Optimal. Leaf size=26 \[ \frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{4 \sqrt {2}}-\frac {1}{4} \tanh ^{-1}(\sin (x)) \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1130, 207} \[ \frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{4 \sqrt {2}}-\frac {1}{4} \tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
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Rule 207
Rule 1130
Rubi steps
\begin {align*} \int \csc (4 x) \sin ^3(x) \, dx &=\operatorname {Subst}\left (\int \frac {x^2}{4-12 x^2+8 x^4} \, dx,x,\sin (x)\right )\\ &=2 \operatorname {Subst}\left (\int \frac {1}{-8+8 x^2} \, dx,x,\sin (x)\right )-\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 )}{4 \sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.34, size = 218, normalized size = 8.38 \[ \frac {2 \log \left (2 \sin (x)+\sqrt {2}\right )+4 \sqrt {2} \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-4 \sqrt {2} \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )-\log \left (-\sqrt {2} \sin (x)-\sqrt {2} \cos (x)+2\right )-\log \left (-\sqrt {2} \sin (x)+\sqrt {2} \cos (x)+2\right )-2 i \tan ^{-1}\left (\frac {\cos \left (\frac {x}{2}\right )-\left (\sqrt {2}-1\right ) \sin \left (\frac {x}{2}\right )}{\left (1+\sqrt {2}\right ) \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}\right )-2 i \tan ^{-1}\left (\frac {\cos \left (\frac {x}{2}\right )-\left (1+\sqrt {2}\right ) \sin \left (\frac {x}{2}\right )}{\left (\sqrt {2}-1\right ) \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}\right )}{16 \sqrt {2}} \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc (4 x) \sin ^3(x) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 1.01, size = 50, normalized size = 1.92 \[ \frac {1}{16} \, \sqrt {2} \log \left (-\frac {2 \, \cos \relax (x)^{2} - 2 \, \sqrt {2} \sin \relax (x) - 3}{2 \, \cos \relax (x)^{2} - 1}\right ) - \frac {1}{8} \, \log \left (\sin \relax (x) + 1\right ) + \frac {1}{8} \, \log \left (-\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.66, size = 48, normalized size = 1.85 \[ -\frac {1}{16} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \relax (x) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \relax (x) \right |}}\right ) - \frac {1}{8} \, \log \left (\sin \relax (x) + 1\right ) + \frac {1}{8} \, \log \left (-\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 28, normalized size = 1.08
method | result | size |
default | \(\frac {\ln \left (-1+\sin \relax (x )\right )}{8}-\frac {\ln \left (1+\sin \relax (x )\right )}{8}+\frac {\arctanh \left (\sin \relax (x ) \sqrt {2}\right ) \sqrt {2}}{8}\) | \(28\) |
risch | \(-\frac {\ln \left ({\mathrm e}^{i x}+i\right )}{4}+\frac {\ln \left ({\mathrm e}^{i x}-i\right )}{4}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{16}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{16}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.04, size = 171, normalized size = 6.58 \[ \frac {1}{32} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) + 2 \, \sqrt {2} \sin \relax (x) + 2\right ) - \frac {1}{32} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) - 2 \, \sqrt {2} \sin \relax (x) + 2\right ) + \frac {1}{32} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) + 2 \, \sqrt {2} \sin \relax (x) + 2\right ) - \frac {1}{32} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) - 2 \, \sqrt {2} \sin \relax (x) + 2\right ) - \frac {1}{8} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right ) + \frac {1}{8} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 27, normalized size = 1.04 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \relax (x)\right )}{8}-\frac {\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 20.13, size = 294, normalized size = 11.31 \[ \frac {4093147632754948 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {2894292447518688 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {4093147632754948 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {2894292447518688 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {1447146223759344 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {1023286908188737 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {1447146223759344 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} + \frac {1023286908188737 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {1447146223759344 \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {1023286908188737 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {1447146223759344 \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} - \frac {1023286908188737 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{16372590531019792 + 11577169790074752 \sqrt {2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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