Optimal. Leaf size=26 \[ \frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}}-\frac {1}{4} \tanh ^{-1}(\sin (x)) \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1093, 207} \[ \frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}}-\frac {1}{4} \tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 207
Rule 1093
Rubi steps
\begin {align*} \int \csc (4 x) \sin (x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{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 )-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}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 26, normalized size = 1.00 \[ \frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}}-\frac {1}{4} \tanh ^{-1}(\sin (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \csc (4 x) \sin (x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.02, size = 50, normalized size = 1.92 \[ \frac {1}{8} \, \sqrt {2} \log \left (-\frac {2 \, \cos \relax (x)^{2} - 2 \, \sqrt {2} \sin \relax (x) - 3}{2 \, \cos \relax (x)^{2} - 1}\right ) - \frac {1}{8} \, \log \left (\sin \relax (x) + 1\right ) + \frac {1}{8} \, \log \left (-\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.64, size = 48, normalized size = 1.85 \[ -\frac {1}{8} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \relax (x) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \relax (x) \right |}}\right ) - \frac {1}{8} \, \log \left (\sin \relax (x) + 1\right ) + \frac {1}{8} \, \log \left (-\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.17, size = 28, normalized size = 1.08
method | result | size |
default | \(-\frac {\ln \left (1+\sin \relax (x )\right )}{8}+\frac {\ln \left (-1+\sin \relax (x )\right )}{8}+\frac {\arctanh \left (\sin \relax (x ) \sqrt {2}\right ) \sqrt {2}}{4}\) | \(28\) |
risch | \(\frac {\ln \left ({\mathrm e}^{i x}-i\right )}{4}-\frac {\ln \left ({\mathrm e}^{i x}+i\right )}{4}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{8}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{8}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.04, size = 171, normalized size = 6.58 \[ \frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) + 2 \, \sqrt {2} \sin \relax (x) + 2\right ) - \frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} + 2 \, \sqrt {2} \cos \relax (x) - 2 \, \sqrt {2} \sin \relax (x) + 2\right ) + \frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) + 2 \, \sqrt {2} \sin \relax (x) + 2\right ) - \frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \relax (x)^{2} + 2 \, \sin \relax (x)^{2} - 2 \, \sqrt {2} \cos \relax (x) - 2 \, \sqrt {2} \sin \relax (x) + 2\right ) - \frac {1}{8} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \sin \relax (x) + 1\right ) + \frac {1}{8} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.47, size = 27, normalized size = 1.04 \[ \frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \relax (x)\right )}{4}-\frac {\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 8.14, size = 294, normalized size = 11.31 \[ \frac {27720 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{110880 \sqrt {2} + 156808} + \frac {39202 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 \right )}}{110880 \sqrt {2} + 156808} - \frac {39202 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{110880 \sqrt {2} + 156808} - \frac {27720 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{110880 \sqrt {2} + 156808} + \frac {27720 \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{110880 \sqrt {2} + 156808} + \frac {19601 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - 1 + \sqrt {2} \right )}}{110880 \sqrt {2} + 156808} + \frac {27720 \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{110880 \sqrt {2} + 156808} + \frac {19601 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} + 1 + \sqrt {2} \right )}}{110880 \sqrt {2} + 156808} - \frac {19601 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{110880 \sqrt {2} + 156808} - \frac {27720 \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} - 1 \right )}}{110880 \sqrt {2} + 156808} - \frac {19601 \sqrt {2} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{110880 \sqrt {2} + 156808} - \frac {27720 \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {2} + 1 \right )}}{110880 \sqrt {2} + 156808} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________