Optimal. Leaf size=26 \[ -\frac {1}{4} \tanh ^{-1}(\sin (x))+\frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}} \]
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Rubi [A]
time = 0.02, 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 = {1107, 213}
\begin {gather*} \frac {\tanh ^{-1}\left (\sqrt {2} \sin (x)\right )}{2 \sqrt {2}}-\frac {1}{4} \tanh ^{-1}(\sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 1107
Rubi steps
\begin {align*} \int \csc (4 x) \sin (x) \, dx &=\text {Subst}\left (\int \frac {1}{4-12 x^2+8 x^4} \, dx,x,\sin (x)\right )\\ &=2 \text {Subst}\left (\int \frac {1}{-8+8 x^2} \, dx,x,\sin (x)\right )-2 \text {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*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.23, size = 218, normalized size = 8.38 \begin {gather*} \frac {-2 i \tan ^{-1}\left (\frac {\cos \left (\frac {x}{2}\right )-\left (-1+\sqrt {2}\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 (-1+\sqrt {2}\right ) \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )}\right )+2 \sqrt {2} \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-2 \sqrt {2} \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )+2 \log \left (\sqrt {2}+2 \sin (x)\right )-\log \left (2-\sqrt {2} \cos (x)-\sqrt {2} \sin (x)\right )-\log \left (2+\sqrt {2} \cos (x)-\sqrt {2} \sin (x)\right )}{8 \sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 28, normalized size = 1.08
method | result | size |
default | \(-\frac {\ln \left (\sin \left (x \right )+1\right )}{8}+\frac {\arctanh \left (\sin \left (x \right ) \sqrt {2}\right ) \sqrt {2}}{4}+\frac {\ln \left (-1+\sin \left (x \right )\right )}{8}\) | \(28\) |
risch | \(\frac {\ln \left ({\mathrm e}^{i x}-i\right )}{4}-\frac {\ln \left ({\mathrm e}^{i x}+i\right )}{4}+\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}+i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{8}-\frac {\sqrt {2}\, \ln \left ({\mathrm e}^{2 i x}-i \sqrt {2}\, {\mathrm e}^{i x}-1\right )}{8}\) | \(72\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 171 vs.
\(2 (18) = 36\).
time = 2.09, size = 171, normalized size = 6.58 \begin {gather*} \frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) + 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) - \frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} + 2 \, \sqrt {2} \cos \left (x\right ) - 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) + \frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt {2} \cos \left (x\right ) + 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) - \frac {1}{16} \, \sqrt {2} \log \left (2 \, \cos \left (x\right )^{2} + 2 \, \sin \left (x\right )^{2} - 2 \, \sqrt {2} \cos \left (x\right ) - 2 \, \sqrt {2} \sin \left (x\right ) + 2\right ) - \frac {1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \sin \left (x\right ) + 1\right ) + \frac {1}{8} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \sin \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. 50 vs.
\(2 (18) = 36\).
time = 1.32, size = 50, normalized size = 1.92 \begin {gather*} \frac {1}{8} \, \sqrt {2} \log \left (-\frac {2 \, \cos \left (x\right )^{2} - 2 \, \sqrt {2} \sin \left (x\right ) - 3}{2 \, \cos \left (x\right )^{2} - 1}\right ) - \frac {1}{8} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac {1}{8} \, \log \left (-\sin \left (x\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 294 vs.
\(2 (22) = 44\).
time = 3.56, size = 294, normalized size = 11.31 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs.
\(2 (18) = 36\).
time = 0.80, size = 48, normalized size = 1.85 \begin {gather*} -\frac {1}{8} \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (x\right ) \right |}}\right ) - \frac {1}{8} \, \log \left (\sin \left (x\right ) + 1\right ) + \frac {1}{8} \, \log \left (-\sin \left (x\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.47, size = 27, normalized size = 1.04 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\sqrt {2}\,\sin \left (x\right )\right )}{4}-\frac {\mathrm {atanh}\left (\frac {\sin \left (\frac {x}{2}\right )}{\cos \left (\frac {x}{2}\right )}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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