3.5.25 \(\int \frac {\sqrt {1-x^6} (2+x^6)}{x^3 (-1+x^4+x^6)} \, dx\)

Optimal. Leaf size=34 \[ \frac {\sqrt {1-x^6}}{x^2}-\tanh ^{-1}\left (\frac {x^2}{\sqrt {1-x^6}}\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 1.08, 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^6} \left (2+x^6\right )}{x^3 \left (-1+x^4+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [C]  time = 5.54, size = 892, normalized size = 26.24

result too large to display

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Sqrt[1 - x^6]/x^2 - (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)*((EllipticPi[(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)]*(2 + Root[-1 + #1^2 + #1^3 & , 1, 0]^3))/((-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) + Root[-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*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 & , 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])*(Root[-1 + #1^2 + #1^3 & , 1, 0] - Root[-1 + #1^2 + #1^3 & , 3, 0]))))/(3*Sqrt[1 - x^6])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 6.99, size = 34, normalized size = 1.00 \begin {gather*} \frac {\sqrt {1-x^6}}{x^2}-\tanh ^{-1}\left (\frac {x^2}{\sqrt {1-x^6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 - x^6]/x^2 - ArcTanh[x^2/Sqrt[1 - x^6]]

________________________________________________________________________________________

fricas [A]  time = 0.50, size = 58, normalized size = 1.71 \begin {gather*} \frac {x^{2} \log \left (-\frac {x^{6} - x^{4} + 2 \, \sqrt {-x^{6} + 1} x^{2} - 1}{x^{6} + x^{4} - 1}\right ) + 2 \, \sqrt {-x^{6} + 1}}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^6+1)^(1/2)*(x^6+2)/x^3/(x^6+x^4-1),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 2\right )} \sqrt {-x^{6} + 1}}{{\left (x^{6} + x^{4} - 1\right )} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^6+1)^(1/2)*(x^6+2)/x^3/(x^6+x^4-1),x, algorithm="giac")

[Out]

integrate((x^6 + 2)*sqrt(-x^6 + 1)/((x^6 + x^4 - 1)*x^3), x)

________________________________________________________________________________________

maple [A]  time = 0.57, size = 53, normalized size = 1.56

method	result	size
trager	\(\frac {\sqrt {-x^{6}+1}}{x^{2}}-\frac {\ln \left (\frac {-x^{6}+x^{4}+2 \sqrt {-x^{6}+1}\, x^{2}+1}{x^{6}+x^{4}-1}\right )}{2}\)	\(53\)
risch	\(-\frac {x^{6}-1}{x^{2} \sqrt {-x^{6}+1}}-\frac {\ln \left (-\frac {-x^{6}+x^{4}+2 \sqrt {-x^{6}+1}\, x^{2}+1}{x^{6}+x^{4}-1}\right )}{2}\)	\(60\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^6+1)^(1/2)*(x^6+2)/x^3/(x^6+x^4-1),x,method=_RETURNVERBOSE)

[Out]

(-x^6+1)^(1/2)/x^2-1/2*ln((-x^6+x^4+2*(-x^6+1)^(1/2)*x^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 )} \sqrt {-x^{6} + 1}}{{\left (x^{6} + x^{4} - 1\right )} x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-x^6+1)^(1/2)*(x^6+2)/x^3/(x^6+x^4-1),x, algorithm="maxima")

[Out]

integrate((x^6 + 2)*sqrt(-x^6 + 1)/((x^6 + x^4 - 1)*x^3), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {1-x^6}\,\left (x^6+2\right )}{x^3\,\left (x^6+x^4-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - x^6)^(1/2)*(x^6 + 2))/(x^3*(x^4 + x^6 - 1)),x)

[Out]

int(((1 - x^6)^(1/2)*(x^6 + 2))/(x^3*(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**6+1)**(1/2)*(x**6+2)/x**3/(x**6+x**4-1),x)

[Out]

Timed out

________________________________________________________________________________________