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

Optimal. Leaf size=14 \[ -\tanh ^{-1}\left (\frac {x}{\sqrt {x^6+1}}\right ) \]

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Rubi [F]  time = 0.56, 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 {-1+2 x^6}{\sqrt {1+x^6} \left (1-x^2+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

Rubi steps

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

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Mathematica [F]  time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-1+2 x^6}{\sqrt {1+x^6} \left (1-x^2+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 3.42, size = 14, normalized size = 1.00 \begin {gather*} -\tanh ^{-1}\left (\frac {x}{\sqrt {1+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTanh[x/Sqrt[1 + x^6]]

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fricas [B]  time = 0.51, size = 34, normalized size = 2.43 \begin {gather*} \frac {1}{2} \, \log \left (\frac {x^{6} + x^{2} - 2 \, \sqrt {x^{6} + 1} x + 1}{x^{6} - x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.19, size = 36, normalized size = 2.57

method	result	size
trager	\(-\frac {\ln \left (-\frac {x^{6}+2 \sqrt {x^{6}+1}\, x +x^{2}+1}{x^{6}-x^{2}+1}\right )}{2}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \begin {gather*} \int \frac {2\,x^6-1}{\sqrt {x^6+1}\,\left (x^6-x^2+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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