Optimal. Leaf size=33 \[ -\frac {3 x^2}{4}-\frac {\cos ^2(x)}{4}-x \cot (x)+\log (\sin (x))-\frac {1}{2} x \cos (x) \sin (x) \]
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Rubi [A]
time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4493, 3391, 30,
3801, 3556} \begin {gather*} -\frac {3 x^2}{4}-\frac {\cos ^2(x)}{4}-x \cot (x)+\log (\sin (x))-\frac {1}{2} x \sin (x) \cos (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 3391
Rule 3556
Rule 3801
Rule 4493
Rubi steps
\begin {align*} \int x \cos ^2(x) \cot ^2(x) \, dx &=-\int x \cos ^2(x) \, dx+\int x \cot ^2(x) \, dx\\ &=-\frac {1}{4} \cos ^2(x)-x \cot (x)-\frac {1}{2} x \cos (x) \sin (x)-\frac {\int x \, dx}{2}-\int x \, dx+\int \cot (x) \, dx\\ &=-\frac {3 x^2}{4}-\frac {\cos ^2(x)}{4}-x \cot (x)+\log (\sin (x))-\frac {1}{2} x \cos (x) \sin (x)\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 33, normalized size = 1.00 \begin {gather*} -\frac {3 x^2}{4}-\frac {1}{8} \cos (2 x)-x \cot (x)+\log (\sin (x))-\frac {1}{4} x \sin (2 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.15, size = 60, normalized size = 1.82
method | result | size |
risch | \(-\frac {3 x^{2}}{4}+\frac {i \left (i+2 x \right ) {\mathrm e}^{2 i x}}{16}-\frac {i \left (-i+2 x \right ) {\mathrm e}^{-2 i x}}{16}-2 i x -\frac {2 i x}{{\mathrm e}^{2 i x}-1}+\ln \left ({\mathrm e}^{2 i x}-1\right )\) | \(60\) |
norman | \(\frac {-\frac {\tan \left (\frac {x}{2}\right )}{2}-\frac {\left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{2}-\frac {x}{2}-\frac {3 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}+\frac {3 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{2}+\frac {x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{2}-\frac {3 x^{2} \tan \left (\frac {x}{2}\right )}{4}-\frac {3 x^{2} \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{2}-\frac {3 x^{2} \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{4}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{2} \tan \left (\frac {x}{2}\right )}-\ln \left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )+\ln \left (\tan \left (\frac {x}{2}\right )\right )\) | \(111\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.94, size = 45, normalized size = 1.36 \begin {gather*} \frac {4 \, x \cos \left (x\right )^{3} - 12 \, x \cos \left (x\right ) - {\left (6 \, x^{2} + 2 \, \cos \left (x\right )^{2} - 1\right )} \sin \left (x\right ) + 8 \, \log \left (\frac {1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right )}{8 \, \sin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 507 vs.
\(2 (32) = 64\).
time = 0.74, size = 507, normalized size = 15.36 \begin {gather*} - \frac {3 x^{2} \tan ^{5}{\left (\frac {x}{2} \right )}}{4 \tan ^{5}{\left (\frac {x}{2} \right )} + 8 \tan ^{3}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )}} - \frac {6 x^{2} \tan ^{3}{\left (\frac {x}{2} \right )}}{4 \tan ^{5}{\left (\frac {x}{2} \right )} + 8 \tan ^{3}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )}} - \frac {3 x^{2} \tan {\left (\frac {x}{2} \right )}}{4 \tan ^{5}{\left (\frac {x}{2} \right )} + 8 \tan ^{3}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )}} + \frac {2 x \tan ^{6}{\left (\frac {x}{2} \right )}}{4 \tan ^{5}{\left (\frac {x}{2} \right )} + 8 \tan ^{3}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )}} + \frac {6 x \tan ^{4}{\left (\frac {x}{2} \right )}}{4 \tan ^{5}{\left (\frac {x}{2} \right )} + 8 \tan ^{3}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )}} - \frac {6 x \tan ^{2}{\left (\frac {x}{2} \right )}}{4 \tan ^{5}{\left (\frac {x}{2} \right )} + 8 \tan ^{3}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )}} - \frac {2 x}{4 \tan ^{5}{\left (\frac {x}{2} \right )} + 8 \tan ^{3}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )}} - \frac {4 \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \tan ^{5}{\left (\frac {x}{2} \right )}}{4 \tan ^{5}{\left (\frac {x}{2} \right )} + 8 \tan ^{3}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )}} - \frac {8 \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \tan ^{3}{\left (\frac {x}{2} \right )}}{4 \tan ^{5}{\left (\frac {x}{2} \right )} + 8 \tan ^{3}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )}} - \frac {4 \log {\left (\tan ^{2}{\left (\frac {x}{2} \right )} + 1 \right )} \tan {\left (\frac {x}{2} \right )}}{4 \tan ^{5}{\left (\frac {x}{2} \right )} + 8 \tan ^{3}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )}} + \frac {4 \log {\left (\tan {\left (\frac {x}{2} \right )} \right )} \tan ^{5}{\left (\frac {x}{2} \right )}}{4 \tan ^{5}{\left (\frac {x}{2} \right )} + 8 \tan ^{3}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )}} + \frac {8 \log {\left (\tan {\left (\frac {x}{2} \right )} \right )} \tan ^{3}{\left (\frac {x}{2} \right )}}{4 \tan ^{5}{\left (\frac {x}{2} \right )} + 8 \tan ^{3}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )}} + \frac {4 \log {\left (\tan {\left (\frac {x}{2} \right )} \right )} \tan {\left (\frac {x}{2} \right )}}{4 \tan ^{5}{\left (\frac {x}{2} \right )} + 8 \tan ^{3}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )}} + \frac {4 \tan ^{3}{\left (\frac {x}{2} \right )}}{4 \tan ^{5}{\left (\frac {x}{2} \right )} + 8 \tan ^{3}{\left (\frac {x}{2} \right )} + 4 \tan {\left (\frac {x}{2} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 206 vs.
\(2 (27) = 54\).
time = 0.84, size = 206, normalized size = 6.24 \begin {gather*} -\frac {6 \, x^{2} \tan \left (\frac {1}{2} \, x\right )^{5} - 4 \, x \tan \left (\frac {1}{2} \, x\right )^{6} - 4 \, \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, x\right )^{2}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{5} + 12 \, x^{2} \tan \left (\frac {1}{2} \, x\right )^{3} - 12 \, x \tan \left (\frac {1}{2} \, x\right )^{4} + \tan \left (\frac {1}{2} \, x\right )^{5} - 8 \, \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, x\right )^{2}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right )^{3} + 6 \, x^{2} \tan \left (\frac {1}{2} \, x\right ) + 12 \, x \tan \left (\frac {1}{2} \, x\right )^{2} - 6 \, \tan \left (\frac {1}{2} \, x\right )^{3} - 4 \, \log \left (\frac {16 \, \tan \left (\frac {1}{2} \, x\right )^{2}}{\tan \left (\frac {1}{2} \, x\right )^{4} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \tan \left (\frac {1}{2} \, x\right ) + 4 \, x + \tan \left (\frac {1}{2} \, x\right )}{8 \, {\left (\tan \left (\frac {1}{2} \, x\right )^{5} + 2 \, \tan \left (\frac {1}{2} \, x\right )^{3} + \tan \left (\frac {1}{2} \, x\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.49, size = 56, normalized size = 1.70 \begin {gather*} \ln \left ({\mathrm {e}}^{x\,2{}\mathrm {i}}-1\right )-{\mathrm {e}}^{-x\,2{}\mathrm {i}}\,\left (\frac {1}{16}+\frac {x\,1{}\mathrm {i}}{8}\right )+{\mathrm {e}}^{x\,2{}\mathrm {i}}\,\left (-\frac {1}{16}+\frac {x\,1{}\mathrm {i}}{8}\right )-\frac {3\,x^2}{4}-x\,2{}\mathrm {i}-\frac {x\,2{}\mathrm {i}}{{\mathrm {e}}^{x\,2{}\mathrm {i}}-1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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