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

Optimal. Leaf size=26 \[ \frac {\sqrt {x^6-1}}{x}+\tan ^{-1}\left (\frac {x}{\sqrt {x^6-1}}\right ) \]

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Rubi [F]  time = 0.77, 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 {-1+x^6} \left (1+2 x^6\right )}{x^2 \left (-1+x^2+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int \frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{x^2 \left (-1+x^2+x^6\right )} \, dx &=\int \left (-\frac {\sqrt {-1+x^6}}{x^2}+\frac {\left (1+3 x^4\right ) \sqrt {-1+x^6}}{-1+x^2+x^6}\right ) \, dx\\ &=-\int \frac {\sqrt {-1+x^6}}{x^2} \, dx+\int \frac {\left (1+3 x^4\right ) \sqrt {-1+x^6}}{-1+x^2+x^6} \, dx\\ &=\frac {\sqrt {-1+x^6}}{x}-3 \int \frac {x^4}{\sqrt {-1+x^6}} \, dx+\int \left (\frac {\sqrt {-1+x^6}}{-1+x^2+x^6}+\frac {3 x^4 \sqrt {-1+x^6}}{-1+x^2+x^6}\right ) \, dx\\ &=\frac {\sqrt {-1+x^6}}{x}+\frac {3}{2} \int \frac {-1+\sqrt {3}-2 x^4}{\sqrt {-1+x^6}} \, dx+3 \int \frac {x^4 \sqrt {-1+x^6}}{-1+x^2+x^6} \, dx+\frac {1}{2} \left (3 \left (1-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {-1+x^6}} \, dx+\int \frac {\sqrt {-1+x^6}}{-1+x^2+x^6} \, dx\\ &=\frac {\sqrt {-1+x^6}}{x}+\frac {3 \left (1+\sqrt {3}\right ) x \sqrt {-1+x^6}}{2 \left (1-\left (1+\sqrt {3}\right ) x^2\right )}+\frac {3 \sqrt [4]{3} x \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} E\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 )}{2 \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {-1+x^6}}+\frac {3^{3/4} \left (1-\sqrt {3}\right ) 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 )}{4 \sqrt {-\frac {x^2 \left (1-x^2\right )}{\left (1-\left (1+\sqrt {3}\right ) x^2\right )^2}} \sqrt {-1+x^6}}+3 \int \frac {x^4 \sqrt {-1+x^6}}{-1+x^2+x^6} \, dx+\int \frac {\sqrt {-1+x^6}}{-1+x^2+x^6} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.24, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-1+x^6} \left (1+2 x^6\right )}{x^2 \left (-1+x^2+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.63, size = 26, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-1+x^6}}{x}+\tan ^{-1}\left (\frac {x}{\sqrt {-1+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[-1 + x^6]/x + ArcTan[x/Sqrt[-1 + x^6]]

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fricas [A]  time = 0.84, size = 40, normalized size = 1.54 \begin {gather*} \frac {x \arctan \left (\frac {2 \, \sqrt {x^{6} - 1} x}{x^{6} - x^{2} - 1}\right ) + 2 \, \sqrt {x^{6} - 1}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.88, size = 71, normalized size = 2.73

method	result	size
trager	\(\frac {\sqrt {x^{6}-1}}{x}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{6}-1}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{6}+x^{2}-1}\right )}{2}\)	\(71\)
risch	\(\frac {\sqrt {x^{6}-1}}{x}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{6}-1}\, x +\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{6}+x^{2}-1}\right )}{2}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^6-1)^(1/2)/x-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 {{\left (2 \, x^{6} + 1\right )} \sqrt {x^{6} - 1}}{{\left (x^{6} + x^{2} - 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {x^6-1}\,\left (2\,x^6+1\right )}{x^2\,\left (x^6+x^2-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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