3.395 \(\int \frac{1}{x^7 (1-3 x^4+x^8)} \, dx\)

Optimal. Leaf size=97 \[ -\frac{3}{2 x^2}-\frac{1}{6 x^6}-\frac{1}{2} \sqrt{\frac{1}{10} \left (123-55 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{2} \sqrt{\frac{1}{10} \left (123+55 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right ) \]

[Out]

-1/(6*x^6) - 3/(2*x^2) - (Sqrt[(123 - 55*Sqrt[5])/10]*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + (Sqrt[(123 + 55*
Sqrt[5])/10]*ArcTanh[Sqrt[(3 + Sqrt[5])/2]*x^2])/2

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Rubi [A]  time = 0.0904821, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1359, 1123, 1281, 1166, 207} \[ -\frac{3}{2 x^2}-\frac{1}{6 x^6}-\frac{1}{2} \sqrt{\frac{1}{10} \left (123-55 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{2} \sqrt{\frac{1}{10} \left (123+55 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^7*(1 - 3*x^4 + x^8)),x]

[Out]

-1/(6*x^6) - 3/(2*x^2) - (Sqrt[(123 - 55*Sqrt[5])/10]*ArcTanh[Sqrt[2/(3 + Sqrt[5])]*x^2])/2 + (Sqrt[(123 + 55*
Sqrt[5])/10]*ArcTanh[Sqrt[(3 + Sqrt[5])/2]*x^2])/2

Rule 1359

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[
1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k) + c*x^((2*n)/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b,
c, p}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 1123

Int[((d_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*x^2 +
c*x^4)^(p + 1))/(a*d*(m + 1)), x] - Dist[1/(a*d^2*(m + 1)), Int[(d*x)^(m + 2)*(b*(m + 2*p + 3) + c*(m + 4*p +
5)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && In
tegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1281

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d*(
f*x)^(m + 1)*(a + b*x^2 + c*x^4)^(p + 1))/(a*f*(m + 1)), x] + Dist[1/(a*f^2*(m + 1)), Int[(f*x)^(m + 2)*(a + b
*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

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{1}{x^7 \left (1-3 x^4+x^8\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^4 \left (1-3 x^2+x^4\right )} \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}+\frac{1}{6} \operatorname{Subst}\left (\int \frac{9-3 x^2}{x^2 \left (1-3 x^2+x^4\right )} \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}-\frac{3}{2 x^2}-\frac{1}{6} \operatorname{Subst}\left (\int \frac{-24+9 x^2}{1-3 x^2+x^4} \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}-\frac{3}{2 x^2}-\frac{1}{20} \left (15-7 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{3}{2}-\frac{\sqrt{5}}{2}+x^2} \, dx,x,x^2\right )-\frac{1}{20} \left (15+7 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{3}{2}+\frac{\sqrt{5}}{2}+x^2} \, dx,x,x^2\right )\\ &=-\frac{1}{6 x^6}-\frac{3}{2 x^2}-\frac{1}{2} \sqrt{\frac{1}{10} \left (123-55 \sqrt{5}\right )} \tanh ^{-1}\left (\sqrt{\frac{2}{3+\sqrt{5}}} x^2\right )+\frac{1}{20} \sqrt{1230+550 \sqrt{5}} \tanh ^{-1}\left (\sqrt{\frac{1}{2} \left (3+\sqrt{5}\right )} x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0739215, size = 111, normalized size = 1.14 \[ \frac{1}{120} \left (-\frac{180}{x^2}-\frac{20}{x^6}-3 \left (25+11 \sqrt{5}\right ) \log \left (-2 x^2+\sqrt{5}-1\right )+3 \left (25-11 \sqrt{5}\right ) \log \left (-2 x^2+\sqrt{5}+1\right )+3 \left (25+11 \sqrt{5}\right ) \log \left (2 x^2+\sqrt{5}-1\right )+3 \left (11 \sqrt{5}-25\right ) \log \left (2 x^2+\sqrt{5}+1\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^7*(1 - 3*x^4 + x^8)),x]

[Out]

(-20/x^6 - 180/x^2 - 3*(25 + 11*Sqrt[5])*Log[-1 + Sqrt[5] - 2*x^2] + 3*(25 - 11*Sqrt[5])*Log[1 + Sqrt[5] - 2*x
^2] + 3*(25 + 11*Sqrt[5])*Log[-1 + Sqrt[5] + 2*x^2] + 3*(-25 + 11*Sqrt[5])*Log[1 + Sqrt[5] + 2*x^2])/120

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Maple [A]  time = 0.01, size = 72, normalized size = 0.7 \begin{align*}{\frac{5\,\ln \left ({x}^{4}-{x}^{2}-1 \right ) }{8}}+{\frac{11\,\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{5}}{5}} \right ) }-{\frac{1}{6\,{x}^{6}}}-{\frac{3}{2\,{x}^{2}}}-{\frac{5\,\ln \left ({x}^{4}+{x}^{2}-1 \right ) }{8}}+{\frac{11\,\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{5}}{5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^7/(x^8-3*x^4+1),x)

[Out]

5/8*ln(x^4-x^2-1)+11/20*5^(1/2)*arctanh(1/5*(2*x^2-1)*5^(1/2))-1/6/x^6-3/2/x^2-5/8*ln(x^4+x^2-1)+11/20*5^(1/2)
*arctanh(1/5*(2*x^2+1)*5^(1/2))

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Maxima [A]  time = 1.47745, size = 134, normalized size = 1.38 \begin{align*} -\frac{11}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5} + 1}{2 \, x^{2} + \sqrt{5} + 1}\right ) - \frac{11}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{2} - \sqrt{5} - 1}{2 \, x^{2} + \sqrt{5} - 1}\right ) - \frac{9 \, x^{4} + 1}{6 \, x^{6}} - \frac{5}{8} \, \log \left (x^{4} + x^{2} - 1\right ) + \frac{5}{8} \, \log \left (x^{4} - x^{2} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^8-3*x^4+1),x, algorithm="maxima")

[Out]

-11/40*sqrt(5)*log((2*x^2 - sqrt(5) + 1)/(2*x^2 + sqrt(5) + 1)) - 11/40*sqrt(5)*log((2*x^2 - sqrt(5) - 1)/(2*x
^2 + sqrt(5) - 1)) - 1/6*(9*x^4 + 1)/x^6 - 5/8*log(x^4 + x^2 - 1) + 5/8*log(x^4 - x^2 - 1)

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Fricas [B]  time = 1.71408, size = 327, normalized size = 3.37 \begin{align*} \frac{33 \, \sqrt{5} x^{6} \log \left (\frac{2 \, x^{4} + 2 \, x^{2} + \sqrt{5}{\left (2 \, x^{2} + 1\right )} + 3}{x^{4} + x^{2} - 1}\right ) + 33 \, \sqrt{5} x^{6} \log \left (\frac{2 \, x^{4} - 2 \, x^{2} + \sqrt{5}{\left (2 \, x^{2} - 1\right )} + 3}{x^{4} - x^{2} - 1}\right ) - 75 \, x^{6} \log \left (x^{4} + x^{2} - 1\right ) + 75 \, x^{6} \log \left (x^{4} - x^{2} - 1\right ) - 180 \, x^{4} - 20}{120 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^8-3*x^4+1),x, algorithm="fricas")

[Out]

1/120*(33*sqrt(5)*x^6*log((2*x^4 + 2*x^2 + sqrt(5)*(2*x^2 + 1) + 3)/(x^4 + x^2 - 1)) + 33*sqrt(5)*x^6*log((2*x
^4 - 2*x^2 + sqrt(5)*(2*x^2 - 1) + 3)/(x^4 - x^2 - 1)) - 75*x^6*log(x^4 + x^2 - 1) + 75*x^6*log(x^4 - x^2 - 1)
- 180*x^4 - 20)/x^6

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Sympy [B]  time = 0.516308, size = 197, normalized size = 2.03 \begin{align*} \left (\frac{11 \sqrt{5}}{40} + \frac{5}{8}\right ) \log{\left (x^{2} - \frac{2207}{22} - \frac{2207 \sqrt{5}}{50} + \frac{1152 \left (\frac{11 \sqrt{5}}{40} + \frac{5}{8}\right )^{3}}{11} \right )} + \left (\frac{5}{8} - \frac{11 \sqrt{5}}{40}\right ) \log{\left (x^{2} - \frac{2207}{22} + \frac{1152 \left (\frac{5}{8} - \frac{11 \sqrt{5}}{40}\right )^{3}}{11} + \frac{2207 \sqrt{5}}{50} \right )} + \left (- \frac{5}{8} + \frac{11 \sqrt{5}}{40}\right ) \log{\left (x^{2} - \frac{2207 \sqrt{5}}{50} + \frac{1152 \left (- \frac{5}{8} + \frac{11 \sqrt{5}}{40}\right )^{3}}{11} + \frac{2207}{22} \right )} + \left (- \frac{5}{8} - \frac{11 \sqrt{5}}{40}\right ) \log{\left (x^{2} + \frac{1152 \left (- \frac{5}{8} - \frac{11 \sqrt{5}}{40}\right )^{3}}{11} + \frac{2207 \sqrt{5}}{50} + \frac{2207}{22} \right )} - \frac{9 x^{4} + 1}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**7/(x**8-3*x**4+1),x)

[Out]

(11*sqrt(5)/40 + 5/8)*log(x**2 - 2207/22 - 2207*sqrt(5)/50 + 1152*(11*sqrt(5)/40 + 5/8)**3/11) + (5/8 - 11*sqr
t(5)/40)*log(x**2 - 2207/22 + 1152*(5/8 - 11*sqrt(5)/40)**3/11 + 2207*sqrt(5)/50) + (-5/8 + 11*sqrt(5)/40)*log
(x**2 - 2207*sqrt(5)/50 + 1152*(-5/8 + 11*sqrt(5)/40)**3/11 + 2207/22) + (-5/8 - 11*sqrt(5)/40)*log(x**2 + 115
2*(-5/8 - 11*sqrt(5)/40)**3/11 + 2207*sqrt(5)/50 + 2207/22) - (9*x**4 + 1)/(6*x**6)

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Giac [A]  time = 1.16999, size = 140, normalized size = 1.44 \begin{align*} -\frac{11}{40} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x^{2} - \sqrt{5} + 1 \right |}}{2 \, x^{2} + \sqrt{5} + 1}\right ) - \frac{11}{40} \, \sqrt{5} \log \left (\frac{{\left | 2 \, x^{2} - \sqrt{5} - 1 \right |}}{{\left | 2 \, x^{2} + \sqrt{5} - 1 \right |}}\right ) - \frac{9 \, x^{4} + 1}{6 \, x^{6}} - \frac{5}{8} \, \log \left ({\left | x^{4} + x^{2} - 1 \right |}\right ) + \frac{5}{8} \, \log \left ({\left | x^{4} - x^{2} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^7/(x^8-3*x^4+1),x, algorithm="giac")

[Out]

-11/40*sqrt(5)*log(abs(2*x^2 - sqrt(5) + 1)/(2*x^2 + sqrt(5) + 1)) - 11/40*sqrt(5)*log(abs(2*x^2 - sqrt(5) - 1
)/abs(2*x^2 + sqrt(5) - 1)) - 1/6*(9*x^4 + 1)/x^6 - 5/8*log(abs(x^4 + x^2 - 1)) + 5/8*log(abs(x^4 - x^2 - 1))