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

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

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Rubi [A]  time = 0.01, 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 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])

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]

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*}

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Mathematica [A]  time = 0.04, 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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IntegrateAlgebraic [A]  time = 0.05, size = 38, normalized size = 1.06 \[ \frac {x \left (x^4+10 x^2-30\right )}{3 \left (x^2-2\right )}-5 \sqrt {2} \tanh ^{-1}\left (\frac {x}{\sqrt {2}}\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^6/(-2 + x^2)^2,x]

[Out]

(x*(-30 + 10*x^2 + x^4))/(3*(-2 + x^2)) - 5*Sqrt[2]*ArcTanh[x/Sqrt[2]]

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fricas [A]  time = 1.32, size = 53, normalized size = 1.47 \[ \frac {2 \, x^{5} + 20 \, x^{3} + 15 \, \sqrt {2} {\left (x^{2} - 2\right )} \log \left (\frac {x^{2} - 2 \, \sqrt {2} x + 2}{x^{2} - 2}\right ) - 60 \, x}{6 \, {\left (x^{2} - 2\right )}} \]

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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giac [A]  time = 1.08, size = 48, normalized size = 1.33 \[ \frac {1}{3} \, x^{3} + \frac {5}{2} \, \sqrt {2} \log \left (\frac {{\left | 2 \, x - 2 \, \sqrt {2} \right |}}{{\left | 2 \, x + 2 \, \sqrt {2} \right |}}\right ) + 4 \, x - \frac {2 \, x}{x^{2} - 2} \]

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)

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maple [A]  time = 0.38, size = 32, normalized size = 0.89

method	result	size
default	\(4 x +\frac {x^{3}}{3}-\frac {2 x}{x^{2}-2}-5 \arctanh \left (\frac {x \sqrt {2}}{2}\right ) \sqrt {2}\)	\(32\)
risch	\(\frac {x^{3}}{3}+4 x -\frac {2 x}{x^{2}-2}+\frac {5 \sqrt {2}\, \ln \left (x -\sqrt {2}\right )}{2}-\frac {5 \sqrt {2}\, \ln \left (x +\sqrt {2}\right )}{2}\)	\(44\)
meijerg	\(i \sqrt {2}\, \left (-\frac {i x \sqrt {2}\, \left (-\frac {7}{2} x^{4}-35 x^{2}+105\right )}{42 \left (-\frac {x^{2}}{2}+1\right )}+5 i \arctanh \left (\frac {x \sqrt {2}}{2}\right )\right )\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(x^2-2)^2,x,method=_RETURNVERBOSE)

[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 = 0.97, size = 40, normalized size = 1.11 \[ \frac {1}{3} \, x^{3} + \frac {5}{2} \, \sqrt {2} \log \left (\frac {x - \sqrt {2}}{x + \sqrt {2}}\right ) + 4 \, x - \frac {2 \, x}{x^{2} - 2} \]

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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mupad [B]  time = 0.25, size = 33, normalized size = 0.92 \[ 4\,x-\frac {2\,x}{x^2-2}+\frac {x^3}{3}+\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,x\,1{}\mathrm {i}}{2}\right )\,5{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6/(x^2 - 2)^2,x)

[Out]

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

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sympy [A]  time = 0.12, size = 49, normalized size = 1.36 \[ \frac {x^{3}}{3} + 4 x - \frac {2 x}{x^{2} - 2} + \frac {5 \sqrt {2} \log {\left (x - \sqrt {2} \right )}}{2} - \frac {5 \sqrt {2} \log {\left (x + \sqrt {2} \right )}}{2} \]

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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