3.376 \(\int \frac{1}{x^5 (1+3 x^4+x^8)} \, dx\)

Optimal. Leaf size=66 \[ -\frac{1}{4 x^4}+\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (2 x^4-\sqrt{5}+3\right )+\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}+3\right )-3 \log (x) \]

[Out]

-1/(4*x^4) - 3*Log[x] + ((15 + 7*Sqrt[5])*Log[3 - Sqrt[5] + 2*x^4])/40 + ((15 - 7*Sqrt[5])*Log[3 + Sqrt[5] + 2
*x^4])/40

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Rubi [A]  time = 0.06666, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {1357, 709, 800, 632, 31} \[ -\frac{1}{4 x^4}+\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (2 x^4-\sqrt{5}+3\right )+\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}+3\right )-3 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(1 + 3*x^4 + x^8)),x]

[Out]

-1/(4*x^4) - 3*Log[x] + ((15 + 7*Sqrt[5])*Log[3 - Sqrt[5] + 2*x^4])/40 + ((15 - 7*Sqrt[5])*Log[3 + Sqrt[5] + 2
*x^4])/40

Rule 1357

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

Rule 709

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

Rule 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + g*x))/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 632

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{x^5 \left (1+3 x^4+x^8\right )} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+3 x+x^2\right )} \, dx,x,x^4\right )\\ &=-\frac{1}{4 x^4}+\frac{1}{4} \operatorname{Subst}\left (\int \frac{-3-x}{x \left (1+3 x+x^2\right )} \, dx,x,x^4\right )\\ &=-\frac{1}{4 x^4}+\frac{1}{4} \operatorname{Subst}\left (\int \left (-\frac{3}{x}+\frac{8+3 x}{1+3 x+x^2}\right ) \, dx,x,x^4\right )\\ &=-\frac{1}{4 x^4}-3 \log (x)+\frac{1}{4} \operatorname{Subst}\left (\int \frac{8+3 x}{1+3 x+x^2} \, dx,x,x^4\right )\\ &=-\frac{1}{4 x^4}-3 \log (x)+\frac{1}{40} \left (15-7 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{3}{2}+\frac{\sqrt{5}}{2}+x} \, dx,x,x^4\right )+\frac{1}{40} \left (15+7 \sqrt{5}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{3}{2}-\frac{\sqrt{5}}{2}+x} \, dx,x,x^4\right )\\ &=-\frac{1}{4 x^4}-3 \log (x)+\frac{1}{40} \left (15+7 \sqrt{5}\right ) \log \left (3-\sqrt{5}+2 x^4\right )+\frac{1}{40} \left (15-7 \sqrt{5}\right ) \log \left (3+\sqrt{5}+2 x^4\right )\\ \end{align*}

Mathematica [A]  time = 0.0337213, size = 60, normalized size = 0.91 \[ \frac{1}{40} \left (-\frac{10}{x^4}+\left (15+7 \sqrt{5}\right ) \log \left (-2 x^4+\sqrt{5}-3\right )+\left (15-7 \sqrt{5}\right ) \log \left (2 x^4+\sqrt{5}+3\right )-120 \log (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(1 + 3*x^4 + x^8)),x]

[Out]

(-10/x^4 - 120*Log[x] + (15 + 7*Sqrt[5])*Log[-3 + Sqrt[5] - 2*x^4] + (15 - 7*Sqrt[5])*Log[3 + Sqrt[5] + 2*x^4]
)/40

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Maple [A]  time = 0.007, size = 42, normalized size = 0.6 \begin{align*}{\frac{3\,\ln \left ({x}^{8}+3\,{x}^{4}+1 \right ) }{8}}-{\frac{7\,\sqrt{5}}{20}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{4}+3 \right ) \sqrt{5}}{5}} \right ) }-{\frac{1}{4\,{x}^{4}}}-3\,\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(x^8+3*x^4+1),x)

[Out]

3/8*ln(x^8+3*x^4+1)-7/20*arctanh(1/5*(2*x^4+3)*5^(1/2))*5^(1/2)-1/4/x^4-3*ln(x)

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Maxima [A]  time = 1.47315, size = 76, normalized size = 1.15 \begin{align*} \frac{7}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) - \frac{1}{4 \, x^{4}} + \frac{3}{8} \, \log \left (x^{8} + 3 \, x^{4} + 1\right ) - \frac{3}{4} \, \log \left (x^{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^8+3*x^4+1),x, algorithm="maxima")

[Out]

7/40*sqrt(5)*log((2*x^4 - sqrt(5) + 3)/(2*x^4 + sqrt(5) + 3)) - 1/4/x^4 + 3/8*log(x^8 + 3*x^4 + 1) - 3/4*log(x
^4)

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Fricas [A]  time = 1.66243, size = 193, normalized size = 2.92 \begin{align*} \frac{7 \, \sqrt{5} x^{4} \log \left (\frac{2 \, x^{8} + 6 \, x^{4} - \sqrt{5}{\left (2 \, x^{4} + 3\right )} + 7}{x^{8} + 3 \, x^{4} + 1}\right ) + 15 \, x^{4} \log \left (x^{8} + 3 \, x^{4} + 1\right ) - 120 \, x^{4} \log \left (x\right ) - 10}{40 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^8+3*x^4+1),x, algorithm="fricas")

[Out]

1/40*(7*sqrt(5)*x^4*log((2*x^8 + 6*x^4 - sqrt(5)*(2*x^4 + 3) + 7)/(x^8 + 3*x^4 + 1)) + 15*x^4*log(x^8 + 3*x^4
+ 1) - 120*x^4*log(x) - 10)/x^4

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Sympy [A]  time = 0.193161, size = 65, normalized size = 0.98 \begin{align*} - 3 \log{\left (x \right )} + \left (\frac{3}{8} + \frac{7 \sqrt{5}}{40}\right ) \log{\left (x^{4} - \frac{\sqrt{5}}{2} + \frac{3}{2} \right )} + \left (\frac{3}{8} - \frac{7 \sqrt{5}}{40}\right ) \log{\left (x^{4} + \frac{\sqrt{5}}{2} + \frac{3}{2} \right )} - \frac{1}{4 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**5/(x**8+3*x**4+1),x)

[Out]

-3*log(x) + (3/8 + 7*sqrt(5)/40)*log(x**4 - sqrt(5)/2 + 3/2) + (3/8 - 7*sqrt(5)/40)*log(x**4 + sqrt(5)/2 + 3/2
) - 1/(4*x**4)

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Giac [A]  time = 1.23136, size = 85, normalized size = 1.29 \begin{align*} \frac{7}{40} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) + \frac{3 \, x^{4} - 1}{4 \, x^{4}} + \frac{3}{8} \, \log \left (x^{8} + 3 \, x^{4} + 1\right ) - \frac{3}{4} \, \log \left (x^{4}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(x^8+3*x^4+1),x, algorithm="giac")

[Out]

7/40*sqrt(5)*log((2*x^4 - sqrt(5) + 3)/(2*x^4 + sqrt(5) + 3)) + 1/4*(3*x^4 - 1)/x^4 + 3/8*log(x^8 + 3*x^4 + 1)
- 3/4*log(x^4)