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

Optimal. Leaf size=41 \[ \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-2}}\right )-\sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{2}{3}} x}{\sqrt{x^2-2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0218235, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {483, 217, 206, 377, 207} \[ \tanh ^{-1}\left (\frac{x}{\sqrt{x^2-2}}\right )-\sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{2}{3}} x}{\sqrt{x^2-2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 483

Int[(((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_)^(n_))^(q_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Dist[e^n/b, Int[
(e*x)^(m - n)*(c + d*x^n)^q, x], x] - Dist[(a*e^n)/b, Int[((e*x)^(m - n)*(c + d*x^n)^q)/(a + b*x^n), x], x] /;
FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1] && IntBinomialQ[a, b
, c, d, e, m, n, -1, q, x]

Rule 217

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

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 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 207

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

Rubi steps

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

Mathematica [A]  time = 0.0284174, size = 41, normalized size = 1. \[ \log \left (\sqrt{x^2-2}+x\right )-\sqrt{\frac{3}{2}} \tanh ^{-1}\left (\frac{\sqrt{\frac{2}{3}} x}{\sqrt{x^2-2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.036, size = 100, normalized size = 2.4 \begin{align*} \ln \left ( x+\sqrt{{x}^{2}-2} \right ) -{\frac{\sqrt{6}}{4}{\it Artanh} \left ({\frac{8+2\, \left ( x-\sqrt{6} \right ) \sqrt{6}}{4}{\frac{1}{\sqrt{ \left ( x-\sqrt{6} \right ) ^{2}+2\, \left ( x-\sqrt{6} \right ) \sqrt{6}+4}}}} \right ) }+{\frac{\sqrt{6}}{4}{\it Artanh} \left ({\frac{8-2\, \left ( x+\sqrt{6} \right ) \sqrt{6}}{4}{\frac{1}{\sqrt{ \left ( x+\sqrt{6} \right ) ^{2}-2\, \left ( x+\sqrt{6} \right ) \sqrt{6}+4}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^2-6)/(x^2-2)^(1/2),x)

[Out]

ln(x+(x^2-2)^(1/2))-1/4*6^(1/2)*arctanh(1/4*(8+2*(x-6^(1/2))*6^(1/2))/((x-6^(1/2))^2+2*(x-6^(1/2))*6^(1/2)+4)^
(1/2))+1/4*6^(1/2)*arctanh(1/4*(8-2*(x+6^(1/2))*6^(1/2))/((x+6^(1/2))^2-2*(x+6^(1/2))*6^(1/2)+4)^(1/2))

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Maxima [B]  time = 1.47848, size = 144, normalized size = 3.51 \begin{align*} \frac{1}{12} \, \sqrt{6}{\left (2 \, \sqrt{6} \log \left (x + \sqrt{x^{2} - 2}\right ) - 3 \, \log \left (\sqrt{6} + \frac{4 \, \sqrt{x^{2} - 2}}{{\left | 2 \, x - 2 \, \sqrt{6} \right |}} + \frac{8}{{\left | 2 \, x - 2 \, \sqrt{6} \right |}}\right ) + 3 \, \log \left (-\sqrt{6} + \frac{4 \, \sqrt{x^{2} - 2}}{{\left | 2 \, x + 2 \, \sqrt{6} \right |}} + \frac{8}{{\left | 2 \, x + 2 \, \sqrt{6} \right |}}\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.8899, size = 211, normalized size = 5.15 \begin{align*} \frac{1}{4} \, \sqrt{3} \sqrt{2} \log \left (-\frac{2 \, \sqrt{3} \sqrt{2}{\left (5 \, x^{2} - 6\right )} - 25 \, x^{2} + 2 \,{\left (5 \, \sqrt{3} \sqrt{2} x - 12 \, x\right )} \sqrt{x^{2} - 2} + 30}{x^{2} - 6}\right ) - \log \left (-x + \sqrt{x^{2} - 2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*sqrt(3)*sqrt(2)*log(-(2*sqrt(3)*sqrt(2)*(5*x^2 - 6) - 25*x^2 + 2*(5*sqrt(3)*sqrt(2)*x - 12*x)*sqrt(x^2 - 2
) + 30)/(x^2 - 6)) - log(-x + sqrt(x^2 - 2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\left (x^{2} - 6\right ) \sqrt{x^{2} - 2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(x**2-6)/(x**2-2)**(1/2),x)

[Out]

Integral(x**2/((x**2 - 6)*sqrt(x**2 - 2)), x)

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Giac [B]  time = 1.11784, size = 97, normalized size = 2.37 \begin{align*} -\frac{1}{4} \, \sqrt{6} \log \left (\frac{{\left | 2 \,{\left (x - \sqrt{x^{2} - 2}\right )}^{2} - 8 \, \sqrt{6} - 20 \right |}}{{\left | 2 \,{\left (x - \sqrt{x^{2} - 2}\right )}^{2} + 8 \, \sqrt{6} - 20 \right |}}\right ) - \frac{1}{2} \, \log \left ({\left (x - \sqrt{x^{2} - 2}\right )}^{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(6)*log(abs(2*(x - sqrt(x^2 - 2))^2 - 8*sqrt(6) - 20)/abs(2*(x - sqrt(x^2 - 2))^2 + 8*sqrt(6) - 20))
- 1/2*log((x - sqrt(x^2 - 2))^2)