Optimal. Leaf size=53 \[ \frac {2 \sqrt {x^3+1} \left (2 x^3+15 x^2+2\right )}{3 x^3}-10 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {x^3+1}}\right ) \]
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Rubi [F] time = 0.99, 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 (-2+x^3\right ) \sqrt {1+x^3} \left (2-x^2+2 x^3\right )}{x^4 \left (1-3 x^2+x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-2+x^3\right ) \sqrt {1+x^3} \left (2-x^2+2 x^3\right )}{x^4 \left (1-3 x^2+x^3\right )} \, dx &=\int \left (-\frac {4 \sqrt {1+x^3}}{x^4}-\frac {10 \sqrt {1+x^3}}{x^2}+\frac {2 \sqrt {1+x^3}}{x}+\frac {15 (-2+x) \sqrt {1+x^3}}{1-3 x^2+x^3}\right ) \, dx\\ &=2 \int \frac {\sqrt {1+x^3}}{x} \, dx-4 \int \frac {\sqrt {1+x^3}}{x^4} \, dx-10 \int \frac {\sqrt {1+x^3}}{x^2} \, dx+15 \int \frac {(-2+x) \sqrt {1+x^3}}{1-3 x^2+x^3} \, dx\\ &=\frac {10 \sqrt {1+x^3}}{x}+\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x} \, dx,x,x^3\right )-\frac {4}{3} \operatorname {Subst}\left (\int \frac {\sqrt {1+x}}{x^2} \, dx,x,x^3\right )-15 \int \frac {x}{\sqrt {1+x^3}} \, dx+15 \int \left (-\frac {2 \sqrt {1+x^3}}{1-3 x^2+x^3}+\frac {x \sqrt {1+x^3}}{1-3 x^2+x^3}\right ) \, dx\\ &=\frac {4 \sqrt {1+x^3}}{3}+\frac {4 \sqrt {1+x^3}}{3 x^3}+\frac {10 \sqrt {1+x^3}}{x}-15 \int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx+15 \int \frac {x \sqrt {1+x^3}}{1-3 x^2+x^3} \, dx-30 \int \frac {\sqrt {1+x^3}}{1-3 x^2+x^3} \, dx-\left (15 \sqrt {2 \left (2-\sqrt {3}\right )}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx\\ &=\frac {4 \sqrt {1+x^3}}{3}+\frac {4 \sqrt {1+x^3}}{3 x^3}+\frac {10 \sqrt {1+x^3}}{x}-\frac {30 \sqrt {1+x^3}}{1+\sqrt {3}+x}+\frac {15 \sqrt [4]{3} \sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} E\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {10 \sqrt {2} 3^{3/4} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+15 \int \frac {x \sqrt {1+x^3}}{1-3 x^2+x^3} \, dx-30 \int \frac {\sqrt {1+x^3}}{1-3 x^2+x^3} \, dx\\ \end {align*}
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Mathematica [C] time = 6.15, size = 1719, normalized size = 32.43
result too large to display
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.67, size = 53, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {1+x^3} \left (2+15 x^2+2 x^3\right )}{3 x^3}-10 \sqrt {3} \tanh ^{-1}\left (\frac {\sqrt {3} x}{\sqrt {1+x^3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 112, normalized size = 2.11 \begin {gather*} \frac {15 \, \sqrt {3} x^{3} \log \left (-\frac {x^{6} + 18 \, x^{5} + 9 \, x^{4} + 2 \, x^{3} - 4 \, \sqrt {3} {\left (x^{4} + 3 \, x^{3} + x\right )} \sqrt {x^{3} + 1} + 18 \, x^{2} + 1}{x^{6} - 6 \, x^{5} + 9 \, x^{4} + 2 \, x^{3} - 6 \, x^{2} + 1}\right ) + 4 \, {\left (2 \, x^{3} + 15 \, x^{2} + 2\right )} \sqrt {x^{3} + 1}}{6 \, x^{3}} \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*} \int \frac {{\left (2 \, x^{3} - x^{2} + 2\right )} \sqrt {x^{3} + 1} {\left (x^{3} - 2\right )}}{{\left (x^{3} - 3 \, x^{2} + 1\right )} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.64, size = 87, normalized size = 1.64
method | result | size |
trager | \(\frac {2 \sqrt {x^{3}+1}\, \left (2 x^{3}+15 x^{2}+2\right )}{3 x^{3}}-5 \RootOf \left (\textit {\_Z}^{2}-3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}-3\right ) x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-3\right ) x^{2}+6 x \sqrt {x^{3}+1}+\RootOf \left (\textit {\_Z}^{2}-3\right )}{x^{3}-3 x^{2}+1}\right )\) | \(87\) |
default | \(\frac {4 \sqrt {x^{3}+1}}{3 x^{3}}+\frac {10 \sqrt {x^{3}+1}}{x}+\frac {4 \sqrt {x^{3}+1}}{3}+\frac {30 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+5 \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-3 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+4 \underline {\hspace {1.25 ex}}\alpha -4\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}-2 \underline {\hspace {1.25 ex}}\alpha +2-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{6}+\frac {2 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{3}-\frac {2 i \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )\) | \(330\) |
elliptic | \(\frac {4 \sqrt {x^{3}+1}}{3 x^{3}}+\frac {10 \sqrt {x^{3}+1}}{x}+\frac {4 \sqrt {x^{3}+1}}{3}+\frac {30 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+5 \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-3 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+4 \underline {\hspace {1.25 ex}}\alpha -4\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}-2 \underline {\hspace {1.25 ex}}\alpha +2-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{6}+\frac {2 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{3}-\frac {2 i \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )\) | \(330\) |
risch | \(\frac {10 x^{5}+10 x^{2}+\frac {8}{3} x^{3}+\frac {4}{3}+\frac {4}{3} x^{6}}{x^{3} \sqrt {x^{3}+1}}+\frac {30 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+5 \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{3}-3 \textit {\_Z}^{2}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-\underline {\hspace {1.25 ex}}\alpha ^{2}+4 \underline {\hspace {1.25 ex}}\alpha -4\right ) \left (3-i \sqrt {3}\right ) \sqrt {\frac {1+x}{3-i \sqrt {3}}}\, \sqrt {\frac {-1+2 x -i \sqrt {3}}{-3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x -1}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\underline {\hspace {1.25 ex}}\alpha ^{2}}{2}-2 \underline {\hspace {1.25 ex}}\alpha +2-\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2}}{6}+\frac {2 i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{3}-\frac {2 i \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}\right )\) | \(331\) |
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*} \int \frac {{\left (2 \, x^{3} - x^{2} + 2\right )} \sqrt {x^{3} + 1} {\left (x^{3} - 2\right )}}{{\left (x^{3} - 3 \, x^{2} + 1\right )} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 76, normalized size = 1.43 \begin {gather*} 5\,\sqrt {3}\,\ln \left (\frac {3\,x^2+x^3-2\,\sqrt {3}\,x\,\sqrt {x^3+1}+1}{x^3-3\,x^2+1}\right )+\frac {4\,\sqrt {x^3+1}}{3}+\frac {10\,\sqrt {x^3+1}}{x}+\frac {4\,\sqrt {x^3+1}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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