Optimal. Leaf size=49 \[ -\frac {1}{12} \log \left (x^4+1\right )-\frac {\tan ^{-1}\left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}}+\frac {1}{24} \log \left (x^8-x^4+1\right ) \]
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Rubi [A] time = 0.04, 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 = {275, 292, 31, 634, 618, 204, 628} \[ -\frac {1}{12} \log \left (x^4+1\right )+\frac {1}{24} \log \left (x^8-x^4+1\right )-\frac {\tan ^{-1}\left (\frac {1-2 x^4}{\sqrt {3}}\right )}{4 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 275
Rule 292
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {x^7}{1+x^{12}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{1+x^3} \, dx,x,x^4\right )\\ &=-\left (\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,x^4\right )\right )+\frac {1}{12} \operatorname {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} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,x^4\right )+\frac {1}{8} \operatorname {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} \operatorname {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] time = 0.12, size = 260, normalized size = 5.31 \[ \frac {1}{24} \left (-2 \log \left (x^2-\sqrt {2} x+1\right )-2 \log \left (x^2+\sqrt {2} x+1\right )+\log \left (2 x^2-\sqrt {6} x+\sqrt {2} x+2\right )+\log \left (2 x^2+\sqrt {2} \left (\sqrt {3}-1\right ) x+2\right )+\log \left (2 x^2-\left (\sqrt {2}+\sqrt {6}\right ) x+2\right )+\log \left (2 x^2+\left (\sqrt {2}+\sqrt {6}\right ) x+2\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {-2 \sqrt {2} x+\sqrt {3}+1}{1-\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt {2} x-\sqrt {3}+1}{1+\sqrt {3}}\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt {2} x+\sqrt {3}-1}{1+\sqrt {3}}\right )-2 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt {2} x+\sqrt {3}+1}{\sqrt {3}-1}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 40, normalized size = 0.82 \[ \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{24} \, \log \left (x^{8} - x^{4} + 1\right ) - \frac {1}{12} \, \log \left (x^{4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 40, normalized size = 0.82 \[ \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{24} \, \log \left (x^{8} - x^{4} + 1\right ) - \frac {1}{12} \, \log \left (x^{4} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 41, normalized size = 0.84 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{4}-1\right ) \sqrt {3}}{3}\right )}{12}-\frac {\ln \left (x^{4}+1\right )}{12}+\frac {\ln \left (x^{8}-x^{4}+1\right )}{24} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.16, size = 40, normalized size = 0.82 \[ \frac {1}{12} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{4} - 1\right )}\right ) + \frac {1}{24} \, \log \left (x^{8} - x^{4} + 1\right ) - \frac {1}{12} \, \log \left (x^{4} + 1\right ) \]
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 \[ -\frac {\ln \left (x^4+1\right )}{12}-\ln \left (x^4-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}-\frac {1}{2}\right )\,\left (-\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right )+\ln \left (x^4+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}-\frac {1}{2}\right )\,\left (\frac {1}{24}+\frac {\sqrt {3}\,1{}\mathrm {i}}{24}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 46, normalized size = 0.94 \[ - \frac {\log {\left (x^{4} + 1 \right )}}{12} + \frac {\log {\left (x^{8} - x^{4} + 1 \right )}}{24} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} x^{4}}{3} - \frac {\sqrt {3}}{3} \right )}}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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