Optimal. Leaf size=42 \[ \frac {2}{3} \sqrt [4]{1+x^6}-\frac {1}{3} \text {ArcTan}\left (\sqrt [4]{1+x^6}\right )-\frac {1}{3} \tanh ^{-1}\left (\sqrt [4]{1+x^6}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {272, 52, 65,
218, 212, 209} \begin {gather*} -\frac {1}{3} \text {ArcTan}\left (\sqrt [4]{x^6+1}\right )+\frac {2}{3} \sqrt [4]{x^6+1}-\frac {1}{3} \tanh ^{-1}\left (\sqrt [4]{x^6+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{1+x^6}}{x} \, dx &=\frac {1}{6} \text {Subst}\left (\int \frac {\sqrt [4]{1+x}}{x} \, dx,x,x^6\right )\\ &=\frac {2}{3} \sqrt [4]{1+x^6}+\frac {1}{6} \text {Subst}\left (\int \frac {1}{x (1+x)^{3/4}} \, dx,x,x^6\right )\\ &=\frac {2}{3} \sqrt [4]{1+x^6}+\frac {2}{3} \text {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{1+x^6}\right )\\ &=\frac {2}{3} \sqrt [4]{1+x^6}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^6}\right )-\frac {1}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^6}\right )\\ &=\frac {2}{3} \sqrt [4]{1+x^6}-\frac {1}{3} \tan ^{-1}\left (\sqrt [4]{1+x^6}\right )-\frac {1}{3} \tanh ^{-1}\left (\sqrt [4]{1+x^6}\right )\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 42, normalized size = 1.00 \begin {gather*} \frac {2}{3} \sqrt [4]{1+x^6}-\frac {1}{3} \text {ArcTan}\left (\sqrt [4]{1+x^6}\right )-\frac {1}{3} \tanh ^{-1}\left (\sqrt [4]{1+x^6}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 5 vs. order
3.
time = 3.88, size = 45, normalized size = 1.07
method | result | size |
meijerg | \(-\frac {-\Gamma \left (\frac {3}{4}\right ) x^{6} \hypergeom \left (\left [\frac {3}{4}, 1, 1\right ], \left [2, 2\right ], -x^{6}\right )-4 \left (4-3 \ln \left (2\right )+\frac {\pi }{2}+6 \ln \left (x \right )\right ) \Gamma \left (\frac {3}{4}\right )}{24 \Gamma \left (\frac {3}{4}\right )}\) | \(45\) |
trager | \(\frac {2 \left (x^{6}+1\right )^{\frac {1}{4}}}{3}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+2 \left (x^{6}+1\right )^{\frac {3}{4}}-2 \sqrt {x^{6}+1}\, \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{6}+1\right )^{\frac {1}{4}}+2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{6}}\right )}{6}+\frac {\ln \left (\frac {-x^{6}+2 \left (x^{6}+1\right )^{\frac {3}{4}}-2 \sqrt {x^{6}+1}+2 \left (x^{6}+1\right )^{\frac {1}{4}}-2}{x^{6}}\right )}{6}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 44, normalized size = 1.05 \begin {gather*} \frac {2}{3} \, {\left (x^{6} + 1\right )}^{\frac {1}{4}} - \frac {1}{3} \, \arctan \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{6} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 44, normalized size = 1.05 \begin {gather*} \frac {2}{3} \, {\left (x^{6} + 1\right )}^{\frac {1}{4}} - \frac {1}{3} \, \arctan \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{6} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.51, size = 37, normalized size = 0.88 \begin {gather*} - \frac {x^{\frac {3}{2}} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{6}}} \right )}}{6 \Gamma \left (\frac {3}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 44, normalized size = 1.05 \begin {gather*} \frac {2}{3} \, {\left (x^{6} + 1\right )}^{\frac {1}{4}} - \frac {1}{3} \, \arctan \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{6} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{6} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 30, normalized size = 0.71 \begin {gather*} \frac {2\,{\left (x^6+1\right )}^{1/4}}{3}-\frac {\mathrm {atanh}\left ({\left (x^6+1\right )}^{1/4}\right )}{3}-\frac {\mathrm {atan}\left ({\left (x^6+1\right )}^{1/4}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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