \right )}}{\sqrt{\csc{\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 )}{\sqrt{\csc \left (b x + a\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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