Optimal. Leaf size=125 \[ \frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}-2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{1+x^3}}\right )+\sqrt {2} \text {ArcTan}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{-x^2+\sqrt {1+x^3}}\right )-2 \tanh ^{-1}\left (\frac {\sqrt [4]{1+x^3}}{x}\right )+\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \]
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Rubi [F]
time = 1.57, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (4+x^3\right ) \left (1+2 x^3+x^6+x^8\right )}{x^4 \sqrt [4]{1+x^3} \left (-1-2 x^3-x^6+x^8\right )} \, dx &=\int \left (-\frac {4}{x^4 \sqrt [4]{1+x^3}}-\frac {1}{x \sqrt [4]{1+x^3}}+\frac {4+x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )}+\frac {4+x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )}\right ) \, dx\\ &=-\left (4 \int \frac {1}{x^4 \sqrt [4]{1+x^3}} \, dx\right )-\int \frac {1}{x \sqrt [4]{1+x^3}} \, dx+\int \frac {4+x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {4+x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ &=-\left (\frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,x^3\right )\right )-\frac {4}{3} \text {Subst}\left (\int \frac {1}{x^2 \sqrt [4]{1+x}} \, dx,x,x^3\right )+\int \left (\frac {4}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )}+\frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )}\right ) \, dx+\int \left (\frac {4}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )}+\frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )}\right ) \, dx\\ &=\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}+\frac {1}{3} \text {Subst}\left (\int \frac {1}{x \sqrt [4]{1+x}} \, dx,x,x^3\right )-\frac {4}{3} \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+x^3}\right )+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ &=\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}+\frac {2}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^3}\right )-\frac {2}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^3}\right )+\frac {4}{3} \text {Subst}\left (\int \frac {x^2}{-1+x^4} \, dx,x,\sqrt [4]{1+x^3}\right )+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ &=\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}-\frac {2}{3} \tan ^{-1}\left (\sqrt [4]{1+x^3}\right )+\frac {2}{3} \tanh ^{-1}\left (\sqrt [4]{1+x^3}\right )-\frac {2}{3} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sqrt [4]{1+x^3}\right )+\frac {2}{3} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt [4]{1+x^3}\right )+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ &=\frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+4 \int \frac {1}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (-1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\sqrt [4]{1+x^3} \left (1+x^3+x^4\right )} \, dx\\ \end {align*}
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Mathematica [A]
time = 11.01, size = 127, normalized size = 1.02 \begin {gather*} \frac {4 \left (1+x^3\right )^{3/4}}{3 x^3}+2 \left (\text {ArcTan}\left (\frac {\sqrt [4]{1+x^3}}{x}\right )-\frac {\text {ArcTan}\left (\frac {-x^2+\sqrt {1+x^3}}{\sqrt {2} x \sqrt [4]{1+x^3}}\right )}{\sqrt {2}}-\tanh ^{-1}\left (\frac {x}{\sqrt [4]{1+x^3}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )}{\sqrt {2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 7.25, size = 383, normalized size = 3.06
method | result | size |
trager | \(\frac {4 \left (x^{3}+1\right )^{\frac {3}{4}}}{3 x^{3}}+\ln \left (-\frac {2 \left (x^{3}+1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-x^{4}-x^{3}-1}{x^{4}-x^{3}-1}\right )+\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{3}+1}\, x^{2}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}-2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}+1}\right )-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{3}+1}\, x^{2}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}+1}\right )+\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {x^{3}+1}\, x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x -2 \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}}{x^{4}-x^{3}-1}\right )\) | \(383\) |
risch | \(\frac {4 \left (x^{3}+1\right )^{\frac {3}{4}}}{3 x^{3}}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{3}+1}\, x^{2}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}+1}\right )-\RootOf \left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{3}+1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\RootOf \left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}+1}\right )-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{4}+1\right )^{2} \sqrt {x^{3}+1}\, x^{2}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2} x^{3}-2 \left (x^{3}+1\right )^{\frac {3}{4}} x +2 \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-\RootOf \left (\textit {\_Z}^{4}+1\right )^{2}}{x^{4}-x^{3}-1}\right )+\ln \left (-\frac {2 \left (x^{3}+1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{3}+1}+2 \left (x^{3}+1\right )^{\frac {1}{4}} x^{3}-x^{4}-x^{3}-1}{x^{4}-x^{3}-1}\right )\) | \(384\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 816 vs.
\(2 (102) = 204\).
time = 23.80, size = 816, normalized size = 6.53 \begin {gather*} -\frac {12 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{4} + 2 \, x^{3} + 2 \, \sqrt {2} {\left (3 \, x^{5} - x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{6} - 3 \, x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{5} + x^{2}\right )} \sqrt {x^{3} + 1} + {\left (16 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x^{5} + 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{5} - x^{2}\right )} \sqrt {x^{3} + 1} + \sqrt {2} {\left (x^{8} + 8 \, x^{7} - x^{6} + 8 \, x^{4} - 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{7} + x^{6} + x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{4} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{4} + x^{3} + 1}} + 1}{x^{8} - 14 \, x^{7} + x^{6} - 14 \, x^{4} + 2 \, x^{3} + 1}\right ) - 12 \, \sqrt {2} x^{3} \arctan \left (\frac {x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{4} + 2 \, x^{3} - 2 \, \sqrt {2} {\left (3 \, x^{5} - x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{6} - 3 \, x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{6} + x^{5} + x^{2}\right )} \sqrt {x^{3} + 1} + {\left (16 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x^{5} - 2 \, \sqrt {2} {\left (3 \, x^{6} - x^{5} - x^{2}\right )} \sqrt {x^{3} + 1} - \sqrt {2} {\left (x^{8} + 8 \, x^{7} - x^{6} + 8 \, x^{4} - 2 \, x^{3} - 1\right )} + 4 \, {\left (x^{7} + x^{6} + x^{3}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{4} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{4} + x^{3} + 1}} + 1}{x^{8} - 14 \, x^{7} + x^{6} - 14 \, x^{4} + 2 \, x^{3} + 1}\right ) - 3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, {\left (x^{4} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} + 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{4} + x^{3} + 1}\right ) + 3 \, \sqrt {2} x^{3} \log \left (\frac {4 \, {\left (x^{4} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 4 \, \sqrt {x^{3} + 1} x^{2} - 2 \, \sqrt {2} {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1\right )}}{x^{4} + x^{3} + 1}\right ) - 12 \, x^{3} \arctan \left (\frac {2 \, {\left ({\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + {\left (x^{3} + 1\right )}^{\frac {3}{4}} x\right )}}{x^{4} - x^{3} - 1}\right ) - 12 \, x^{3} \log \left (\frac {x^{4} - 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}} x^{3} + x^{3} + 2 \, \sqrt {x^{3} + 1} x^{2} - 2 \, {\left (x^{3} + 1\right )}^{\frac {3}{4}} x + 1}{x^{4} - x^{3} - 1}\right ) - 16 \, {\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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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {\left (x^3+4\right )\,\left (x^8+x^6+2\,x^3+1\right )}{x^4\,{\left (x^3+1\right )}^{1/4}\,\left (-x^8+x^6+2\,x^3+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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