Optimal. Leaf size=63 \[ -\text {ArcTan}\left (\frac {x}{\sqrt {2-x^3-x^4}}\right )-\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x \sqrt {2-x^3-x^4}}{-2+x^3+x^4}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 1.12, 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.
[In]
[Out]
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*}
________________________________________________________________________________________
Mathematica [A]
time = 0.85, size = 52, normalized size = 0.83 \begin {gather*} -\text {ArcTan}\left (\frac {x}{\sqrt {2-x^3-x^4}}\right )+\sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} x}{\sqrt {2-x^3-x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 7.06, size = 5959, normalized size = 94.59
method | result | size |
trager | \(-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{4}+\RootOf \left (\textit {\_Z}^{2}+3\right ) x^{3}+3 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}+6 \sqrt {-x^{4}-x^{3}+2}\, x -2 \RootOf \left (\textit {\_Z}^{2}+3\right )}{x^{4}+x^{3}-3 x^{2}-2}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {-x^{4}-x^{3}+2}\, x +2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}+x^{3}-x^{2}-2}\right )}{2}\) | \(170\) |
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.43, 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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________