Optimal. Leaf size=38 \[ -\frac {1}{16} \tanh ^{-1}(\cos (x))-\frac {1}{16} \cot (x) \csc (x)+\frac {1}{8} \cot (x) \csc ^3(x)-\frac {1}{6} \cot ^3(x) \csc ^3(x) \]
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Rubi [A]
time = 0.04, 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 = {2691, 3853,
3855} \begin {gather*} -\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) \end {gather*}
Antiderivative was successfully verified.
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Rule 2691
Rule 3853
Rule 3855
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] Leaf count is larger than twice the leaf count of optimal. \(95\) vs. \(2(38)=76\).
time = 0.02, size = 95, normalized size = 2.50 \begin {gather*} -\frac {1}{64} \csc ^2\left (\frac {x}{2}\right )+\frac {1}{64} \csc ^4\left (\frac {x}{2}\right )-\frac {1}{384} \csc ^6\left (\frac {x}{2}\right )-\frac {1}{16} \log \left (\cos \left (\frac {x}{2}\right )\right )+\frac {1}{16} \log \left (\sin \left (\frac {x}{2}\right )\right )+\frac {1}{64} \sec ^2\left (\frac {x}{2}\right )-\frac {1}{64} \sec ^4\left (\frac {x}{2}\right )+\frac {1}{384} \sec ^6\left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 52, normalized size = 1.37
method | result | size |
default | \(-\frac {\cos ^{5}\left (x \right )}{6 \sin \left (x \right )^{6}}-\frac {\cos ^{5}\left (x \right )}{24 \sin \left (x \right )^{4}}+\frac {\cos ^{5}\left (x \right )}{48 \sin \left (x \right )^{2}}+\frac {\left (\cos ^{3}\left (x \right )\right )}{48}+\frac {\cos \left (x \right )}{16}+\frac {\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )}{16}\) | \(52\) |
risch | \(\frac {3 \,{\mathrm e}^{11 i x}+47 \,{\mathrm e}^{9 i x}+78 \,{\mathrm e}^{7 i x}+78 \,{\mathrm e}^{5 i x}+47 \,{\mathrm e}^{3 i x}+3 \,{\mathrm e}^{i x}}{24 \left ({\mathrm e}^{2 i x}-1\right )^{6}}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{16}-\frac {\ln \left (1+{\mathrm e}^{i x}\right )}{16}\) | \(76\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.86, size = 54, normalized size = 1.42 \begin {gather*} \frac {3 \, \cos \left (x\right )^{5} + 8 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{48 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} - \frac {1}{32} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac {1}{32} \, \log \left (\cos \left (x\right ) - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs.
\(2 (30) = 60\).
time = 1.23, size = 93, normalized size = 2.45 \begin {gather*} \frac {6 \, \cos \left (x\right )^{5} + 16 \, \cos \left (x\right )^{3} - 3 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) + 3 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )} \log \left (-\frac {1}{2} \, \cos \left (x\right ) + \frac {1}{2}\right ) - 6 \, \cos \left (x\right )}{96 \, {\left (\cos \left (x\right )^{6} - 3 \, \cos \left (x\right )^{4} + 3 \, \cos \left (x\right )^{2} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 56, normalized size = 1.47 \begin {gather*} - \frac {- 3 \cos ^{5}{\left (x \right )} - 8 \cos ^{3}{\left (x \right )} + 3 \cos {\left (x \right )}}{48 \cos ^{6}{\left (x \right )} - 144 \cos ^{4}{\left (x \right )} + 144 \cos ^{2}{\left (x \right )} - 48} + \frac {\log {\left (\cos {\left (x \right )} - 1 \right )}}{32} - \frac {\log {\left (\cos {\left (x \right )} + 1 \right )}}{32} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.41, size = 44, normalized size = 1.16 \begin {gather*} \frac {3 \, \cos \left (x\right )^{5} + 8 \, \cos \left (x\right )^{3} - 3 \, \cos \left (x\right )}{48 \, {\left (\cos \left (x\right )^{2} - 1\right )}^{3}} - \frac {1}{32} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac {1}{32} \, \log \left (-\cos \left (x\right ) + 1\right ) \end {gather*}
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 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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