Optimal. Leaf size=52 \[ \frac {1}{16} \tan ^{-1}\left (\sqrt [4]{x^3+1}\right )+\frac {1}{16} \tanh ^{-1}\left (\sqrt [4]{x^3+1}\right )+\frac {\sqrt [4]{x^3+1} \left (-x^3-4\right )}{24 x^6} \]
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Rubi [A] time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.17, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {266, 47, 51, 63, 212, 206, 203} \begin {gather*} -\frac {\sqrt [4]{x^3+1}}{24 x^3}+\frac {1}{16} \tan ^{-1}\left (\sqrt [4]{x^3+1}\right )+\frac {1}{16} \tanh ^{-1}\left (\sqrt [4]{x^3+1}\right )-\frac {\sqrt [4]{x^3+1}}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 47
Rule 51
Rule 63
Rule 203
Rule 206
Rule 212
Rule 266
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{1+x^3}}{x^7} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [4]{1+x}}{x^3} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [4]{1+x^3}}{6 x^6}+\frac {1}{24} \operatorname {Subst}\left (\int \frac {1}{x^2 (1+x)^{3/4}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [4]{1+x^3}}{6 x^6}-\frac {\sqrt [4]{1+x^3}}{24 x^3}-\frac {1}{32} \operatorname {Subst}\left (\int \frac {1}{x (1+x)^{3/4}} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [4]{1+x^3}}{6 x^6}-\frac {\sqrt [4]{1+x^3}}{24 x^3}-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{-1+x^4} \, dx,x,\sqrt [4]{1+x^3}\right )\\ &=-\frac {\sqrt [4]{1+x^3}}{6 x^6}-\frac {\sqrt [4]{1+x^3}}{24 x^3}+\frac {1}{16} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^3}\right )+\frac {1}{16} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^3}\right )\\ &=-\frac {\sqrt [4]{1+x^3}}{6 x^6}-\frac {\sqrt [4]{1+x^3}}{24 x^3}+\frac {1}{16} \tan ^{-1}\left (\sqrt [4]{1+x^3}\right )+\frac {1}{16} \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.50 \begin {gather*} -\frac {4}{15} \left (x^3+1\right )^{5/4} \, _2F_1\left (\frac {5}{4},3;\frac {9}{4};x^3+1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 52, normalized size = 1.00 \begin {gather*} \frac {\left (-4-x^3\right ) \sqrt [4]{1+x^3}}{24 x^6}+\frac {1}{16} \tan ^{-1}\left (\sqrt [4]{1+x^3}\right )+\frac {1}{16} \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 63, normalized size = 1.21 \begin {gather*} \frac {6 \, x^{6} \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) + 3 \, x^{6} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) - 3 \, x^{6} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} - 1\right ) - 4 \, {\left (x^{3} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{96 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 58, normalized size = 1.12 \begin {gather*} -\frac {{\left (x^{3} + 1\right )}^{\frac {5}{4}} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{24 \, x^{6}} + \frac {1}{16} \, \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{32} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{32} \, \log \left ({\left | {\left (x^{3} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.68, size = 58, normalized size = 1.12
method | result | size |
meijerg | \(-\frac {-\frac {7 \Gamma \left (\frac {3}{4}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {11}{4}\right ], \left [2, 4\right ], -x^{3}\right )}{32}+\frac {3 \left (-\frac {1}{6}-3 \ln \relax (2)+\frac {\pi }{2}+3 \ln \relax (x )\right ) \Gamma \left (\frac {3}{4}\right )}{8}+\frac {2 \Gamma \left (\frac {3}{4}\right )}{x^{6}}+\frac {\Gamma \left (\frac {3}{4}\right )}{x^{3}}}{12 \Gamma \left (\frac {3}{4}\right )}\) | \(58\) |
risch | \(-\frac {x^{6}+5 x^{3}+4}{24 x^{6} \left (x^{3}+1\right )^{\frac {3}{4}}}-\frac {-\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], -x^{3}\right )}{4}+\left (-3 \ln \relax (2)+\frac {\pi }{2}+3 \ln \relax (x )\right ) \Gamma \left (\frac {3}{4}\right )}{32 \Gamma \left (\frac {3}{4}\right )}\) | \(66\) |
trager | \(-\frac {\left (x^{3}+4\right ) \left (x^{3}+1\right )^{\frac {1}{4}}}{24 x^{6}}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {3}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{3}+1\right )^{\frac {1}{4}}}{x^{3}}\right )}{32}-\frac {\ln \left (\frac {2 \left (x^{3}+1\right )^{\frac {3}{4}}-x^{3}-2 \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {1}{4}}-2}{x^{3}}\right )}{32}\) | \(126\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 72, normalized size = 1.38 \begin {gather*} \frac {{\left (x^{3} + 1\right )}^{\frac {5}{4}} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{24 \, {\left (2 \, x^{3} - {\left (x^{3} + 1\right )}^{2} + 1\right )}} + \frac {1}{16} \, \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{32} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{32} \, \log \left ({\left (x^{3} + 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 = 0.69, size = 45, normalized size = 0.87 \begin {gather*} \frac {\mathrm {atan}\left ({\left (x^3+1\right )}^{1/4}\right )}{16}+\frac {\mathrm {atanh}\left ({\left (x^3+1\right )}^{1/4}\right )}{16}-\frac {{\left (x^3+1\right )}^{1/4}}{8\,x^6}-\frac {{\left (x^3+1\right )}^{5/4}}{24\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.16, size = 34, normalized size = 0.65 \begin {gather*} - \frac {\Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{\frac {21}{4}} \Gamma \left (\frac {11}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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