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

Optimal. Leaf size=86 \[ -\frac{\sqrt{x^6-3 x^4+3 x^2} \left (3-2 x^2\right )}{8 x}-\frac{3 \sqrt{x^6-3 x^4+3 x^2} \sinh ^{-1}\left (\frac{3-2 x^2}{\sqrt{3}}\right )}{16 x \sqrt{x^4-3 x^2+3}} \]

[Out]

-((3 - 2*x^2)*Sqrt[3*x^2 - 3*x^4 + x^6])/(8*x) - (3*Sqrt[3*x^2 - 3*x^4 + x^6]*ArcSinh[(3 - 2*x^2)/Sqrt[3]])/(1
6*x*Sqrt[3 - 3*x^2 + x^4])

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Rubi [A]  time = 0.0413086, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {1903, 1107, 612, 619, 215} \[ -\frac{\sqrt{x^6-3 x^4+3 x^2} \left (3-2 x^2\right )}{8 x}-\frac{3 \sqrt{x^6-3 x^4+3 x^2} \sinh ^{-1}\left (\frac{3-2 x^2}{\sqrt{3}}\right )}{16 x \sqrt{x^4-3 x^2+3}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3*x^2 - 3*x^4 + x^6],x]

[Out]

-((3 - 2*x^2)*Sqrt[3*x^2 - 3*x^4 + x^6])/(8*x) - (3*Sqrt[3*x^2 - 3*x^4 + x^6]*ArcSinh[(3 - 2*x^2)/Sqrt[3]])/(1
6*x*Sqrt[3 - 3*x^2 + x^4])

Rule 1903

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

Rule 1107

Int[(x_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(a + b*x + c*x^2)^p, x],
x, x^2], x] /; FreeQ[{a, b, c, p}, x]

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \sqrt{3 x^2-3 x^4+x^6} \, dx &=\frac{\sqrt{3 x^2-3 x^4+x^6} \int x \sqrt{3-3 x^2+x^4} \, dx}{x \sqrt{3-3 x^2+x^4}}\\ &=\frac{\sqrt{3 x^2-3 x^4+x^6} \operatorname{Subst}\left (\int \sqrt{3-3 x+x^2} \, dx,x,x^2\right )}{2 x \sqrt{3-3 x^2+x^4}}\\ &=-\frac{\left (3-2 x^2\right ) \sqrt{3 x^2-3 x^4+x^6}}{8 x}+\frac{\left (3 \sqrt{3 x^2-3 x^4+x^6}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{3-3 x+x^2}} \, dx,x,x^2\right )}{16 x \sqrt{3-3 x^2+x^4}}\\ &=-\frac{\left (3-2 x^2\right ) \sqrt{3 x^2-3 x^4+x^6}}{8 x}+\frac{\left (\sqrt{3} \sqrt{3 x^2-3 x^4+x^6}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{3}}} \, dx,x,-3+2 x^2\right )}{16 x \sqrt{3-3 x^2+x^4}}\\ &=-\frac{\left (3-2 x^2\right ) \sqrt{3 x^2-3 x^4+x^6}}{8 x}-\frac{3 \sqrt{3 x^2-3 x^4+x^6} \sinh ^{-1}\left (\frac{3-2 x^2}{\sqrt{3}}\right )}{16 x \sqrt{3-3 x^2+x^4}}\\ \end{align*}

Mathematica [A]  time = 0.034346, size = 70, normalized size = 0.81 \[ \frac{x \left (4 x^6-18 x^4+30 x^2+3 \sqrt{x^4-3 x^2+3} \sinh ^{-1}\left (\frac{2 x^2-3}{\sqrt{3}}\right )-18\right )}{16 \sqrt{x^2 \left (x^4-3 x^2+3\right )}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3*x^2 - 3*x^4 + x^6],x]

[Out]

(x*(-18 + 30*x^2 - 18*x^4 + 4*x^6 + 3*Sqrt[3 - 3*x^2 + x^4]*ArcSinh[(-3 + 2*x^2)/Sqrt[3]]))/(16*Sqrt[x^2*(3 -
3*x^2 + x^4)])

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Maple [A]  time = 0.007, size = 81, normalized size = 0.9 \begin{align*}{\frac{1}{16\,x}\sqrt{{x}^{6}-3\,{x}^{4}+3\,{x}^{2}} \left ( 4\,\sqrt{{x}^{4}-3\,{x}^{2}+3}{x}^{2}-6\,\sqrt{{x}^{4}-3\,{x}^{2}+3}+3\,{\it Arcsinh} \left ( 1/3\,\sqrt{3} \left ( 2\,{x}^{2}-3 \right ) \right ) \right ){\frac{1}{\sqrt{{x}^{4}-3\,{x}^{2}+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(x^6-3*x^4+3*x^2)^(1/2)*(4*(x^4-3*x^2+3)^(1/2)*x^2-6*(x^4-3*x^2+3)^(1/2)+3*arcsinh(1/3*3^(1/2)*(2*x^2-3))
)/x/(x^4-3*x^2+3)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x^{6} - 3 \, x^{4} + 3 \, x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.29569, size = 157, normalized size = 1.83 \begin{align*} -\frac{12 \, x \log \left (-\frac{2 \, x^{3} - 3 \, x - 2 \, \sqrt{x^{6} - 3 \, x^{4} + 3 \, x^{2}}}{x}\right ) - 8 \, \sqrt{x^{6} - 3 \, x^{4} + 3 \, x^{2}}{\left (2 \, x^{2} - 3\right )} - 9 \, x}{64 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(12*x*log(-(2*x^3 - 3*x - 2*sqrt(x^6 - 3*x^4 + 3*x^2))/x) - 8*sqrt(x^6 - 3*x^4 + 3*x^2)*(2*x^2 - 3) - 9*
x)/x

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{x^{6} - 3 x^{4} + 3 x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x**6 - 3*x**4 + 3*x**2), x)

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Giac [A]  time = 1.08852, size = 93, normalized size = 1.08 \begin{align*} \frac{1}{16} \,{\left (2 \, \sqrt{x^{4} - 3 \, x^{2} + 3}{\left (2 \, x^{2} - 3\right )} - 3 \, \log \left (-2 \, x^{2} + 2 \, \sqrt{x^{4} - 3 \, x^{2} + 3} + 3\right )\right )} \mathrm{sgn}\left (x\right ) + \frac{3}{16} \,{\left (2 \, \sqrt{3} + \log \left (2 \, \sqrt{3} + 3\right )\right )} \mathrm{sgn}\left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(2*sqrt(x^4 - 3*x^2 + 3)*(2*x^2 - 3) - 3*log(-2*x^2 + 2*sqrt(x^4 - 3*x^2 + 3) + 3))*sgn(x) + 3/16*(2*sqrt
(3) + log(2*sqrt(3) + 3))*sgn(x)