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

Optimal. Leaf size=51 \[ \tan ^{-1}\left (\frac {x}{\sqrt {x^6-x^2-1}}\right )-\sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {x^6-x^2-1}}\right ) \]

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Rubi [F]  time = 1.56, 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+x^6} \left (1+2 x^6\right )}{\left (-1+x^6\right ) \left (-2+x^2+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [F]  time = 0.44, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {-1-x^2+x^6} \left (1+2 x^6\right )}{\left (-1+x^6\right ) \left (-2+x^2+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 2.49, size = 51, normalized size = 1.00 \begin {gather*} \tan ^{-1}\left (\frac {x}{\sqrt {-1-x^2+x^6}}\right )-\sqrt {\frac {3}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1-x^2+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.61, size = 75, normalized size = 1.47 \begin {gather*} -\frac {1}{4} \, \sqrt {3} \sqrt {2} \arctan \left (\frac {2 \, \sqrt {3} \sqrt {2} \sqrt {x^{6} - x^{2} - 1} x}{2 \, x^{6} - 5 \, x^{2} - 2}\right ) + \frac {1}{2} \, \arctan \left (\frac {2 \, \sqrt {x^{6} - x^{2} - 1} x}{x^{6} - 2 \, x^{2} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(3)*sqrt(2)*arctan(2*sqrt(3)*sqrt(2)*sqrt(x^6 - x^2 - 1)*x/(2*x^6 - 5*x^2 - 2)) + 1/2*arctan(2*sqrt(x
^6 - x^2 - 1)*x/(x^6 - 2*x^2 - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.54, size = 154, normalized size = 3.02

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{2}+6\right ) \ln \left (-\frac {2 \RootOf \left (\textit {\_Z}^{2}+6\right ) x^{6}-5 \RootOf \left (\textit {\_Z}^{2}+6\right ) x^{2}+12 \sqrt {x^{6}-x^{2}-1}\, x -2 \RootOf \left (\textit {\_Z}^{2}+6\right )}{2 x^{6}+x^{2}-2}\right )}{4}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \sqrt {x^{6}-x^{2}-1}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+x +1\right ) \left (x^{2}-x +1\right )}\right )}{2}\)	\(154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*RootOf(_Z^2+6)*ln(-(2*RootOf(_Z^2+6)*x^6-5*RootOf(_Z^2+6)*x^2+12*(x^6-x^2-1)^(1/2)*x-2*RootOf(_Z^2+6))/(2
*x^6+x^2-2))-1/2*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)*x^6-2*RootOf(_Z^2+1)*x^2-2*(x^6-x^2-1)^(1/2)*x-RootOf(_Z^2+
1))/(-1+x)/(1+x)/(x^2+x+1)/(x^2-x+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} - x^{2} - 1}}{{\left (2 \, x^{6} + x^{2} - 2\right )} {\left (x^{6} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (2\,x^6+1\right )\,\sqrt {x^6-x^2-1}}{\left (x^6-1\right )\,\left (2\,x^6+x^2-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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