3.9.86 \(\int \frac {\sqrt {1+x^2-2 x^6} (1+4 x^6)}{(-1-4 x^2+2 x^6) (-1-2 x^2+2 x^6)} \, dx\)

Optimal. Leaf size=67 \[ -\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt {-2 x^6+x^2+1}}\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {-2 x^6+x^2+1}}{2 x^6-x^2-1}\right ) \]

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Rubi [F]  time = 1.39, 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^2-2 x^6} \left (1+4 x^6\right )}{\left (-1-4 x^2+2 x^6\right ) \left (-1-2 x^2+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.31, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1+x^2-2 x^6} \left (1+4 x^6\right )}{\left (-1-4 x^2+2 x^6\right ) \left (-1-2 x^2+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(Sqrt[1 + x^2 - 2*x^6]*(1 + 4*x^6))/((-1 - 4*x^2 + 2*x^6)*(-1 - 2*x^2 + 2*x^6)),x]

[Out]

Integrate[(Sqrt[1 + x^2 - 2*x^6]*(1 + 4*x^6))/((-1 - 4*x^2 + 2*x^6)*(-1 - 2*x^2 + 2*x^6)), x]

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IntegrateAlgebraic [A]  time = 2.80, size = 67, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^2-2 x^6}}\right )-\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x \sqrt {1+x^2-2 x^6}}{-1-x^2+2 x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[1 + x^2 - 2*x^6]*(1 + 4*x^6))/((-1 - 4*x^2 + 2*x^6)*(-1 - 2*x^2 + 2*x^6)),x]

[Out]

-1/2*ArcTan[x/Sqrt[1 + x^2 - 2*x^6]] - (Sqrt[3]*ArcTan[(Sqrt[3]*x*Sqrt[1 + x^2 - 2*x^6])/(-1 - x^2 + 2*x^6)])/
2

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fricas [A]  time = 0.78, size = 66, normalized size = 0.99 \begin {gather*} -\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {-2 \, x^{6} + x^{2} + 1} x}{2 \, x^{6} + 2 \, x^{2} - 1}\right ) + \frac {1}{4} \, \arctan \left (\frac {2 \, \sqrt {-2 \, x^{6} + x^{2} + 1} x}{2 \, x^{6} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^6+x^2+1)^(1/2)*(4*x^6+1)/(2*x^6-4*x^2-1)/(2*x^6-2*x^2-1),x, algorithm="fricas")

[Out]

-1/4*sqrt(3)*arctan(2*sqrt(3)*sqrt(-2*x^6 + x^2 + 1)*x/(2*x^6 + 2*x^2 - 1)) + 1/4*arctan(2*sqrt(-2*x^6 + x^2 +
1)*x/(2*x^6 - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{6} + 1\right )} \sqrt {-2 \, x^{6} + x^{2} + 1}}{{\left (2 \, x^{6} - 2 \, x^{2} - 1\right )} {\left (2 \, x^{6} - 4 \, x^{2} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^6+x^2+1)^(1/2)*(4*x^6+1)/(2*x^6-4*x^2-1)/(2*x^6-2*x^2-1),x, algorithm="giac")

[Out]

integrate((4*x^6 + 1)*sqrt(-2*x^6 + x^2 + 1)/((2*x^6 - 2*x^2 - 1)*(2*x^6 - 4*x^2 - 1)), x)

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maple [C]  time = 0.58, size = 132, normalized size = 1.97

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}-2 \sqrt {-2 x^{6}+x^{2}+1}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{2 x^{6}-2 x^{2}-1}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{2}+3\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{6}+2 \RootOf \left (\textit {\_Z}^{2}+3\right ) x^{2}+6 \sqrt {-2 x^{6}+x^{2}+1}\, x -\RootOf \left (\textit {\_Z}^{2}+3\right )}{2 x^{6}-4 x^{2}-1}\right )}{4}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^6+x^2+1)^(1/2)*(4*x^6+1)/(2*x^6-4*x^2-1)/(2*x^6-2*x^2-1),x,method=_RETURNVERBOSE)

[Out]

-1/4*RootOf(_Z^2+1)*ln((2*RootOf(_Z^2+1)*x^6-2*(-2*x^6+x^2+1)^(1/2)*x-RootOf(_Z^2+1))/(2*x^6-2*x^2-1))-1/4*Roo
tOf(_Z^2+3)*ln(-(2*RootOf(_Z^2+3)*x^6+2*RootOf(_Z^2+3)*x^2+6*(-2*x^6+x^2+1)^(1/2)*x-RootOf(_Z^2+3))/(2*x^6-4*x
^2-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{6} + 1\right )} \sqrt {-2 \, x^{6} + x^{2} + 1}}{{\left (2 \, x^{6} - 2 \, x^{2} - 1\right )} {\left (2 \, x^{6} - 4 \, x^{2} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^6+x^2+1)^(1/2)*(4*x^6+1)/(2*x^6-4*x^2-1)/(2*x^6-2*x^2-1),x, algorithm="maxima")

[Out]

integrate((4*x^6 + 1)*sqrt(-2*x^6 + x^2 + 1)/((2*x^6 - 2*x^2 - 1)*(2*x^6 - 4*x^2 - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (4\,x^6+1\right )\,\sqrt {-2\,x^6+x^2+1}}{\left (-2\,x^6+2\,x^2+1\right )\,\left (-2\,x^6+4\,x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^6 + 1)*(x^2 - 2*x^6 + 1)^(1/2))/((2*x^2 - 2*x^6 + 1)*(4*x^2 - 2*x^6 + 1)),x)

[Out]

int(((4*x^6 + 1)*(x^2 - 2*x^6 + 1)^(1/2))/((2*x^2 - 2*x^6 + 1)*(4*x^2 - 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+x**2+1)**(1/2)*(4*x**6+1)/(2*x**6-4*x**2-1)/(2*x**6-2*x**2-1),x)

[Out]

Timed out

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