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

Optimal. Leaf size=24 \[ -\frac {x \sqrt {x^6+1}}{x^4-x^2+1} \]

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Rubi [F]  time = 0.97, 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 {-1-2 x^2+2 x^4}{\left (1-x^2+x^4\right ) \sqrt {1+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.35, size = 24, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt {x^6+1}}{x^4-x^2+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((x*Sqrt[1 + x^6])/(1 - x^2 + x^4))

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IntegrateAlgebraic [A]  time = 8.42, size = 24, normalized size = 1.00 \begin {gather*} -\frac {x \sqrt {1+x^6}}{1-x^2+x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((x*Sqrt[1 + x^6])/(1 - x^2 + x^4))

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fricas [A]  time = 0.47, size = 22, normalized size = 0.92 \begin {gather*} -\frac {\sqrt {x^{6} + 1} x}{x^{4} - x^{2} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(x^6 + 1)*x/(x^4 - x^2 + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 16, normalized size = 0.67

method	result	size
gosper	\(-\frac {\left (x^{2}+1\right ) x}{\sqrt {x^{6}+1}}\)	\(16\)
risch	\(-\frac {\left (x^{2}+1\right ) x}{\sqrt {x^{6}+1}}\)	\(16\)
trager	\(-\frac {x \sqrt {x^{6}+1}}{x^{4}-x^{2}+1}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.54, size = 26, normalized size = 1.08 \begin {gather*} -\frac {x^{3} + x}{\sqrt {x^{4} - x^{2} + 1} \sqrt {x^{2} + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.08, size = 22, normalized size = 0.92 \begin {gather*} -\frac {x\,\sqrt {x^6+1}}{x^4-x^2+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x*(x^6 + 1)^(1/2))/(x^4 - x^2 + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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