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

Optimal. Leaf size=30 \[ \frac {\sqrt {x^6-1}}{x^2}-\tanh ^{-1}\left (\frac {x^2}{\sqrt {x^6-1}}\right ) \]

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Rubi [F]  time = 1.01, 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*}

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Mathematica [C]  time = 5.57, size = 947, normalized size = 31.57

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)]*(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) + 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*(Roo
t[-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])

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IntegrateAlgebraic [A]  time = 7.06, size = 30, 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]]

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fricas [B]  time = 0.50, size = 53, normalized size = 1.77 \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

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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)

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maple [A]  time = 0.40, size = 53, normalized size = 1.77

method	result	size
trager	\(\frac {\sqrt {x^{6}-1}}{x^{2}}+\frac {\ln \left (\frac {-x^{6}-x^{4}+2 x^{2} \sqrt {x^{6}-1}+1}{x^{6}-x^{4}-1}\right )}{2}\)	\(53\)
risch	\(\frac {\sqrt {x^{6}-1}}{x^{2}}+\frac {\ln \left (-\frac {-x^{6}-x^{4}+2 x^{2} \sqrt {x^{6}-1}+1}{x^{6}-x^{4}-1}\right )}{2}\)	\(54\)

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

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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)

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

[Out]

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

[Out]

Timed out

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