Optimal. Leaf size=22 \[ \sqrt {\sin ^2(x)}-x \sqrt {\sin ^2(x)} \cot (x) \]
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Rubi [A] time = 0.05, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6720, 3296, 2637} \[ \sqrt {\sin ^2(x)}-x \sqrt {\sin ^2(x)} \cot (x) \]
Antiderivative was successfully verified.
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Rule 2637
Rule 3296
Rule 6720
Rubi steps
\begin {align*} \int x \sqrt {\sin ^2(x)} \, dx &=\left (\csc (x) \sqrt {\sin ^2(x)}\right ) \int x \sin (x) \, dx\\ &=-x \cot (x) \sqrt {\sin ^2(x)}+\left (\csc (x) \sqrt {\sin ^2(x)}\right ) \int \cos (x) \, dx\\ &=\sqrt {\sin ^2(x)}-x \cot (x) \sqrt {\sin ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 16, normalized size = 0.73 \[ \sqrt {\sin ^2(x)} (1-x \cot (x)) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 8, normalized size = 0.36 \[ -x \cos \relax (x) + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 15, normalized size = 0.68 \[ -x \cos \relax (x) \mathrm {sgn}\left (\sin \relax (x)\right ) + \mathrm {sgn}\left (\sin \relax (x)\right ) \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 75, normalized size = 3.41 \[ -\frac {i \sqrt {-\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}\, {\mathrm e}^{2 i x} \left (x +i\right )}{2 \left ({\mathrm e}^{2 i x}-1\right )}-\frac {i \sqrt {-\left ({\mathrm e}^{2 i x}-1\right )^{2} {\mathrm e}^{-2 i x}}\, \left (x -i\right )}{2 \left ({\mathrm e}^{2 i x}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 9, normalized size = 0.41 \[ x \cos \relax (x) - \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.63, size = 48, normalized size = 2.18 \[ \sqrt {{\left (\frac {{\mathrm {e}}^{-x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2}-x\,\mathrm {cot}\relax (x)\,\sqrt {{\left (\frac {{\mathrm {e}}^{-x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{x\,1{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.72, size = 24, normalized size = 1.09 \[ - \frac {x \sqrt {\sin ^{2}{\relax (x )}} \cos {\relax (x )}}{\sin {\relax (x )}} + \sqrt {\sin ^{2}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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