Optimal. Leaf size=82 \[ \sin (x)-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \sin (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sin (x)\right ) \]
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Rubi [A] time = 0.20, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1676, 1166, 207} \[ \sin (x)-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \sin (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sin (x)\right ) \]
Antiderivative was successfully verified.
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Rule 207
Rule 1166
Rule 1676
Rubi steps
\begin {align*} \int \cot (5 x) \sin (x) \, dx &=\operatorname {Subst}\left (\int \frac {1-12 x^2+16 x^4}{5-20 x^2+16 x^4} \, dx,x,\sin (x)\right )\\ &=\operatorname {Subst}\left (\int \left (1-\frac {4 \left (1-2 x^2\right )}{5-20 x^2+16 x^4}\right ) \, dx,x,\sin (x)\right )\\ &=\sin (x)-4 \operatorname {Subst}\left (\int \frac {1-2 x^2}{5-20 x^2+16 x^4} \, dx,x,\sin (x)\right )\\ &=\sin (x)+\frac {1}{5} \left (4 \left (5-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-10+2 \sqrt {5}+16 x^2} \, dx,x,\sin (x)\right )+\frac {1}{5} \left (4 \left (5+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-10-2 \sqrt {5}+16 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac {1}{5} \sqrt {\frac {1}{2} \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \sin (x)\right )-\frac {1}{5} \sqrt {\frac {1}{2} \left (5-\sqrt {5}\right )} \tanh ^{-1}\left (\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} \sin (x)\right )+\sin (x)\\ \end {align*}
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Mathematica [A] time = 0.24, size = 76, normalized size = 0.93 \[ \frac {1}{10} \left (10 \sin (x)-\sqrt {10-2 \sqrt {5}} \tanh ^{-1}\left (\sqrt {2+\frac {2}{\sqrt {5}}} \sin (x)\right )-\sqrt {2 \left (5+\sqrt {5}\right )} \tanh ^{-1}\left (2 \sqrt {\frac {2}{5+\sqrt {5}}} \sin (x)\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 127, normalized size = 1.55 \[ -\frac {1}{20} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {\sqrt {5} + 5} + 4 \, \sin \relax (x)\right ) + \frac {1}{20} \, \sqrt {2} \sqrt {\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {\sqrt {5} + 5} - 4 \, \sin \relax (x)\right ) - \frac {1}{20} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {-\sqrt {5} + 5} + 4 \, \sin \relax (x)\right ) + \frac {1}{20} \, \sqrt {2} \sqrt {-\sqrt {5} + 5} \log \left (\sqrt {2} \sqrt {-\sqrt {5} + 5} - 4 \, \sin \relax (x)\right ) + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.33, size = 111, normalized size = 1.35 \[ -\frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \log \left ({\left | \frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} + 5} + \sin \relax (x) \right |}\right ) + \frac {1}{20} \, \sqrt {2 \, \sqrt {5} + 10} \log \left ({\left | -\frac {1}{2} \, \sqrt {\frac {1}{2}} \sqrt {\sqrt {5} + 5} + \sin \relax (x) \right |}\right ) - \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left ({\left | \sqrt {-\frac {1}{8} \, \sqrt {5} + \frac {5}{8}} + \sin \relax (x) \right |}\right ) + \frac {1}{20} \, \sqrt {-2 \, \sqrt {5} + 10} \log \left ({\left | -\sqrt {-\frac {1}{8} \, \sqrt {5} + \frac {5}{8}} + \sin \relax (x) \right |}\right ) + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 70, normalized size = 0.85 \[ \sin \relax (x )-\frac {\left (\sqrt {5}-1\right ) \sqrt {5}\, \arctanh \left (\frac {4 \sin \relax (x )}{\sqrt {10-2 \sqrt {5}}}\right )}{5 \sqrt {10-2 \sqrt {5}}}-\frac {\left (\sqrt {5}+1\right ) \sqrt {5}\, \arctanh \left (\frac {4 \sin \relax (x )}{\sqrt {10+2 \sqrt {5}}}\right )}{5 \sqrt {10+2 \sqrt {5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.61, size = 119, normalized size = 1.45 \[ \sin \relax (x)-\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\frac {25\,\sqrt {2}\,\sin \relax (x)\,\sqrt {\sqrt {5}+5}}{2}+\frac {11\,\sqrt {2}\,\sqrt {5}\,\sin \relax (x)\,\sqrt {\sqrt {5}+5}}{2}}{20\,\sqrt {5}+45}\right )\,\sqrt {\sqrt {5}+5}}{10}+\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\frac {25\,\sqrt {2}\,\sin \relax (x)\,\sqrt {5-\sqrt {5}}}{2}-\frac {11\,\sqrt {2}\,\sqrt {5}\,\sin \relax (x)\,\sqrt {5-\sqrt {5}}}{2}}{20\,\sqrt {5}-45}\right )\,\sqrt {5-\sqrt {5}}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sin {\relax (x )} \cot {\left (5 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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