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

Optimal. Leaf size=20 \[ \frac {\sqrt {x^6+1}}{x \left (x^2+1\right )} \]

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Rubi [F]  time = 0.59, 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}{x^2 \left (1+x^2\right ) \sqrt {1+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.11, size = 17, normalized size = 0.85 \begin {gather*} \frac {\sqrt {x^6+1}}{x^3+x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 + x^6]/(x + x^3)

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IntegrateAlgebraic [A]  time = 9.81, size = 20, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1+x^6}}{x \left (1+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 + x^6]/(x*(1 + x^2))

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fricas [A]  time = 0.46, size = 15, normalized size = 0.75 \begin {gather*} \frac {\sqrt {x^{6} + 1}}{x^{3} + x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^6 + 1)/(x^3 + x)

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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^{2} + 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [A]  time = 0.07, size = 19, normalized size = 0.95

method	result	size
trager	\(\frac {\sqrt {x^{6}+1}}{x \left (x^{2}+1\right )}\)	\(19\)
gosper	\(\frac {x^{4}-x^{2}+1}{x \sqrt {x^{6}+1}}\)	\(22\)
risch	\(\frac {x^{4}-x^{2}+1}{x \sqrt {x^{6}+1}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.82, size = 23, normalized size = 1.15 \begin {gather*} \frac {\sqrt {x^{4} - x^{2} + 1}}{\sqrt {x^{2} + 1} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x^4 - x^2 + 1)/(sqrt(x^2 + 1)*x)

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mupad [B]  time = 0.17, size = 18, normalized size = 0.90 \begin {gather*} \frac {\sqrt {x^6+1}}{x\,\left (x^2+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^6 + 1)^(1/2)/(x*(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}{x^{2} \sqrt {\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 1\right )} \left (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**2/(x**2+1)/(x**6+1)**(1/2),x)

[Out]

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

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