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

Optimal. Leaf size=47 \[ -\frac {\sqrt {x^4-x^2+x-1} x}{x^4+x-1}-\tan ^{-1}\left (\frac {x}{\sqrt {x^4-x^2+x-1}}\right ) \]

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Rubi [F]  time = 0.60, 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 )^2} \, 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)^2,x]

[Out]

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

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Mathematica [C]  time = 6.32, size = 12187, normalized size = 259.30 \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)^2,x]

[Out]

Result too large to show

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IntegrateAlgebraic [A]  time = 0.14, size = 47, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt {-1+x-x^2+x^4}}{-1+x+x^4}-\tan ^{-1}\left (\frac {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)^2,x]

[Out]

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

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fricas [A]  time = 0.53, size = 64, normalized size = 1.36 \begin {gather*} -\frac {{\left (x^{4} + x - 1\right )} \arctan \left (\frac {2 \, \sqrt {x^{4} - x^{2} + x - 1} x}{x^{4} - 2 \, x^{2} + x - 1}\right ) + 2 \, \sqrt {x^{4} - x^{2} + x - 1} x}{2 \, {\left (x^{4} + 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)^2,x, algorithm="fricas")

[Out]

-1/2*((x^4 + x - 1)*arctan(2*sqrt(x^4 - x^2 + x - 1)*x/(x^4 - 2*x^2 + x - 1)) + 2*sqrt(x^4 - x^2 + x - 1)*x)/(
x^4 + 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 - 1\right )}^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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

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maple [C]  time = 12.64, size = 99, normalized size = 2.11

method	result	size
trager	\(-\frac {x \sqrt {x^{4}-x^{2}+x -1}}{x^{4}+x -1}+\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \sqrt {x^{4}-x^{2}+x -1}\, x -\RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{4}+x -1}\right )}{2}\)	\(99\)
risch	\(\text {Expression too large to display}\)	\(3619\)
elliptic	\(\text {Expression too large to display}\)	\(3619\)
default	\(\text {Expression too large to display}\)	\(6119\)

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)^2,x,method=_RETURNVERBOSE)

[Out]

-x*(x^4-x^2+x-1)^(1/2)/(x^4+x-1)+1/2*RootOf(_Z^2+1)*ln(-(RootOf(_Z^2+1)*x^4-2*RootOf(_Z^2+1)*x^2+RootOf(_Z^2+1
)*x-2*(x^4-x^2+x-1)^(1/2)*x-RootOf(_Z^2+1))/(x^4+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 - 1\right )}^{2}}\,{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)^2,x, algorithm="maxima")

[Out]

integrate((2*x^4 - x + 2)*sqrt(x^4 - x^2 + x - 1)/(x^4 + x - 1)^2, 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 )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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