Optimal. Leaf size=49 \[ -\frac {\tan ^{-1}\left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{12} \log \left (1+x^4\right )+\frac {1}{24} \log \left (1-x^4+x^8\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {281, 298, 31,
648, 632, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{12} \log \left (x^4+1\right )+\frac {1}{24} \log \left (x^8-x^4+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 210
Rule 281
Rule 298
Rule 632
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {x^7}{1+x^{12}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x}{1+x^3} \, dx,x,x^4\right )\\ &=-\left (\frac {1}{12} \text {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^4\right )\right )+\frac {1}{12} \text {Subst}\left (\int \frac {1+x}{1-x+x^2} \, dx,x,x^4\right )\\ &=-\frac {1}{12} \log \left (1+x^4\right )+\frac {1}{24} \text {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,x^4\right )+\frac {1}{8} \text {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^4\right )\\ &=-\frac {1}{12} \log \left (1+x^4\right )+\frac {1}{24} \log \left (1-x^4+x^8\right )-\frac {1}{4} \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^4\right )\\ &=-\frac {\tan ^{-1}\left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}-\frac {1}{12} \log \left (1+x^4\right )+\frac {1}{24} \log \left (1-x^4+x^8\right )\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(260\) vs. \(2(49)=98\).
time = 0.08, size = 260, normalized size = 5.31 \begin {gather*} \frac {1}{24} \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+\sqrt {3}-2 \sqrt {2} x}{1-\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {1-\sqrt {3}+2 \sqrt {2} x}{1+\sqrt {3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {-1+\sqrt {3}+2 \sqrt {2} x}{1+\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {1+\sqrt {3}+2 \sqrt {2} x}{-1+\sqrt {3}}\right )-2 \log \left (1-\sqrt {2} x+x^2\right )-2 \log \left (1+\sqrt {2} x+x^2\right )+\log \left (2+\sqrt {2} x-\sqrt {6} x+2 x^2\right )+\log \left (2+\sqrt {2} \left (-1+\sqrt {3}\right ) x+2 x^2\right )+\log \left (2-\left (\sqrt {2}+\sqrt {6}\right ) x+2 x^2\right )+\log \left (2+\left (\sqrt {2}+\sqrt {6}\right ) x+2 x^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 41, normalized size = 0.84
method | result | size |
risch | \(-\frac {\ln \left (x^{4}+1\right )}{12}+\frac {\ln \left (x^{8}-x^{4}+1\right )}{24}+\frac {\sqrt {3}\, \arctan \left (\frac {2 \left (x^{4}-\frac {1}{2}\right ) \sqrt {3}}{3}\right )}{12}\) | \(39\) |
default | \(-\frac {\ln \left (x^{4}+1\right )}{12}+\frac {\ln \left (x^{8}-x^{4}+1\right )}{24}+\frac {\arctan \left (\frac {\left (2 x^{4}-1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{12}\) | \(41\) |
meijerg | \(-\frac {x^{8} \ln \left (1+\left (x^{12}\right )^{\frac {1}{3}}\right )}{12 \left (x^{12}\right )^{\frac {2}{3}}}+\frac {x^{8} \ln \left (1-\left (x^{12}\right )^{\frac {1}{3}}+\left (x^{12}\right )^{\frac {2}{3}}\right )}{24 \left (x^{12}\right )^{\frac {2}{3}}}+\frac {x^{8} \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{12}\right )^{\frac {1}{3}}}{2-\left (x^{12}\right )^{\frac {1}{3}}}\right )}{12 \left (x^{12}\right )^{\frac {2}{3}}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.79, size = 40, normalized size = 0.82 \begin {gather*} \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{24} \, \log \left (x^{8} - x^{4} + 1\right ) - \frac {1}{12} \, \log \left (x^{4} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.60, size = 40, normalized size = 0.82 \begin {gather*} \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{24} \, \log \left (x^{8} - x^{4} + 1\right ) - \frac {1}{12} \, \log \left (x^{4} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 46, normalized size = 0.94 \begin {gather*} - \frac {\log {\left (x^{4} + 1 \right )}}{12} + \frac {\log {\left (x^{8} - x^{4} + 1 \right )}}{24} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} - \frac {\sqrt {3}}{3} \right )}}{12} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 40, normalized size = 0.82 \begin {gather*} \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{24} \, \log \left (x^{8} - x^{4} + 1\right ) - \frac {1}{12} \, \log \left (x^{4} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.14, size = 52, normalized size = 1.06 \begin {gather*} -\frac {\ln \left (x^4+1\right )}{12}-\ln \left (x^4-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}-\frac {1}{2}\right )\,\left (-\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )+\ln \left (x^4+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}-\frac {1}{2}\right )\,\left (\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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