Optimal. Leaf size=230 \[ \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^2-x^3}}\right )-\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1+x^2-x^3}}\right )}{\sqrt [3]{2}}+\log \left (x+\sqrt [3]{1+x^2-x^3}\right )-\frac {\log \left (2 x+2^{2/3} \sqrt [3]{1+x^2-x^3}\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^2-x^3}+\left (1+x^2-x^3\right )^{2/3}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1+x^2-x^3}-\sqrt [3]{2} \left (1+x^2-x^3\right )^{2/3}\right )}{2 \sqrt [3]{2}} \]
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Rubi [F]
time = 3.75, 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 {x^3 \left (3+x^2\right )}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3} \left (1+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 {x^3 \left (3+x^2\right )}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx &=\int \left (\frac {1}{\sqrt [3]{1+x^2-x^3}}+\frac {2}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3}}+\frac {-3-x^2}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )}\right ) \, dx\\ &=2 \int \frac {1}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3}} \, dx+\int \frac {1}{\sqrt [3]{1+x^2-x^3}} \, dx+\int \frac {-3-x^2}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx\\ &=2 \int \left (\frac {i}{2 (i-x) \sqrt [3]{1+x^2-x^3}}+\frac {i}{2 (i+x) \sqrt [3]{1+x^2-x^3}}\right ) \, dx+\int \left (-\frac {3}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )}-\frac {x^2}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )}\right ) \, dx+\text {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {29}{27}+\frac {x}{3}-x^3}} \, dx,x,-\frac {1}{3}+x\right )\\ &=i \int \frac {1}{(i-x) \sqrt [3]{1+x^2-x^3}} \, dx+i \int \frac {1}{(i+x) \sqrt [3]{1+x^2-x^3}} \, dx-3 \int \frac {1}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx+\frac {\left (\sqrt [3]{2+2 \sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}-6 x} \sqrt [3]{-2+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+18 \left (-\frac {1}{3}+x\right )^2+2 \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) (-1+3 x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}}{3\ 2^{2/3}}-x} \sqrt [3]{\frac {1}{18} \left (-2+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )+\frac {1}{3} \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) x+x^2}} \, dx,x,-\frac {1}{3}+x\right )}{3\ 2^{2/3} \sqrt [3]{1+x^2-x^3}}-\int \frac {x^2}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx\\ &=i \text {Subst}\left (\int \frac {1}{\left (\left (-\frac {1}{3}+i\right )-x\right ) \sqrt [3]{\frac {29}{27}+\frac {x}{3}-x^3}} \, dx,x,-\frac {1}{3}+x\right )+i \text {Subst}\left (\int \frac {1}{\left (\left (\frac {1}{3}+i\right )+x\right ) \sqrt [3]{\frac {29}{27}+\frac {x}{3}-x^3}} \, dx,x,-\frac {1}{3}+x\right )-3 \int \frac {1}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx-\frac {\left (\sqrt [3]{2+2 \sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}-6 x} \sqrt [3]{1+\frac {-2 \left (2+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )+6\ 2^{2/3} \sqrt [3]{29+3 \sqrt {93}} x}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+\sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (4-2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}} \sqrt [3]{1+\frac {-2 \left (2+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )+6\ 2^{2/3} \sqrt [3]{29+3 \sqrt {93}} x}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}-i \sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (-4+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{x} \sqrt [3]{1-\frac {6\ 2^{2/3} \sqrt [3]{29+3 \sqrt {93}} x}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+\sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (4-2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}} \sqrt [3]{1-\frac {6\ 2^{2/3} \sqrt [3]{29+3 \sqrt {93}} x}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}-i \sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (-4+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}}} \, dx,x,\frac {\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}+2^{2/3} (1-3 x)}{3\ 2^{2/3}}\right )}{\sqrt [3]{6} \sqrt [3]{1+x^2-x^3}}-\int \frac {x^2}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx\\ &=-\frac {\left (2+2 \sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}-6 x\right ) \sqrt [3]{1-\frac {2 \left (2+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}-3\ 2^{2/3} \sqrt [3]{29+3 \sqrt {93}} x\right )}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+\sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (4-2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}} \sqrt [3]{1-\frac {2 \left (2+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}-3\ 2^{2/3} \sqrt [3]{29+3 \sqrt {93}} x\right )}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}-i \sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (-4+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {2 \sqrt [3]{29+3 \sqrt {93}} \left (\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}+2^{2/3} (1-3 x)\right )}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+\sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (4-2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}-\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}},\frac {2 \sqrt [3]{29+3 \sqrt {93}} \left (\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}+2^{2/3} (1-3 x)\right )}{6+3 \sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}-i \sqrt [6]{2} \sqrt [3]{29+3 \sqrt {93}} \sqrt {3 \left (-4+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )}}\right )}{4 \sqrt [3]{1+x^2-x^3}}-3 \int \frac {1}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx+\frac {\left (i \sqrt [3]{2+2 \sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}-6 x} \sqrt [3]{-2+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+18 \left (-\frac {1}{3}+x\right )^2+2 \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) (-1+3 x)}\right ) \text {Subst}\left (\int \frac {1}{\left (\left (-\frac {1}{3}+i\right )-x\right ) \sqrt [3]{\frac {\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}}{3\ 2^{2/3}}-x} \sqrt [3]{\frac {1}{18} \left (-2+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )+\frac {1}{3} \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) x+x^2}} \, dx,x,-\frac {1}{3}+x\right )}{3\ 2^{2/3} \sqrt [3]{1+x^2-x^3}}+\frac {\left (i \sqrt [3]{2+2 \sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+2^{2/3} \sqrt [3]{29+3 \sqrt {93}}-6 x} \sqrt [3]{-2+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}+18 \left (-\frac {1}{3}+x\right )^2+2 \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) (-1+3 x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{\frac {\frac {2}{\sqrt [3]{29+3 \sqrt {93}}}+\sqrt [3]{58+6 \sqrt {93}}}{3\ 2^{2/3}}-x} \left (\left (\frac {1}{3}+i\right )+x\right ) \sqrt [3]{\frac {1}{18} \left (-2+2 \left (\frac {2}{29+3 \sqrt {93}}\right )^{2/3}+\sqrt [3]{2} \left (29+3 \sqrt {93}\right )^{2/3}\right )+\frac {1}{3} \left (\sqrt [3]{\frac {2}{29+3 \sqrt {93}}}+\sqrt [3]{\frac {1}{2} \left (29+3 \sqrt {93}\right )}\right ) x+x^2}} \, dx,x,-\frac {1}{3}+x\right )}{3\ 2^{2/3} \sqrt [3]{1+x^2-x^3}}-\int \frac {x^2}{\sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.62, size = 230, normalized size = 1.00 \begin {gather*} \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^2-x^3}}\right )-\frac {\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1+x^2-x^3}}\right )}{\sqrt [3]{2}}+\log \left (x+\sqrt [3]{1+x^2-x^3}\right )-\frac {\log \left (2 x+2^{2/3} \sqrt [3]{1+x^2-x^3}\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (x^2-x \sqrt [3]{1+x^2-x^3}+\left (1+x^2-x^3\right )^{2/3}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1+x^2-x^3}-\sqrt [3]{2} \left (1+x^2-x^3\right )^{2/3}\right )}{2 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{3} \left (x^{2}+3\right )}{\left (x^{2}+1\right ) \left (-x^{3}+x^{2}+1\right )^{\frac {1}{3}} \left (x^{3}+x^{2}+1\right )}\, dx\]
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 [C] Result contains complex when optimal does not.
time = 1.38, size = 471, normalized size = 2.05 \begin {gather*} \frac {1}{4} \cdot 2^{\frac {2}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} \log \left (-\frac {x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{3} - 6 \cdot 2^{\frac {1}{3}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2} + 8 \, x - 24 \, {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}}{8 \, x}\right ) - \frac {1}{8} \, {\left (2^{\frac {2}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} - 2 \, \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}}\right )} \log \left (-\frac {3 \, {\left (2^{\frac {2}{3}} \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} + 2^{\frac {1}{3}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2} - 8 \, {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}\right )}}{8 \, x}\right ) - \frac {1}{8} \, {\left (2^{\frac {2}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} + 2 \, \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}}\right )} \log \left (\frac {3 \, {\left (2^{\frac {2}{3}} \sqrt {\frac {3}{2}} \sqrt {-2^{\frac {1}{3}} {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )} - 2^{\frac {1}{3}} x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{2} + 8 \, {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}\right )}}{8 \, x}\right ) - \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \log \left (\frac {x {\left (i \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} - \left (-1\right )^{\frac {1}{3}}\right )}^{3} + 32 \, x + 24 \, {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}}}{8 \, x}\right ) - \frac {1}{2} \, \log \left (\frac {x^{2} - {\left (-x^{3} + x^{2} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + x^{2} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \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 {x^{3} \left (x^{2} + 3\right )}{\left (x^{2} + 1\right ) \sqrt [3]{- x^{3} + x^{2} + 1} \left (x^{3} + 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 = -0.00 \begin {gather*} \int \frac {x^3\,\left (x^2+3\right )}{\left (x^2+1\right )\,\left (x^3+x^2+1\right )\,{\left (-x^3+x^2+1\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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