3.307 \(\int \frac{1}{x^6 (1-2 x^4+x^8)} \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{4 x^5 \left (1-x^4\right )}-\frac{9}{20 x^5}-\frac{9}{4 x}-\frac{9}{8} \tan ^{-1}(x)+\frac{9}{8} \tanh ^{-1}(x) \]

[Out]

-9/(20*x^5) - 9/(4*x) + 1/(4*x^5*(1 - x^4)) - (9*ArcTan[x])/8 + (9*ArcTanh[x])/8

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Rubi [A]  time = 0.0122302, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {28, 290, 325, 298, 203, 206} \[ \frac{1}{4 x^5 \left (1-x^4\right )}-\frac{9}{20 x^5}-\frac{9}{4 x}-\frac{9}{8} \tan ^{-1}(x)+\frac{9}{8} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^6*(1 - 2*x^4 + x^8)),x]

[Out]

-9/(20*x^5) - 9/(4*x) + 1/(4*x^5*(1 - x^4)) - (9*ArcTan[x])/8 + (9*ArcTanh[x])/8

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 290

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

Rule 325

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

Rule 298

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
&  !GtQ[a/b, 0]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{1}{x^6 \left (1-2 x^4+x^8\right )} \, dx &=\int \frac{1}{x^6 \left (-1+x^4\right )^2} \, dx\\ &=\frac{1}{4 x^5 \left (1-x^4\right )}-\frac{9}{4} \int \frac{1}{x^6 \left (-1+x^4\right )} \, dx\\ &=-\frac{9}{20 x^5}+\frac{1}{4 x^5 \left (1-x^4\right )}-\frac{9}{4} \int \frac{1}{x^2 \left (-1+x^4\right )} \, dx\\ &=-\frac{9}{20 x^5}-\frac{9}{4 x}+\frac{1}{4 x^5 \left (1-x^4\right )}-\frac{9}{4} \int \frac{x^2}{-1+x^4} \, dx\\ &=-\frac{9}{20 x^5}-\frac{9}{4 x}+\frac{1}{4 x^5 \left (1-x^4\right )}+\frac{9}{8} \int \frac{1}{1-x^2} \, dx-\frac{9}{8} \int \frac{1}{1+x^2} \, dx\\ &=-\frac{9}{20 x^5}-\frac{9}{4 x}+\frac{1}{4 x^5 \left (1-x^4\right )}-\frac{9}{8} \tan ^{-1}(x)+\frac{9}{8} \tanh ^{-1}(x)\\ \end{align*}

Mathematica [A]  time = 0.0209894, size = 51, normalized size = 1.19 \[ -\frac{x^3}{4 \left (x^4-1\right )}-\frac{1}{5 x^5}-\frac{2}{x}-\frac{9}{16} \log (1-x)+\frac{9}{16} \log (x+1)-\frac{9}{8} \tan ^{-1}(x) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^6*(1 - 2*x^4 + x^8)),x]

[Out]

-1/(5*x^5) - 2/x - x^3/(4*(-1 + x^4)) - (9*ArcTan[x])/8 - (9*Log[1 - x])/16 + (9*Log[1 + x])/16

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Maple [A]  time = 0.016, size = 52, normalized size = 1.2 \begin{align*} -{\frac{x}{8\,{x}^{2}+8}}-{\frac{9\,\arctan \left ( x \right ) }{8}}-{\frac{1}{5\,{x}^{5}}}-2\,{x}^{-1}-{\frac{1}{16+16\,x}}+{\frac{9\,\ln \left ( 1+x \right ) }{16}}-{\frac{1}{16\,x-16}}-{\frac{9\,\ln \left ( x-1 \right ) }{16}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^6/(x^8-2*x^4+1),x)

[Out]

-1/8*x/(x^2+1)-9/8*arctan(x)-1/5/x^5-2/x-1/16/(1+x)+9/16*ln(1+x)-1/16/(x-1)-9/16*ln(x-1)

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Maxima [A]  time = 1.51062, size = 57, normalized size = 1.33 \begin{align*} -\frac{45 \, x^{8} - 36 \, x^{4} - 4}{20 \,{\left (x^{9} - x^{5}\right )}} - \frac{9}{8} \, \arctan \left (x\right ) + \frac{9}{16} \, \log \left (x + 1\right ) - \frac{9}{16} \, \log \left (x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^8-2*x^4+1),x, algorithm="maxima")

[Out]

-1/20*(45*x^8 - 36*x^4 - 4)/(x^9 - x^5) - 9/8*arctan(x) + 9/16*log(x + 1) - 9/16*log(x - 1)

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Fricas [B]  time = 1.47535, size = 171, normalized size = 3.98 \begin{align*} -\frac{180 \, x^{8} - 144 \, x^{4} + 90 \,{\left (x^{9} - x^{5}\right )} \arctan \left (x\right ) - 45 \,{\left (x^{9} - x^{5}\right )} \log \left (x + 1\right ) + 45 \,{\left (x^{9} - x^{5}\right )} \log \left (x - 1\right ) - 16}{80 \,{\left (x^{9} - x^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^8-2*x^4+1),x, algorithm="fricas")

[Out]

-1/80*(180*x^8 - 144*x^4 + 90*(x^9 - x^5)*arctan(x) - 45*(x^9 - x^5)*log(x + 1) + 45*(x^9 - x^5)*log(x - 1) -
16)/(x^9 - x^5)

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Sympy [A]  time = 0.223132, size = 44, normalized size = 1.02 \begin{align*} - \frac{9 \log{\left (x - 1 \right )}}{16} + \frac{9 \log{\left (x + 1 \right )}}{16} - \frac{9 \operatorname{atan}{\left (x \right )}}{8} - \frac{45 x^{8} - 36 x^{4} - 4}{20 x^{9} - 20 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**6/(x**8-2*x**4+1),x)

[Out]

-9*log(x - 1)/16 + 9*log(x + 1)/16 - 9*atan(x)/8 - (45*x**8 - 36*x**4 - 4)/(20*x**9 - 20*x**5)

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Giac [A]  time = 1.1035, size = 58, normalized size = 1.35 \begin{align*} -\frac{x^{3}}{4 \,{\left (x^{4} - 1\right )}} - \frac{10 \, x^{4} + 1}{5 \, x^{5}} - \frac{9}{8} \, \arctan \left (x\right ) + \frac{9}{16} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{9}{16} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^6/(x^8-2*x^4+1),x, algorithm="giac")

[Out]

-1/4*x^3/(x^4 - 1) - 1/5*(10*x^4 + 1)/x^5 - 9/8*arctan(x) + 9/16*log(abs(x + 1)) - 9/16*log(abs(x - 1))