3.1.64 \(\int \frac {1+2 x^6}{\sqrt {-1+x^6} (-1+x^2+x^6)} \, dx\)

Optimal. Leaf size=14 \[ -\tan ^{-1}\left (\frac {x}{\sqrt {x^6-1}}\right ) \]

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Rubi [F]  time = 0.52, 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]

Int[(1 + 2*x^6)/(Sqrt[-1 + x^6]*(-1 + x^2 + x^6)),x]

[Out]

(x*(1 - x^2)*Sqrt[(1 + x^2 + x^4)/(1 - (1 + Sqrt[3])*x^2)^2]*EllipticF[ArcCos[(1 - (1 - Sqrt[3])*x^2)/(1 - (1
+ Sqrt[3])*x^2)], (2 + Sqrt[3])/4])/(3^(1/4)*Sqrt[-((x^2*(1 - x^2))/(1 - (1 + Sqrt[3])*x^2)^2)]*Sqrt[-1 + x^6]
) + 3*Defer[Int][1/(Sqrt[-1 + x^6]*(-1 + x^2 + x^6)), x] - 2*Defer[Int][x^2/(Sqrt[-1 + x^6]*(-1 + x^2 + x^6)),
x]

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*}

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Mathematica [F]  time = 0.13, size = 0, normalized size = 0.00 \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]

Integrate[(1 + 2*x^6)/(Sqrt[-1 + x^6]*(-1 + x^2 + x^6)),x]

[Out]

Integrate[(1 + 2*x^6)/(Sqrt[-1 + x^6]*(-1 + x^2 + x^6)), x]

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IntegrateAlgebraic [A]  time = 3.38, size = 14, normalized size = 1.00 \begin {gather*} -\tan ^{-1}\left (\frac {x}{\sqrt {-1+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + 2*x^6)/(Sqrt[-1 + x^6]*(-1 + x^2 + x^6)),x]

[Out]

-ArcTan[x/Sqrt[-1 + x^6]]

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fricas [B]  time = 0.60, 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]

integrate((2*x^6+1)/(x^6-1)^(1/2)/(x^6+x^2-1),x, algorithm="fricas")

[Out]

-1/2*arctan(2*sqrt(x^6 - 1)*x/(x^6 - x^2 - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{6} + 1}{{\left (x^{6} + x^{2} - 1\right )} \sqrt {x^{6} - 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+1)/(x^6-1)^(1/2)/(x^6+x^2-1),x, algorithm="giac")

[Out]

integrate((2*x^6 + 1)/((x^6 + x^2 - 1)*sqrt(x^6 - 1)), x)

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maple [C]  time = 0.33, size = 62, normalized size = 4.43

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{6}-1}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{6}+x^{2}-1}\right )}{2}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^6+1)/(x^6-1)^(1/2)/(x^6+x^2-1),x,method=_RETURNVERBOSE)

[Out]

-1/2*RootOf(_Z^2+1)*ln(-(RootOf(_Z^2+1)*x^6-RootOf(_Z^2+1)*x^2+2*(x^6-1)^(1/2)*x-RootOf(_Z^2+1))/(x^6+x^2-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{6} + 1}{{\left (x^{6} + x^{2} - 1\right )} \sqrt {x^{6} - 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^6+1)/(x^6-1)^(1/2)/(x^6+x^2-1),x, algorithm="maxima")

[Out]

integrate((2*x^6 + 1)/((x^6 + x^2 - 1)*sqrt(x^6 - 1)), x)

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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]

int((2*x^6 + 1)/((x^6 - 1)^(1/2)*(x^2 + x^6 - 1)),x)

[Out]

int((2*x^6 + 1)/((x^6 - 1)^(1/2)*(x^2 + x^6 - 1)), x)

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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.

[In]

integrate((2*x**6+1)/(x**6-1)**(1/2)/(x**6+x**2-1),x)

[Out]

Timed out

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