Optimal. Leaf size=28 \[ \sin (x)-\frac {1}{4} \tanh ^{-1}(\sin (x))-\frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}} \]
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Rubi [A] time = 0.05, antiderivative size = 28, 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} \[ \sin (x)-\frac {1}{4} \tanh ^{-1}(\sin (x))-\frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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Rule 207
Rule 1166
Rule 1676
Rubi steps
\begin {align*} \int \cot (4 x) \sin (x) \, dx &=\operatorname {Subst}\left (\int \frac {1-8 x^2+8 x^4}{4-12 x^2+8 x^4} \, dx,x,\sin (x)\right )\\ &=\operatorname {Subst}\left (\int \left (1-\frac {3-4 x^2}{4-12 x^2+8 x^4}\right ) \, dx,x,\sin (x)\right )\\ &=\sin (x)-\operatorname {Subst}\left (\int \frac {3-4 x^2}{4-12 x^2+8 x^4} \, dx,x,\sin (x)\right )\\ &=\sin (x)+2 \operatorname {Subst}\left (\int \frac {1}{-8+8 x^2} \, dx,x,\sin (x)\right )+2 \operatorname {Subst}\left (\int \frac {1}{-4+8 x^2} \, dx,x,\sin (x)\right )\\ &=-\frac {1}{4} \tanh ^{-1}(\sin (x))-\frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}}+\sin (x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 28, normalized size = 1.00 \[ \sin (x)-\frac {1}{4} \tanh ^{-1}(\sin (x))-\frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 52, normalized size = 1.86 \[ \frac {1}{8} \, \sqrt {2} \log \left (-\frac {2 \, \cos \relax (x)^{2} + 2 \, \sqrt {2} \sin \relax (x) - 3}{2 \, \cos \relax (x)^{2} - 1}\right ) - \frac {1}{8} \, \log \left (\sin \relax (x) + 1\right ) + \frac {1}{8} \, \log \left (-\sin \relax (x) + 1\right ) + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 50, normalized size = 1.79 \[ \frac {1}{8} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \relax (x) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \relax (x) \right |}}\right ) - \frac {1}{8} \, \log \left (\sin \relax (x) + 1\right ) + \frac {1}{8} \, \log \left (-\sin \relax (x) + 1\right ) + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 30, normalized size = 1.07 \[ \sin \relax (x )+\frac {\ln \left (\sin \relax (x )-1\right )}{8}-\frac {\arctanh \left (\sin \relax (x ) \sqrt {2}\right ) \sqrt {2}}{4}-\frac {\ln \left (1+\sin \relax (x )\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.54, size = 173, normalized size = 6.18 \[ -\frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) + 2 \, \sqrt {2} \sin \relax (x) + 2\right ) + \frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) - 2 \, \sqrt {2} \sin \relax (x) + 2\right ) - \frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) + 2 \, \sqrt {2} \sin \relax (x) + 2\right ) + \frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) - 2 \, \sqrt {2} \sin \relax (x) + 2\right ) - \frac {1}{8} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right ) + \frac {1}{8} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right ) + \sin \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.38, size = 29, normalized size = 1.04 \[ \sin \relax (x)-\frac {\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{2}-\frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \relax (x)\right )}{4} \]
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 \[ \int \sin {\relax (x )} \cot {\left (4 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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