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

Optimal. Leaf size=16 \[ -\tanh ^{-1}\left (\frac {x^2}{\sqrt {x^6-1}}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 0.75, 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 {x \left (2+x^6\right )}{\sqrt {-1+x^6} \left (-1-x^4+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-((Sqrt[2 - Sqrt[3]]*(1 - x^2)*Sqrt[(1 + x^2 + x^4)/(1 - Sqrt[3] - x^2)^2]*EllipticF[ArcSin[(1 + Sqrt[3] - x^2
)/(1 - Sqrt[3] - x^2)], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqrt[-((1 - x^2)/(1 - Sqrt[3] - x^2)^2)]*Sqrt[-1 + x^6])) +
(3*Defer[Subst][Defer[Int][1/(Sqrt[-1 + x^3]*(-1 - x^2 + x^3)), x], x, x^2])/2 + Defer[Subst][Defer[Int][x^2/(
Sqrt[-1 + x^3]*(-1 - x^2 + x^3)), x], x, x^2]/2

Rubi steps

\begin {align*} \int \frac {x \left (2+x^6\right )}{\sqrt {-1+x^6} \left (-1-x^4+x^6\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {2+x^3}{\sqrt {-1+x^3} \left (-1-x^2+x^3\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{\sqrt {-1+x^3}}+\frac {3+x^2}{\sqrt {-1+x^3} \left (-1-x^2+x^3\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {3+x^2}{\sqrt {-1+x^3} \left (-1-x^2+x^3\right )} \, dx,x,x^2\right )\\ &=-\frac {\sqrt {2-\sqrt {3}} \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\sqrt {3}-x^2\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x^2}{1-\sqrt {3}-x^2}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x^2}{\left (1-\sqrt {3}-x^2\right )^2}} \sqrt {-1+x^6}}+\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {3}{\sqrt {-1+x^3} \left (-1-x^2+x^3\right )}+\frac {x^2}{\sqrt {-1+x^3} \left (-1-x^2+x^3\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {\sqrt {2-\sqrt {3}} \left (1-x^2\right ) \sqrt {\frac {1+x^2+x^4}{\left (1-\sqrt {3}-x^2\right )^2}} F\left (\sin ^{-1}\left (\frac {1+\sqrt {3}-x^2}{1-\sqrt {3}-x^2}\right )|-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-\frac {1-x^2}{\left (1-\sqrt {3}-x^2\right )^2}} \sqrt {-1+x^6}}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {-1+x^3} \left (-1-x^2+x^3\right )} \, dx,x,x^2\right )+\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^3} \left (-1-x^2+x^3\right )} \, dx,x,x^2\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 5.62, size = 933, normalized size = 58.31

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[(1 - x^2)/(1 + (-1)^(1/3))]*Sqrt[1 + x^2 + x^4]*(-((Sqrt[3]*(I*Sqrt[3] + (1 + (-1)^(1/3))*x^2)*EllipticF
[ArcSin[Sqrt[(1 - (-1)^(2/3)*x^2)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-1 + (-1)^(2/3)*x^2)) - ((3*I)*((EllipticP
i[(I*Sqrt[3])/((-1)^(1/3) + Root[-1 - #1^2 + #1^3 & , 1, 0]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x^2)/(1 + (-1)^(1/3)
)]], (-1)^(1/3)]*(3 + Root[-1 - #1^2 + #1^3 & , 1, 0]^2))/((-1)^(1/3) + Root[-1 - #1^2 + #1^3 & , 1, 0]) + (2*
EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + Root[-1 - #1^2 + #1^3 & , 3, 0]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x^2)/(1 + (
-1)^(1/3))]], (-1)^(1/3)]*(Root[-1 - #1^2 + #1^3 & , 1, 0] - Root[-1 - #1^2 + #1^3 & , 2, 0])*((-1)^(1/3) + Ro
ot[-1 - #1^2 + #1^3 & , 2, 0]) + EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + Root[-1 - #1^2 + #1^3 & , 3, 0]), ArcSin
[Sqrt[(1 - (-1)^(2/3)*x^2)/(1 + (-1)^(1/3))]], (-1)^(1/3)]*(Root[-1 - #1^2 + #1^3 & , 1, 0] - Root[-1 - #1^2 +
#1^3 & , 2, 0])*((-1)^(1/3) + Root[-1 - #1^2 + #1^3 & , 2, 0])*Root[-1 - #1^2 + #1^3 & , 3, 0]^3 - 2*Elliptic
Pi[(I*Sqrt[3])/((-1)^(1/3) + Root[-1 - #1^2 + #1^3 & , 2, 0]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x^2)/(1 + (-1)^(1/3
))]], (-1)^(1/3)]*(Root[-1 - #1^2 + #1^3 & , 1, 0] - Root[-1 - #1^2 + #1^3 & , 3, 0])*((-1)^(1/3) + Root[-1 -
#1^2 + #1^3 & , 3, 0]) - EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + Root[-1 - #1^2 + #1^3 & , 2, 0]), ArcSin[Sqrt[(1
- (-1)^(2/3)*x^2)/(1 + (-1)^(1/3))]], (-1)^(1/3)]*Root[-1 - #1^2 + #1^3 & , 2, 0]^3*(Root[-1 - #1^2 + #1^3 &
, 1, 0] - Root[-1 - #1^2 + #1^3 & , 3, 0])*((-1)^(1/3) + Root[-1 - #1^2 + #1^3 & , 3, 0]))/(((-1)^(1/3) + Root
[-1 - #1^2 + #1^3 & , 2, 0])*(Root[-1 - #1^2 + #1^3 & , 2, 0] - Root[-1 - #1^2 + #1^3 & , 3, 0])*((-1)^(1/3) +
Root[-1 - #1^2 + #1^3 & , 3, 0]))))/((Root[-1 - #1^2 + #1^3 & , 1, 0] - Root[-1 - #1^2 + #1^3 & , 2, 0])*(Roo
t[-1 - #1^2 + #1^3 & , 1, 0] - Root[-1 - #1^2 + #1^3 & , 3, 0]))))/(3*Sqrt[-1 + x^6])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 15.17, size = 16, normalized size = 1.00 \begin {gather*} -\tanh ^{-1}\left (\frac {x^2}{\sqrt {-1+x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTanh[x^2/Sqrt[-1 + x^6]]

________________________________________________________________________________________

fricas [B]  time = 0.50, size = 36, normalized size = 2.25 \begin {gather*} \frac {1}{2} \, \log \left (\frac {x^{6} + x^{4} - 2 \, \sqrt {x^{6} - 1} x^{2} - 1}{x^{6} - x^{4} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.20, size = 42, normalized size = 2.62

method	result	size
trager	\(\frac {\ln \left (-\frac {-x^{6}-x^{4}+2 x^{2} \sqrt {x^{6}-1}+1}{x^{6}-x^{4}-1}\right )}{2}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int -\frac {x\,\left (x^6+2\right )}{\sqrt {x^6-1}\,\left (-x^6+x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________