Optimal. Leaf size=14 \[ -\text {ArcTan}\left (\frac {x}{\sqrt {-1+x^6}}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F]
time = 0.36, 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 {1+2 x^6}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {1+2 x^6}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx &=\int \left (\frac {2}{\sqrt {-1+x^6}}+\frac {3-2 x^2}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt {-1+x^6}} \, dx+\int \frac {3-2 x^2}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx\\ &=\frac {x \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {-1+x^6}}+\int \left (\frac {3}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )}-\frac {2 x^2}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )}\right ) \, dx\\ &=\frac {x \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} F\left (\cos ^{-1}\left (\frac {1-\left (1-\sqrt {3}\right ) x^2}{1-\left (1+\sqrt {3}\right ) x^2}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{\sqrt [4]{3} \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {-1+x^6}}-2 \int \frac {x^2}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx+3 \int \frac {1}{\sqrt {-1+x^6} \left (-1+x^2+x^6\right )} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 2.74, size = 14, normalized size = 1.00 \begin {gather*} -\text {ArcTan}\left (\frac {x}{\sqrt {-1+x^6}}\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 = 0.59, size = 61, normalized size = 4.36
method | result | size |
trager | \(\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \sqrt {x^{6}-1}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{6}+x^{2}-1}\right )}{2}\) | \(61\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs.
\(2 (12) = 24\).
time = 0.41, size = 25, normalized size = 1.79 \begin {gather*} -\frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{6} - 1} x}{x^{6} - x^{2} - 1}\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 = -0.07 \begin {gather*} \int \frac {2\,x^6+1}{\sqrt {x^6-1}\,\left (x^6+x^2-1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________