Optimal. Leaf size=38 \[ -\frac {1}{16} \tanh ^{-1}(\cos (x))-\frac {1}{6} \cot ^3(x) \csc ^3(x)+\frac {1}{8} \cot (x) \csc ^3(x)-\frac {1}{16} \cot (x) \csc (x) \]
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Rubi [A] time = 0.05, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2611, 3768, 3770} \[ -\frac {1}{16} \tanh ^{-1}(\cos (x))-\frac {1}{6} \cot ^3(x) \csc ^3(x)+\frac {1}{8} \cot (x) \csc ^3(x)-\frac {1}{16} \cot (x) \csc (x) \]
Antiderivative was successfully verified.
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Rule 2611
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \cot ^4(x) \csc ^3(x) \, dx &=-\frac {1}{6} \cot ^3(x) \csc ^3(x)-\frac {1}{2} \int \cot ^2(x) \csc ^3(x) \, dx\\ &=\frac {1}{8} \cot (x) \csc ^3(x)-\frac {1}{6} \cot ^3(x) \csc ^3(x)+\frac {1}{8} \int \csc ^3(x) \, dx\\ &=-\frac {1}{16} \cot (x) \csc (x)+\frac {1}{8} \cot (x) \csc ^3(x)-\frac {1}{6} \cot ^3(x) \csc ^3(x)+\frac {1}{16} \int \csc (x) \, dx\\ &=-\frac {1}{16} \tanh ^{-1}(\cos (x))-\frac {1}{16} \cot (x) \csc (x)+\frac {1}{8} \cot (x) \csc ^3(x)-\frac {1}{6} \cot ^3(x) \csc ^3(x)\\ \end {align*}
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Mathematica [B] time = 0.02, size = 95, normalized size = 2.50 \[ -\frac {1}{384} \csc ^6\left (\frac {x}{2}\right )+\frac {1}{64} \csc ^4\left (\frac {x}{2}\right )-\frac {1}{64} \csc ^2\left (\frac {x}{2}\right )+\frac {1}{384} \sec ^6\left (\frac {x}{2}\right )-\frac {1}{64} \sec ^4\left (\frac {x}{2}\right )+\frac {1}{64} \sec ^2\left (\frac {x}{2}\right )+\frac {1}{16} \log \left (\sin \left (\frac {x}{2}\right )\right )-\frac {1}{16} \log \left (\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cot ^4(x) \csc ^3(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.68, size = 93, normalized size = 2.45 \[ \frac {6 \, \cos \relax (x)^{5} + 16 \, \cos \relax (x)^{3} - 3 \, {\left (\cos \relax (x)^{6} - 3 \, \cos \relax (x)^{4} + 3 \, \cos \relax (x)^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + 3 \, {\left (\cos \relax (x)^{6} - 3 \, \cos \relax (x)^{4} + 3 \, \cos \relax (x)^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) - 6 \, \cos \relax (x)}{96 \, {\left (\cos \relax (x)^{6} - 3 \, \cos \relax (x)^{4} + 3 \, \cos \relax (x)^{2} - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 44, normalized size = 1.16 \[ \frac {3 \, \cos \relax (x)^{5} + 8 \, \cos \relax (x)^{3} - 3 \, \cos \relax (x)}{48 \, {\left (\cos \relax (x)^{2} - 1\right )}^{3}} - \frac {1}{32} \, \log \left (\cos \relax (x) + 1\right ) + \frac {1}{32} \, \log \left (-\cos \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 52, normalized size = 1.37
method | result | size |
default | \(-\frac {\cos ^{5}\relax (x )}{6 \sin \relax (x )^{6}}-\frac {\cos ^{5}\relax (x )}{24 \sin \relax (x )^{4}}+\frac {\cos ^{5}\relax (x )}{48 \sin \relax (x )^{2}}+\frac {\left (\cos ^{3}\relax (x )\right )}{48}+\frac {\cos \relax (x )}{16}+\frac {\ln \left (\csc \relax (x )-\cot \relax (x )\right )}{16}\) | \(52\) |
risch | \(\frac {3 \,{\mathrm e}^{11 i x}+47 \,{\mathrm e}^{9 i x}+78 \,{\mathrm e}^{7 i x}+78 \,{\mathrm e}^{5 i x}+47 \,{\mathrm e}^{3 i x}+3 \,{\mathrm e}^{i x}}{24 \left ({\mathrm e}^{2 i x}-1\right )^{6}}-\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{16}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{16}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 54, normalized size = 1.42 \[ \frac {3 \, \cos \relax (x)^{5} + 8 \, \cos \relax (x)^{3} - 3 \, \cos \relax (x)}{48 \, {\left (\cos \relax (x)^{6} - 3 \, \cos \relax (x)^{4} + 3 \, \cos \relax (x)^{2} - 1\right )}} - \frac {1}{32} \, \log \left (\cos \relax (x) + 1\right ) + \frac {1}{32} \, \log \left (\cos \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.28, size = 57, normalized size = 1.50 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{16}+\frac {\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{128}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{128}-\frac {1}{384}}{{\mathrm {tan}\left (\frac {x}{2}\right )}^6}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{128}-\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^4}{128}+\frac {{\mathrm {tan}\left (\frac {x}{2}\right )}^6}{384} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 56, normalized size = 1.47 \[ - \frac {- 3 \cos ^{5}{\relax (x )} - 8 \cos ^{3}{\relax (x )} + 3 \cos {\relax (x )}}{48 \cos ^{6}{\relax (x )} - 144 \cos ^{4}{\relax (x )} + 144 \cos ^{2}{\relax (x )} - 48} + \frac {\log {\left (\cos {\relax (x )} - 1 \right )}}{32} - \frac {\log {\left (\cos {\relax (x )} + 1 \right )}}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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