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

Optimal. Leaf size=55 \[ 2 \tan ^{-1}\left (\frac {x}{\sqrt {x^4-x^2-x-1}}\right )-2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4-x^2-x-1}}\right ) \]

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Rubi [F]  time = 1.71, 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-x^2+x^4} \left (2+x+2 x^4\right )}{\left (-1-x+x^4\right ) \left (-1-x+x^2+x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 6.66, size = 59389, normalized size = 1079.80 \begin {gather*} \text {Result too large to show} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.22, size = 55, normalized size = 1.00 \begin {gather*} 2 \tan ^{-1}\left (\frac {x}{\sqrt {-1-x-x^2+x^4}}\right )-2 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1-x-x^2+x^4}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

2*ArcTan[x/Sqrt[-1 - x - x^2 + x^4]] - 2*Sqrt[2]*ArcTan[(Sqrt[2]*x)/Sqrt[-1 - x - x^2 + x^4]]

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fricas [A]  time = 0.60, size = 77, normalized size = 1.40 \begin {gather*} -\sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} \sqrt {x^{4} - x^{2} - x - 1} x}{x^{4} - 3 \, x^{2} - x - 1}\right ) + \arctan \left (\frac {2 \, \sqrt {x^{4} - x^{2} - x - 1} x}{x^{4} - 2 \, x^{2} - x - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 17.70, size = 162, normalized size = 2.95

method	result	size
trager	\(\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \sqrt {x^{4}-x^{2}-x -1}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right ) x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}-x -1}\right )+\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {-\RootOf \left (\textit {\_Z}^{2}+2\right ) x^{4}+3 \RootOf \left (\textit {\_Z}^{2}+2\right ) x^{2}+4 \sqrt {x^{4}-x^{2}-x -1}\, x +\RootOf \left (\textit {\_Z}^{2}+2\right ) x +\RootOf \left (\textit {\_Z}^{2}+2\right )}{\left (-1+x \right ) \left (x^{3}+x^{2}+2 x +1\right )}\right )\)	\(162\)
default	\(\text {Expression too large to display}\)	\(7840\)
elliptic	\(\text {Expression too large to display}\)	\(1138832\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootOf(_Z^2+1)*ln(-(RootOf(_Z^2+1)*x^4-2*RootOf(_Z^2+1)*x^2+2*(x^4-x^2-x-1)^(1/2)*x-RootOf(_Z^2+1)*x-RootOf(_Z
^2+1))/(x^4-x-1))+RootOf(_Z^2+2)*ln(-(-RootOf(_Z^2+2)*x^4+3*RootOf(_Z^2+2)*x^2+4*(x^4-x^2-x-1)^(1/2)*x+RootOf(
_Z^2+2)*x+RootOf(_Z^2+2))/(-1+x)/(x^3+x^2+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^{4} + x + 2\right )} \sqrt {x^{4} - x^{2} - x - 1}}{{\left (x^{4} + x^{2} - x - 1\right )} {\left (x^{4} - x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\left (2\,x^4+x+2\right )\,\sqrt {x^4-x^2-x-1}}{\left (-x^4+x+1\right )\,\left (-x^4-x^2+x+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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