Optimal. Leaf size=115 \[ \frac {3 \left (-1-x^3+x^4\right )^{2/3}}{2 x^2}+\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1-x^3+x^4}}\right )+\log \left (x+\sqrt [3]{-1-x^3+x^4}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1-x^3+x^4}+\left (-1-x^3+x^4\right )^{2/3}\right ) \]
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Rubi [F]
time = 0.56, 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 (3+x^4\right ) \left (-1-x^3+x^4\right )^{2/3}}{x^3 \left (-1+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (3+x^4\right ) \left (-1-x^3+x^4\right )^{2/3}}{x^3 \left (-1+x^4\right )} \, dx &=\int \left (\frac {\left (-1-x^3+x^4\right )^{2/3}}{-1+x}-\frac {3 \left (-1-x^3+x^4\right )^{2/3}}{x^3}+\frac {\left (-1-x^3+x^4\right )^{2/3}}{1+x}-\frac {2 x \left (-1-x^3+x^4\right )^{2/3}}{1+x^2}\right ) \, dx\\ &=-\left (2 \int \frac {x \left (-1-x^3+x^4\right )^{2/3}}{1+x^2} \, dx\right )-3 \int \frac {\left (-1-x^3+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (-1-x^3+x^4\right )^{2/3}}{-1+x} \, dx+\int \frac {\left (-1-x^3+x^4\right )^{2/3}}{1+x} \, dx\\ &=-\left (2 \int \left (-\frac {\left (-1-x^3+x^4\right )^{2/3}}{2 (i-x)}+\frac {\left (-1-x^3+x^4\right )^{2/3}}{2 (i+x)}\right ) \, dx\right )-3 \int \frac {\left (-1-x^3+x^4\right )^{2/3}}{x^3} \, dx+\int \frac {\left (-1-x^3+x^4\right )^{2/3}}{-1+x} \, dx+\int \frac {\left (-1-x^3+x^4\right )^{2/3}}{1+x} \, dx\\ &=-\left (3 \int \frac {\left (-1-x^3+x^4\right )^{2/3}}{x^3} \, dx\right )+\int \frac {\left (-1-x^3+x^4\right )^{2/3}}{i-x} \, dx+\int \frac {\left (-1-x^3+x^4\right )^{2/3}}{-1+x} \, dx-\int \frac {\left (-1-x^3+x^4\right )^{2/3}}{i+x} \, dx+\int \frac {\left (-1-x^3+x^4\right )^{2/3}}{1+x} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.44, size = 115, normalized size = 1.00 \begin {gather*} \frac {3 \left (-1-x^3+x^4\right )^{2/3}}{2 x^2}+\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{-1-x^3+x^4}}\right )+\log \left (x+\sqrt [3]{-1-x^3+x^4}\right )-\frac {1}{2} \log \left (x^2-x \sqrt [3]{-1-x^3+x^4}+\left (-1-x^3+x^4\right )^{2/3}\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 = 8.17, size = 297, normalized size = 2.58
method | result | size |
risch | \(\frac {3 \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}}}{2 x^{2}}+\ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+3 \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -5 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{4}+3 x \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}}+3 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-2 x^{3}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{4}+3 \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +3 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-2 x^{4}+3 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+2 x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )\) | \(297\) |
trager | \(\frac {3 \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}}}{2 x^{2}}-12 \ln \left (-\frac {-337536 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}+632880 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-30972 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}+30384 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x -30384 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}+83124 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}-286 x^{4}-1821 x \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}}+1821 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+2574 x^{3}+337536 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+30972 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+286}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right ) \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+12 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \ln \left (\frac {337536 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}-632880 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}+25284 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}+30384 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x -30384 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}-22356 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}+49 x^{4}+4353 x \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}}-4353 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}-42 x^{3}-337536 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}-25284 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )-49}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )-\ln \left (-\frac {-337536 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{4}+632880 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2} x^{3}-30972 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{4}+30384 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}} x -30384 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{2}+83124 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right ) x^{3}-286 x^{4}-1821 x \left (x^{4}-x^{3}-1\right )^{\frac {2}{3}}+1821 \left (x^{4}-x^{3}-1\right )^{\frac {1}{3}} x^{2}+2574 x^{3}+337536 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )^{2}+30972 \RootOf \left (144 \textit {\_Z}^{2}+12 \textit {\_Z} +1\right )+286}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right )}\right )\) | \(688\) |
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 [A]
time = 2.17, size = 150, normalized size = 1.30 \begin {gather*} \frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {728574532 \, \sqrt {3} {\left (x^{4} - x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 812477430 \, \sqrt {3} {\left (x^{4} - x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (355231575 \, x^{4} + 41951449 \, x^{3} - 355231575\right )}}{3 \, {\left (447697125 \, x^{4} - 770525981 \, x^{3} - 447697125\right )}}\right ) + x^{2} \log \left (\frac {x^{4} + 3 \, {\left (x^{4} - x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{4} - x^{3} - 1\right )}^{\frac {2}{3}} x - 1}{x^{4} - 1}\right ) + 3 \, {\left (x^{4} - x^{3} - 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{4} + 3\right ) \left (x^{4} - x^{3} - 1\right )^{\frac {2}{3}}}{x^{3} \left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}\, dx \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^4+3\right )\,{\left (x^4-x^3-1\right )}^{2/3}}{x^3\,\left (x^4-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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