Optimal. Leaf size=20 \[ \frac{2 x}{\sqrt{\csc (x)}}-\frac{4 \sec (x)}{\csc ^{\frac{3}{2}}(x)} \]
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Rubi [A] time = 0.151274, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {6742, 4213, 3771, 2639, 2626} \[ \frac{2 x}{\sqrt{\csc (x)}}-\frac{4 \sec (x)}{\csc ^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 6742
Rule 4213
Rule 3771
Rule 2639
Rule 2626
Rubi steps
\begin{align*} \int \sqrt{\csc (x)} (x \cos (x)-4 \sec (x) \tan (x)) \, dx &=\int \left (x \cos (x) \sqrt{\csc (x)}-\frac{4 \sec ^2(x)}{\sqrt{\csc (x)}}\right ) \, dx\\ &=-\left (4 \int \frac{\sec ^2(x)}{\sqrt{\csc (x)}} \, dx\right )+\int x \cos (x) \sqrt{\csc (x)} \, dx\\ &=\frac{2 x}{\sqrt{\csc (x)}}-\frac{4 \sec (x)}{\csc ^{\frac{3}{2}}(x)}\\ \end{align*}
Mathematica [A] time = 0.447148, size = 17, normalized size = 0.85 \[ \frac{2 (x \csc (x)-2 \sec (x))}{\csc ^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.188, size = 0, normalized size = 0. \begin{align*} \int \sqrt{\csc \left ( x \right ) } \left ( x\cos \left ( x \right ) -4\,\sec \left ( x \right ) \tan \left ( x \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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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{\left (x \cos \left (x\right ) - 4 \, \sec \left (x\right ) \tan \left (x\right )\right )} \sqrt{\csc \left (x\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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