Optimal. Leaf size=33 \[ -\frac {285 x}{2}+(3-2 \cot (x))^3+5 (3-2 \cot (x))^2-42 \cot (x)+4 \log (\sin (x)) \]
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Rubi [A] time = 0.06, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3528, 3525, 3475} \[ -\frac {285 x}{2}+(3-2 \cot (x))^3+5 (3-2 \cot (x))^2-42 \cot (x)+4 \log (\sin (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3525
Rule 3528
Rubi steps
\begin {align*} \int \left (\frac {1}{2}-3 \cot (x)\right ) (3-2 \cot (x))^3 \, dx &=(3-2 \cot (x))^3+\int \left (-\frac {9}{2}-10 \cot (x)\right ) (3-2 \cot (x))^2 \, dx\\ &=5 (3-2 \cot (x))^2+(3-2 \cot (x))^3+\int \left (-\frac {67}{2}-21 \cot (x)\right ) (3-2 \cot (x)) \, dx\\ &=-\frac {285 x}{2}+5 (3-2 \cot (x))^2+(3-2 \cot (x))^3-42 \cot (x)+4 \int \cot (x) \, dx\\ &=-\frac {285 x}{2}+5 (3-2 \cot (x))^2+(3-2 \cot (x))^3-42 \cot (x)+4 \log (\sin (x))\\ \end {align*}
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Mathematica [A] time = 0.03, size = 29, normalized size = 0.88 \[ -\frac {285 x}{2}-148 \cot (x)+56 \csc ^2(x)+4 \log (\sin (x))-8 \cot (x) \csc ^2(x) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\frac {1}{2}-3 \cot (x)\right ) (3-2 \cot (x))^3 \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.86, size = 71, normalized size = 2.15 \[ \frac {4 \, {\left (\cos \left (2 \, x\right ) - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, x\right ) + \frac {1}{2}\right ) \sin \left (2 \, x\right ) - 296 \, \cos \left (2 \, x\right )^{2} - {\left (285 \, x \cos \left (2 \, x\right ) - 285 \, x + 224\right )} \sin \left (2 \, x\right ) + 32 \, \cos \left (2 \, x\right ) + 328}{2 \, {\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.67, size = 75, normalized size = 2.27 \[ \tan \left (\frac {1}{2} \, x\right )^{3} + 14 \, \tan \left (\frac {1}{2} \, x\right )^{2} - \frac {285}{2} \, x - \frac {22 \, \tan \left (\frac {1}{2} \, x\right )^{3} + 225 \, \tan \left (\frac {1}{2} \, x\right )^{2} - 42 \, \tan \left (\frac {1}{2} \, x\right ) + 3}{3 \, \tan \left (\frac {1}{2} \, x\right )^{3}} - 4 \, \log \left (\tan \left (\frac {1}{2} \, x\right )^{2} + 1\right ) + 4 \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) + 75 \, \tan \left (\frac {1}{2} \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 35, normalized size = 1.06
method | result | size |
derivativedivides | \(-8 \left (\cot ^{3}\relax (x )\right )+56 \left (\cot ^{2}\relax (x )\right )-156 \cot \relax (x )-2 \ln \left (\cot ^{2}\relax (x )+1\right )+\frac {285 \pi }{4}-\frac {285 \,\mathrm {arccot}\left (\cot \relax (x )\right )}{2}\) | \(35\) |
default | \(-8 \left (\cot ^{3}\relax (x )\right )+56 \left (\cot ^{2}\relax (x )\right )-156 \cot \relax (x )-2 \ln \left (\cot ^{2}\relax (x )+1\right )+\frac {285 \pi }{4}-\frac {285 \,\mathrm {arccot}\left (\cot \relax (x )\right )}{2}\) | \(35\) |
norman | \(\frac {-8-156 \left (\tan ^{2}\relax (x )\right )-\frac {285 x \left (\tan ^{3}\relax (x )\right )}{2}+56 \tan \relax (x )}{\tan \relax (x )^{3}}+4 \ln \left (\tan \relax (x )\right )-2 \ln \left (1+\tan ^{2}\relax (x )\right )\) | \(40\) |
risch | \(-\frac {285 x}{2}-4 i x +\frac {\left (-\frac {224}{1873}-\frac {264 i}{1873}\right ) \left (1873 \,{\mathrm e}^{4 i x}-1260 i {\mathrm e}^{2 i x}-3358 \,{\mathrm e}^{2 i x}+1221+1036 i\right )}{\left ({\mathrm e}^{2 i x}-1\right )^{3}}+4 \ln \left ({\mathrm e}^{2 i x}-1\right )\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.27, size = 36, normalized size = 1.09 \[ -\frac {285}{2} \, x - \frac {4 \, {\left (39 \, \tan \relax (x)^{2} - 14 \, \tan \relax (x) + 2\right )}}{\tan \relax (x)^{3}} - 2 \, \log \left (\tan \relax (x)^{2} + 1\right ) + 4 \, \log \left (\tan \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 75, normalized size = 2.27 \[ x\,\left (-\frac {285}{2}-4{}\mathrm {i}\right )+4\,\ln \left ({\mathrm {e}}^{x\,2{}\mathrm {i}}-1\right )+\frac {64{}\mathrm {i}}{3\,{\mathrm {e}}^{x\,2{}\mathrm {i}}-3\,{\mathrm {e}}^{x\,4{}\mathrm {i}}+{\mathrm {e}}^{x\,6{}\mathrm {i}}-1}+\frac {-224+96{}\mathrm {i}}{1+{\mathrm {e}}^{x\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{x\,2{}\mathrm {i}}}+\frac {-224-264{}\mathrm {i}}{{\mathrm {e}}^{x\,2{}\mathrm {i}}-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 39, normalized size = 1.18 \[ - \frac {285 x}{2} - 2 \log {\left (\tan ^{2}{\relax (x )} + 1 \right )} + 4 \log {\left (\tan {\relax (x )} \right )} - \frac {156}{\tan {\relax (x )}} + \frac {56}{\tan ^{2}{\relax (x )}} - \frac {8}{\tan ^{3}{\relax (x )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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