3.4.76 \(\int \frac {1}{x^5 (1+3 x^4+x^8)} \, dx\) [376]

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

[Out]

-1/4/x^4-3*ln(x)+1/40*ln(2*x^4+5^(1/2)+3)*(15-7*5^(1/2))+1/40*ln(2*x^4-5^(1/2)+3)*(15+7*5^(1/2))

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Rubi [A]
time = 0.04, antiderivative size = 66, normalized size of antiderivative = 1.00, 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 = {1371, 723, 814, 646, 31} \begin {gather*} -\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) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(1 + 3*x^4 + x^8)),x]

[Out]

-1/4*1/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 31

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

Rule 646

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 723

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 814

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 1371

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]]

Rubi steps

\begin {align*} \int \frac {1}{x^5 \left (1+3 x^4+x^8\right )} \, dx &=\frac {1}{4} \text {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} \text {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} \text {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} \text {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 ) \text {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 ) \text {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*}

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Mathematica [A]
time = 0.02, size = 60, normalized size = 0.91 \begin {gather*} \frac {1}{40} \left (-\frac {10}{x^4}-120 \log (x)+\left (15+7 \sqrt {5}\right ) \log \left (-3+\sqrt {5}-2 x^4\right )+\left (15-7 \sqrt {5}\right ) \log \left (3+\sqrt {5}+2 x^4\right )\right ) \end {gather*}

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 = 0.03, size = 42, normalized size = 0.64

method	result	size
default	\(-\frac {1}{4 x^{4}}-3 \ln \left (x \right )+\frac {3 \ln \left (x^{8}+3 x^{4}+1\right )}{8}-\frac {7 \arctanh \left (\frac {\left (2 x^{4}+3\right ) \sqrt {5}}{5}\right ) \sqrt {5}}{20}\)	\(42\)
risch	\(-\frac {1}{4 x^{4}}-3 \ln \left (x \right )+\frac {3 \ln \left (7 x^{4}+\frac {21}{2}-\frac {7 \sqrt {5}}{2}\right )}{8}+\frac {7 \ln \left (7 x^{4}+\frac {21}{2}-\frac {7 \sqrt {5}}{2}\right ) \sqrt {5}}{40}+\frac {3 \ln \left (7 x^{4}+\frac {21}{2}+\frac {7 \sqrt {5}}{2}\right )}{8}-\frac {7 \ln \left (7 x^{4}+\frac {21}{2}+\frac {7 \sqrt {5}}{2}\right ) \sqrt {5}}{40}\)	\(77\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.64, size = 56, normalized size = 0.85 \begin {gather*} \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 {gather*}

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 = 0.40, size = 76, normalized size = 1.15 \begin {gather*} \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 {gather*}

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 = 0.07, size = 65, normalized size = 0.98 \begin {gather*} - 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 {gather*}

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 = 4.00, size = 63, normalized size = 0.95 \begin {gather*} \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 {gather*}

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)

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Mupad [B]
time = 1.36, size = 49, normalized size = 0.74 \begin {gather*} \ln \left (x^4-\frac {\sqrt {5}}{2}+\frac {3}{2}\right )\,\left (\frac {7\,\sqrt {5}}{40}+\frac {3}{8}\right )-\frac {1}{4\,x^4}-3\,\ln \left (x\right )-\ln \left (x^4+\frac {\sqrt {5}}{2}+\frac {3}{2}\right )\,\left (\frac {7\,\sqrt {5}}{40}-\frac {3}{8}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^5*(3*x^4 + x^8 + 1)),x)

[Out]

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

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