Optimal. Leaf size=31 \[ \frac {x^2}{2}-\frac {\sin ^2(x)}{4}+\frac {3 \cos ^2(x)}{4}+2 x \sin (x) \cos (x) \]
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Rubi [A] time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4431, 3310, 30} \[ \frac {x^2}{2}-\frac {\sin ^2(x)}{4}+\frac {3 \cos ^2(x)}{4}+2 x \sin (x) \cos (x) \]
Antiderivative was successfully verified.
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Rule 30
Rule 3310
Rule 4431
Rubi steps
\begin {align*} \int x \csc (x) \sin (3 x) \, dx &=\int \left (3 x \cos ^2(x)-x \sin ^2(x)\right ) \, dx\\ &=3 \int x \cos ^2(x) \, dx-\int x \sin ^2(x) \, dx\\ &=\frac {3 \cos ^2(x)}{4}+2 x \cos (x) \sin (x)-\frac {\sin ^2(x)}{4}-\frac {\int x \, dx}{2}+\frac {3 \int x \, dx}{2}\\ &=\frac {x^2}{2}+\frac {3 \cos ^2(x)}{4}+2 x \cos (x) \sin (x)-\frac {\sin ^2(x)}{4}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 0.71 \[ \frac {x^2}{2}+x \sin (2 x)+\frac {1}{2} \cos (2 x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 17, normalized size = 0.55 \[ 2 \, x \cos \relax (x) \sin \relax (x) + \frac {1}{2} \, x^{2} + \cos \relax (x)^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 18, normalized size = 0.58 \[ \frac {1}{2} \, x^{2} + x \sin \left (2 \, x\right ) + \frac {1}{2} \, \cos \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 26, normalized size = 0.84 \[ 4 x \left (\frac {\cos \relax (x ) \sin \relax (x )}{2}+\frac {x}{2}\right )-\frac {3 x^{2}}{2}-\left (\sin ^{2}\relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 18, normalized size = 0.58 \[ \frac {1}{2} \, x^{2} + x \sin \left (2 \, x\right ) + \frac {1}{2} \, \cos \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 18, normalized size = 0.58 \[ \frac {\cos \left (2\,x\right )}{2}+x\,\sin \left (2\,x\right )+\frac {x^2}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.88, size = 37, normalized size = 1.19 \[ - x^{2} \sin ^{2}{\relax (x )} - x^{2} \cos ^{2}{\relax (x )} + \frac {3 x^{2}}{2} + 2 x \sin {\relax (x )} \cos {\relax (x )} + \cos ^{2}{\relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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