3.125 \(\int \frac {1}{\sqrt {1-(1-x^2)^3}} \, dx\)

Optimal. Leaf size=45 \[ -\frac {\tanh ^{-1}\left (\frac {x \left (6-3 x^2\right )}{2 \sqrt {3} \sqrt {x^6-3 x^4+3 x^2}}\right )}{2 \sqrt {3}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {1996, 1904, 206} \[ -\frac {\tanh ^{-1}\left (\frac {x \left (6-3 x^2\right )}{2 \sqrt {3} \sqrt {x^6-3 x^4+3 x^2}}\right )}{2 \sqrt {3}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[1 - (1 - x^2)^3],x]

[Out]

-ArcTanh[(x*(6 - 3*x^2))/(2*Sqrt[3]*Sqrt[3*x^2 - 3*x^4 + x^6])]/(2*Sqrt[3])

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 1904

Int[1/Sqrt[(a_.)*(x_)^2 + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(r_.)], x_Symbol] :> Dist[-2/(n - 2), Subst[Int[1/(4*a
- x^2), x], x, (x*(2*a + b*x^(n - 2)))/Sqrt[a*x^2 + b*x^n + c*x^r]], x] /; FreeQ[{a, b, c, n, r}, x] && EqQ[r
, 2*n - 2] && PosQ[n - 2] && NeQ[b^2 - 4*a*c, 0]

Rule 1996

Int[(u_)^(p_), x_Symbol] :> Int[ExpandToSum[u, x]^p, x] /; FreeQ[p, x] && GeneralizedTrinomialQ[u, x] &&  !Gen
eralizedTrinomialMatchQ[u, x]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1-\left (1-x^2\right )^3}} \, dx &=\int \frac {1}{\sqrt {3 x^2-3 x^4+x^6}} \, dx\\ &=-\operatorname {Subst}\left (\int \frac {1}{12-x^2} \, dx,x,\frac {x \left (6-3 x^2\right )}{\sqrt {3 x^2-3 x^4+x^6}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {x \left (6-3 x^2\right )}{2 \sqrt {3} \sqrt {3 x^2-3 x^4+x^6}}\right )}{2 \sqrt {3}}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 73, normalized size = 1.62 \[ -\frac {x \sqrt {x^4-3 x^2+3} \tanh ^{-1}\left (\frac {6-3 x^2}{2 \sqrt {3} \sqrt {x^4-3 x^2+3}}\right )}{2 \sqrt {3} \sqrt {x^2 \left (x^4-3 x^2+3\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[1 - (1 - x^2)^3],x]

[Out]

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

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fricas [A]  time = 0.87, size = 55, normalized size = 1.22 \[ \frac {1}{6} \, \sqrt {3} \log \left (-\frac {3 \, x^{3} + 2 \, \sqrt {3} {\left (x^{3} - 2 \, x\right )} + 2 \, \sqrt {x^{6} - 3 \, x^{4} + 3 \, x^{2}} {\left (\sqrt {3} + 2\right )} - 6 \, x}{x^{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.49, size = 60, normalized size = 1.33 \[ \frac {\sqrt {3} \log \left (x^{2} + \sqrt {3} - \sqrt {x^{4} - 3 \, x^{2} + 3}\right ) - \sqrt {3} \log \left (-x^{2} + \sqrt {3} + \sqrt {x^{4} - 3 \, x^{2} + 3}\right )}{6 \, \mathrm {sgn}\relax (x)} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(sqrt(3)*log(x^2 + sqrt(3) - sqrt(x^4 - 3*x^2 + 3)) - sqrt(3)*log(-x^2 + sqrt(3) + sqrt(x^4 - 3*x^2 + 3)))
/sgn(x)

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maple [A]  time = 0.00, size = 58, normalized size = 1.29 \[ \frac {\sqrt {x^{4}-3 x^{2}+3}\, \sqrt {3}\, x \arctanh \left (\frac {\left (x^{2}-2\right ) \sqrt {3}}{2 \sqrt {x^{4}-3 x^{2}+3}}\right )}{6 \sqrt {x^{6}-3 x^{4}+3 x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {{\left (x^{2} - 1\right )}^{3} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt((x^2 - 1)^3 + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {{\left (x^2-1\right )}^3+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((x^2 - 1)^3 + 1)^(1/2),x)

[Out]

int(1/((x^2 - 1)^3 + 1)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {1 - \left (1 - x^{2}\right )^{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-(-x**2+1)**3)**(1/2),x)

[Out]

Integral(1/sqrt(1 - (1 - x**2)**3), x)

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