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 = 0.15, antiderivative size = 20, normalized size of antiderivative = 1.00, 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 2626
Rule 2639
Rule 3771
Rule 4213
Rule 6742
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*}
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Mathematica [A] time = 0.44, 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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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x \cos \relax (x) - 4 \, \sec \relax (x) \tan \relax (x)\right )} \sqrt {\csc \relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.50, size = 0, normalized size = 0.00 \[ \int \left (\sqrt {\csc }\relax (x )\right ) \left (x \cos \relax (x )-4 \sec \relax (x ) \tan \relax (x )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (x \cos \relax (x) - 4 \, \sec \relax (x) \tan \relax (x)\right )} \sqrt {\csc \relax (x)}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.46, size = 77, normalized size = 3.85 \[ \frac {\left (4\,{\cos \relax (x)}^3-4\,\cos \relax (x)+2\,x\,{\cos \relax (x)}^2\,\sin \relax (x)-\sin \relax (x)\,4{}\mathrm {i}-x\,{\cos \relax (x)}^3\,2{}\mathrm {i}+{\cos \relax (x)}^2\,\sin \relax (x)\,4{}\mathrm {i}+x\,\cos \relax (x)\,2{}\mathrm {i}\right )\,1{}\mathrm {i}}{\cos \relax (x)\,\sin \relax (x)\,\sqrt {\frac {1}{\sin \relax (x)}}\,\left (-\sin \relax (x)+\cos \relax (x)\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (x \cos {\relax (x )} - 4 \tan {\relax (x )} \sec {\relax (x )}\right ) \sqrt {\csc {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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