Optimal. Leaf size=45 \[ -\frac {\left (x^3+1\right )^{3/4}}{3 x^3}-\frac {1}{6} \tan ^{-1}\left (\sqrt [4]{x^3+1}\right )+\frac {1}{6} \tanh ^{-1}\left (\sqrt [4]{x^3+1}\right ) \]
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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, 51, 63, 298, 203, 206} \begin {gather*} -\frac {\left (x^3+1\right )^{3/4}}{3 x^3}-\frac {1}{6} \tan ^{-1}\left (\sqrt [4]{x^3+1}\right )+\frac {1}{6} \tanh ^{-1}\left (\sqrt [4]{x^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 203
Rule 206
Rule 266
Rule 298
Rubi steps
\begin {align*} \int \frac {1}{x^4 \sqrt [4]{1+x^3}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{1+x}} \, dx,x,x^3\right )\\ &=-\frac {\left (1+x^3\right )^{3/4}}{3 x^3}-\frac {1}{12} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,x^3\right )\\ &=-\frac {\left (1+x^3\right )^{3/4}}{3 x^3}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+x^3}\right )\\ &=-\frac {\left (1+x^3\right )^{3/4}}{3 x^3}+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^3}\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^3}\right )\\ &=-\frac {\left (1+x^3\right )^{3/4}}{3 x^3}-\frac {1}{6} \tan ^{-1}\left (\sqrt [4]{1+x^3}\right )+\frac {1}{6} \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.00, size = 26, normalized size = 0.58 \begin {gather*} \frac {4}{9} \left (x^3+1\right )^{3/4} \, _2F_1\left (\frac {3}{4},2;\frac {7}{4};x^3+1\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 45, normalized size = 1.00 \begin {gather*} -\frac {\left (x^3+1\right )^{3/4}}{3 x^3}-\frac {1}{6} \tan ^{-1}\left (\sqrt [4]{x^3+1}\right )+\frac {1}{6} \tanh ^{-1}\left (\sqrt [4]{x^3+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 57, normalized size = 1.27 \begin {gather*} -\frac {2 \, x^{3} \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) - x^{3} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) + x^{3} \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} - 1\right ) + 4 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}}}{12 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 48, normalized size = 1.07 \begin {gather*} -\frac {{\left (x^{3} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} - \frac {1}{6} \, \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{12} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{12} \, \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 = 0.23, size = 72, normalized size = 1.60 \begin {gather*} -\frac {\left (x^{3}+1\right )^{\frac {3}{4}}}{3 x^{3}}-\frac {\sqrt {2}\, \Gamma \left (\frac {3}{4}\right ) \left (-\frac {\pi \sqrt {2}\, x^{3} \hypergeom \left (\left [1, 1, \frac {5}{4}\right ], \left [2, 2\right ], -x^{3}\right )}{4 \Gamma \left (\frac {3}{4}\right )}+\frac {\left (-3 \ln \relax (2)-\frac {\pi }{2}+3 \ln \relax (x )\right ) \pi \sqrt {2}}{\Gamma \left (\frac {3}{4}\right )}\right )}{24 \pi } \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 47, normalized size = 1.04 \begin {gather*} -\frac {{\left (x^{3} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} - \frac {1}{6} \, \arctan \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{12} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {1}{12} \, \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 = 0.58, size = 33, normalized size = 0.73 \begin {gather*} \frac {\mathrm {atanh}\left ({\left (x^3+1\right )}^{1/4}\right )}{6}-\frac {\mathrm {atan}\left ({\left (x^3+1\right )}^{1/4}\right )}{6}-\frac {{\left (x^3+1\right )}^{3/4}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.90, size = 32, normalized size = 0.71 \begin {gather*} - \frac {\Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{\frac {15}{4}} \Gamma \left (\frac {9}{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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