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

Optimal. Leaf size=36 \[ \frac{x^3}{3}-\frac{2 x}{x^2-2}+4 x-5 \sqrt{2} \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

[Out]

4*x + x^3/3 - (2*x)/(-2 + x^2) - 5*Sqrt[2]*ArcTanh[x/Sqrt[2]]

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Rubi [A]  time = 0.0147501, antiderivative size = 42, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {288, 302, 207} \[ \frac{x^5}{2 \left (2-x^2\right )}+\frac{5 x^3}{6}+5 x-5 \sqrt{2} \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[x^6/(-2 + x^2)^2,x]

[Out]

5*x + (5*x^3)/6 + x^5/(2*(2 - x^2)) - 5*Sqrt[2]*ArcTanh[x/Sqrt[2]]

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

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^6}{\left (-2+x^2\right )^2} \, dx &=\frac{x^5}{2 \left (2-x^2\right )}+\frac{5}{2} \int \frac{x^4}{-2+x^2} \, dx\\ &=\frac{x^5}{2 \left (2-x^2\right )}+\frac{5}{2} \int \left (2+x^2+\frac{4}{-2+x^2}\right ) \, dx\\ &=5 x+\frac{5 x^3}{6}+\frac{x^5}{2 \left (2-x^2\right )}+10 \int \frac{1}{-2+x^2} \, dx\\ &=5 x+\frac{5 x^3}{6}+\frac{x^5}{2 \left (2-x^2\right )}-5 \sqrt{2} \tanh ^{-1}\left (\frac{x}{\sqrt{2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0354436, size = 53, normalized size = 1.47 \[ \frac{x^3}{3}-\frac{2 x}{x^2-2}+4 x+\frac{5 \log \left (\sqrt{2}-x\right )}{\sqrt{2}}-\frac{5 \log \left (x+\sqrt{2}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(-2 + x^2)^2,x]

[Out]

4*x + x^3/3 - (2*x)/(-2 + x^2) + (5*Log[Sqrt[2] - x])/Sqrt[2] - (5*Log[Sqrt[2] + x])/Sqrt[2]

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Maple [A]  time = 0.006, size = 32, normalized size = 0.9 \begin{align*} 4\,x+{\frac{{x}^{3}}{3}}-2\,{\frac{x}{{x}^{2}-2}}-5\,{\it Artanh} \left ( 1/2\,x\sqrt{2} \right ) \sqrt{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x+1/3*x^3-2*x/(x^2-2)-5*arctanh(1/2*x*2^(1/2))*2^(1/2)

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Maxima [A]  time = 1.4058, size = 54, normalized size = 1.5 \begin{align*} \frac{1}{3} \, x^{3} + \frac{5}{2} \, \sqrt{2} \log \left (\frac{x - \sqrt{2}}{x + \sqrt{2}}\right ) + 4 \, x - \frac{2 \, x}{x^{2} - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 + 5/2*sqrt(2)*log((x - sqrt(2))/(x + sqrt(2))) + 4*x - 2*x/(x^2 - 2)

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Fricas [A]  time = 2.07184, size = 136, normalized size = 3.78 \begin{align*} \frac{2 \, x^{5} + 20 \, x^{3} + 15 \, \sqrt{2}{\left (x^{2} - 2\right )} \log \left (\frac{x^{2} - 2 \, \sqrt{2} x + 2}{x^{2} - 2}\right ) - 60 \, x}{6 \,{\left (x^{2} - 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*x^5 + 20*x^3 + 15*sqrt(2)*(x^2 - 2)*log((x^2 - 2*sqrt(2)*x + 2)/(x^2 - 2)) - 60*x)/(x^2 - 2)

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Sympy [A]  time = 0.102646, size = 49, normalized size = 1.36 \begin{align*} \frac{x^{3}}{3} + 4 x - \frac{2 x}{x^{2} - 2} + \frac{5 \sqrt{2} \log{\left (x - \sqrt{2} \right )}}{2} - \frac{5 \sqrt{2} \log{\left (x + \sqrt{2} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3/3 + 4*x - 2*x/(x**2 - 2) + 5*sqrt(2)*log(x - sqrt(2))/2 - 5*sqrt(2)*log(x + sqrt(2))/2

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Giac [A]  time = 1.054, size = 65, normalized size = 1.81 \begin{align*} \frac{1}{3} \, x^{3} + \frac{5}{2} \, \sqrt{2} \log \left (\frac{{\left | 2 \, x - 2 \, \sqrt{2} \right |}}{{\left | 2 \, x + 2 \, \sqrt{2} \right |}}\right ) + 4 \, x - \frac{2 \, x}{x^{2} - 2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 + 5/2*sqrt(2)*log(abs(2*x - 2*sqrt(2))/abs(2*x + 2*sqrt(2))) + 4*x - 2*x/(x^2 - 2)