Optimal. Leaf size=45 \[ -\frac {\sqrt [4]{x^6+1}}{6 x^6}-\frac {1}{12} \tan ^{-1}\left (\sqrt [4]{x^6+1}\right )-\frac {1}{12} \tanh ^{-1}\left (\sqrt [4]{x^6+1}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 45, 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 = {266, 47, 63, 212, 206, 203} \begin {gather*} -\frac {\sqrt [4]{x^6+1}}{6 x^6}-\frac {1}{12} \tan ^{-1}\left (\sqrt [4]{x^6+1}\right )-\frac {1}{12} \tanh ^{-1}\left (\sqrt [4]{x^6+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 63
Rule 203
Rule 206
Rule 212
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{1+x^6}}{x^7} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {\sqrt [4]{1+x}}{x^2} \, dx,x,x^6\right )\\ &=-\frac {\sqrt [4]{1+x^6}}{6 x^6}+\frac {1}{24} \operatorname {Subst}\left (\int \frac {1}{x (1+x)^{3/4}} \, dx,x,x^6\right )\\ &=-\frac {\sqrt [4]{1+x^6}}{6 x^6}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{1+x^6}\right )\\ &=-\frac {\sqrt [4]{1+x^6}}{6 x^6}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^6}\right )-\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^6}\right )\\ &=-\frac {\sqrt [4]{1+x^6}}{6 x^6}-\frac {1}{12} \tan ^{-1}\left (\sqrt [4]{1+x^6}\right )-\frac {1}{12} \tanh ^{-1}\left (\sqrt [4]{1+x^6}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.58 \begin {gather*} \frac {2}{15} \left (x^6+1\right )^{5/4} \, _2F_1\left (\frac {5}{4},2;\frac {9}{4};x^6+1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.04, size = 45, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [4]{x^6+1}}{6 x^6}-\frac {1}{12} \tan ^{-1}\left (\sqrt [4]{x^6+1}\right )-\frac {1}{12} \tanh ^{-1}\left (\sqrt [4]{x^6+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 57, normalized size = 1.27 \begin {gather*} -\frac {2 \, x^{6} \arctan \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}}\right ) + x^{6} \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} + 1\right ) - x^{6} \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} - 1\right ) + 4 \, {\left (x^{6} + 1\right )}^{\frac {1}{4}}}{24 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 47, normalized size = 1.04 \begin {gather*} -\frac {{\left (x^{6} + 1\right )}^{\frac {1}{4}}}{6 \, x^{6}} - \frac {1}{12} \, \arctan \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{24} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{24} \, \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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maple [C] time = 0.29, size = 56, normalized size = 1.24 \begin {gather*} -\frac {\left (x^{6}+1\right )^{\frac {1}{4}}}{6 x^{6}}+\frac {-\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{6} \hypergeom \left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], -x^{6}\right )}{4}+\left (-3 \ln \relax (2)+\frac {\pi }{2}+6 \ln \relax (x )\right ) \Gamma \left (\frac {3}{4}\right )}{24 \Gamma \left (\frac {3}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 47, normalized size = 1.04 \begin {gather*} -\frac {{\left (x^{6} + 1\right )}^{\frac {1}{4}}}{6 \, x^{6}} - \frac {1}{12} \, \arctan \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{24} \, \log \left ({\left (x^{6} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{24} \, \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 = 0.60, size = 33, normalized size = 0.73 \begin {gather*} -\frac {\mathrm {atan}\left ({\left (x^6+1\right )}^{1/4}\right )}{12}-\frac {\mathrm {atanh}\left ({\left (x^6+1\right )}^{1/4}\right )}{12}-\frac {{\left (x^6+1\right )}^{1/4}}{6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.05, size = 34, normalized size = 0.76 \begin {gather*} - \frac {\Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{6}}} \right )}}{6 x^{\frac {9}{2}} \Gamma \left (\frac {7}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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