Optimal. Leaf size=52 \[ \frac {\left (-4-x^3\right ) \sqrt [4]{1+x^3}}{24 x^6}+\frac {1}{16} \text {ArcTan}\left (\sqrt [4]{1+x^3}\right )+\frac {1}{16} \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right ) \]
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Rubi [A]
time = 0.01, 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 = {272, 43, 44, 65,
218, 212, 209} \begin {gather*} \frac {1}{16} \text {ArcTan}\left (\sqrt [4]{x^3+1}\right )-\frac {\sqrt [4]{x^3+1}}{24 x^3}+\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 43
Rule 44
Rule 65
Rule 209
Rule 212
Rule 218
Rule 272
Rubi steps
\begin {align*} \int \frac {\sqrt [4]{1+x^3}}{x^7} \, dx &=\frac {1}{3} \text {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} \text {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} \text {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} \text {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} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^3}\right )+\frac {1}{16} \text {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 [A]
time = 0.05, size = 48, normalized size = 0.92 \begin {gather*} \frac {1}{48} \left (-\frac {2 \sqrt [4]{1+x^3} \left (4+x^3\right )}{x^6}+3 \text {ArcTan}\left (\sqrt [4]{1+x^3}\right )+3 \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right )\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 = 1.64, size = 58, normalized size = 1.12
method | result | size |
meijerg | \(-\frac {-\frac {7 \Gamma \left (\frac {3}{4}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {11}{4}\right ], \left [2, 4\right ], -x^{3}\right )}{32}+\frac {3 \left (-\frac {1}{6}-3 \ln \left (2\right )+\frac {\pi }{2}+3 \ln \left (x \right )\right ) \Gamma \left (\frac {3}{4}\right )}{8}+\frac {2 \Gamma \left (\frac {3}{4}\right )}{x^{6}}+\frac {\Gamma \left (\frac {3}{4}\right )}{x^{3}}}{12 \Gamma \left (\frac {3}{4}\right )}\) | \(58\) |
risch | \(-\frac {x^{6}+5 x^{3}+4}{24 x^{6} \left (x^{3}+1\right )^{\frac {3}{4}}}-\frac {-\frac {3 \Gamma \left (\frac {3}{4}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {7}{4}\right ], \left [2, 2\right ], -x^{3}\right )}{4}+\left (-3 \ln \left (2\right )+\frac {\pi }{2}+3 \ln \left (x \right )\right ) \Gamma \left (\frac {3}{4}\right )}{32 \Gamma \left (\frac {3}{4}\right )}\) | \(66\) |
trager | \(-\frac {\left (x^{3}+4\right ) \left (x^{3}+1\right )^{\frac {1}{4}}}{24 x^{6}}-\frac {\ln \left (\frac {2 \left (x^{3}+1\right )^{\frac {3}{4}}-x^{3}-2 \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {1}{4}}-2}{x^{3}}\right )}{32}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{3}+1\right )^{\frac {3}{4}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}+1}-2 \left (x^{3}+1\right )^{\frac {1}{4}}+2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{3}}\right )}{32}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, 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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Fricas [A]
time = 0.38, 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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Sympy [C] Result contains complex when optimal does not.
time = 1.11, 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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Giac [A]
time = 0.38, 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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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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