Optimal. Leaf size=63 \[ -\tan ^{-1}\left (\frac {x}{\sqrt {-x^4-x^3+2}}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {-x^4-x^3+2}}{x^4+x^3-2}\right ) \]
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Rubi [F] time = 1.69, 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 {\sqrt {2-x^3-x^4} \left (4+x^3+2 x^4\right )}{\left (-2-3 x^2+x^3+x^4\right ) \left (-2-x^2+x^3+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\sqrt {2-x^3-x^4} \left (4+x^3+2 x^4\right )}{\left (-2-3 x^2+x^3+x^4\right ) \left (-2-x^2+x^3+x^4\right )} \, dx &=\int \left (\frac {\left (-6+3 x+4 x^2\right ) \sqrt {2-x^3-x^4}}{2 \left (-2-3 x^2+x^3+x^4\right )}+\frac {\left (2-3 x-4 x^2\right ) \sqrt {2-x^3-x^4}}{2 \left (-2-x^2+x^3+x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\left (-6+3 x+4 x^2\right ) \sqrt {2-x^3-x^4}}{-2-3 x^2+x^3+x^4} \, dx+\frac {1}{2} \int \frac {\left (2-3 x-4 x^2\right ) \sqrt {2-x^3-x^4}}{-2-x^2+x^3+x^4} \, dx\\ &=\frac {1}{2} \int \left (-\frac {6 \sqrt {2-x^3-x^4}}{-2-3 x^2+x^3+x^4}+\frac {3 x \sqrt {2-x^3-x^4}}{-2-3 x^2+x^3+x^4}+\frac {4 x^2 \sqrt {2-x^3-x^4}}{-2-3 x^2+x^3+x^4}\right ) \, dx+\frac {1}{2} \int \left (\frac {2 \sqrt {2-x^3-x^4}}{-2-x^2+x^3+x^4}-\frac {3 x \sqrt {2-x^3-x^4}}{-2-x^2+x^3+x^4}-\frac {4 x^2 \sqrt {2-x^3-x^4}}{-2-x^2+x^3+x^4}\right ) \, dx\\ &=\frac {3}{2} \int \frac {x \sqrt {2-x^3-x^4}}{-2-3 x^2+x^3+x^4} \, dx-\frac {3}{2} \int \frac {x \sqrt {2-x^3-x^4}}{-2-x^2+x^3+x^4} \, dx+2 \int \frac {x^2 \sqrt {2-x^3-x^4}}{-2-3 x^2+x^3+x^4} \, dx-2 \int \frac {x^2 \sqrt {2-x^3-x^4}}{-2-x^2+x^3+x^4} \, dx-3 \int \frac {\sqrt {2-x^3-x^4}}{-2-3 x^2+x^3+x^4} \, dx+\int \frac {\sqrt {2-x^3-x^4}}{-2-x^2+x^3+x^4} \, dx\\ \end {align*}
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Mathematica [C] time = 6.63, size = 61074, normalized size = 969.43 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.35, size = 63, normalized size = 1.00 \begin {gather*} -\tan ^{-1}\left (\frac {x}{\sqrt {2-x^3-x^4}}\right )-\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {2-x^3-x^4}}{-2+x^3+x^4}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 75, normalized size = 1.19 \begin {gather*} -\frac {1}{2} \, \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {-x^{4} - x^{3} + 2} x}{x^{4} + x^{3} + 3 \, x^{2} - 2}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {-x^{4} - x^{3} + 2} x}{x^{4} + x^{3} + x^{2} - 2}\right ) \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^{4} + x^{3} + 4\right )} \sqrt {-x^{4} - x^{3} + 2}}{{\left (x^{4} + x^{3} - x^{2} - 2\right )} {\left (x^{4} + x^{3} - 3 \, x^{2} - 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.77, size = 167, normalized size = 2.65
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}+6 \sqrt {-x^{4}-x^{3}+2}\, x -2 \RootOf \left (\textit {\_Z}^{2}+3\right )}{x^{4}+x^{3}-3 x^{2}-2}\right )}{2}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {-x^{4}-x^{3}+2}\, x -2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}+x^{3}-x^{2}-2}\right )}{2}\) | \(167\) |
default | \(\text {Expression too large to display}\) | \(5959\) |
elliptic | \(\text {Expression too large to display}\) | \(433723\) |
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^{4} + x^{3} + 4\right )} \sqrt {-x^{4} - x^{3} + 2}}{{\left (x^{4} + x^{3} - x^{2} - 2\right )} {\left (x^{4} + x^{3} - 3 \, x^{2} - 2\right )}}\,{d x} \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 = -0.02 \begin {gather*} \int \frac {\sqrt {-x^4-x^3+2}\,\left (2\,x^4+x^3+4\right )}{\left (-x^4-x^3+3\,x^2+2\right )\,\left (-x^4-x^3+x^2+2\right )} \,d x \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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