\(\int \frac {x^7}{1-3 x^4+x^8} \, dx\) [388]

Optimal result

Integrand size = 16, antiderivative size = 55 \[ \int \frac {x^7}{1-3 x^4+x^8} \, dx=\frac {1}{40} \left (5-3 \sqrt {5}\right ) \log \left (3-\sqrt {5}-2 x^4\right )+\frac {1}{40} \left (5+3 \sqrt {5}\right ) \log \left (3+\sqrt {5}-2 x^4\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1371, 646, 31} \[ \int \frac {x^7}{1-3 x^4+x^8} \, dx=\frac {1}{40} \left (5-3 \sqrt {5}\right ) \log \left (-2 x^4-\sqrt {5}+3\right )+\frac {1}{40} \left (5+3 \sqrt {5}\right ) \log \left (-2 x^4+\sqrt {5}+3\right ) \]

[In]

Int[x^7/(1 - 3*x^4 + x^8),x]

[Out]

((5 - 3*Sqrt[5])*Log[3 - Sqrt[5] - 2*x^4])/40 + ((5 + 3*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 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*} \text {integral}& = \frac {1}{4} \text {Subst}\left (\int \frac {x}{1-3 x+x^2} \, dx,x,x^4\right ) \\ & = \frac {1}{40} \left (5-3 \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 (5+3 \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 (5-3 \sqrt {5}\right ) \log \left (3-\sqrt {5}-2 x^4\right )+\frac {1}{40} \left (5+3 \sqrt {5}\right ) \log \left (3+\sqrt {5}-2 x^4\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \frac {x^7}{1-3 x^4+x^8} \, dx=\frac {1}{40} \left (5+3 \sqrt {5}\right ) \log \left (3+\sqrt {5}-2 x^4\right )+\frac {1}{40} \left (5-3 \sqrt {5}\right ) \log \left (-3+\sqrt {5}+2 x^4\right ) \]

[In]

Integrate[x^7/(1 - 3*x^4 + x^8),x]

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.60

method	result	size
default	\(\frac {\ln \left (x^{8}-3 x^{4}+1\right )}{8}-\frac {3 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\left (2 x^{4}-3\right ) \sqrt {5}}{5}\right )}{20}\)	\(33\)
risch	\(\frac {\ln \left (2 x^{4}-\sqrt {5}-3\right )}{8}+\frac {3 \ln \left (2 x^{4}-\sqrt {5}-3\right ) \sqrt {5}}{40}+\frac {\ln \left (2 x^{4}+\sqrt {5}-3\right )}{8}-\frac {3 \ln \left (2 x^{4}+\sqrt {5}-3\right ) \sqrt {5}}{40}\)	\(64\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.04 \[ \int \frac {x^7}{1-3 x^4+x^8} \, dx=\frac {3}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{8} - 6 \, x^{4} - \sqrt {5} {\left (2 \, x^{4} - 3\right )} + 7}{x^{8} - 3 \, x^{4} + 1}\right ) + \frac {1}{8} \, \log \left (x^{8} - 3 \, x^{4} + 1\right ) \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \frac {x^7}{1-3 x^4+x^8} \, dx=\left (\frac {1}{8} + \frac {3 \sqrt {5}}{40}\right ) \log {\left (x^{4} - \frac {3}{2} - \frac {\sqrt {5}}{2} \right )} + \left (\frac {1}{8} - \frac {3 \sqrt {5}}{40}\right ) \log {\left (x^{4} - \frac {3}{2} + \frac {\sqrt {5}}{2} \right )} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.82 \[ \int \frac {x^7}{1-3 x^4+x^8} \, dx=\frac {3}{40} \, \sqrt {5} \log \left (\frac {2 \, x^{4} - \sqrt {5} - 3}{2 \, x^{4} + \sqrt {5} - 3}\right ) + \frac {1}{8} \, \log \left (x^{8} - 3 \, x^{4} + 1\right ) \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87 \[ \int \frac {x^7}{1-3 x^4+x^8} \, dx=\frac {3}{40} \, \sqrt {5} \log \left (\frac {{\left | 2 \, x^{4} - \sqrt {5} - 3 \right |}}{{\left | 2 \, x^{4} + \sqrt {5} - 3 \right |}}\right ) + \frac {1}{8} \, \log \left ({\left | x^{8} - 3 \, x^{4} + 1 \right |}\right ) \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.07 \[ \int \frac {x^7}{1-3 x^4+x^8} \, dx=\frac {\ln \left (x^4-\frac {\sqrt {5}}{2}-\frac {3}{2}\right )}{8}+\frac {\ln \left (x^4+\frac {\sqrt {5}}{2}-\frac {3}{2}\right )}{8}+\frac {3\,\sqrt {5}\,\ln \left (x^4-\frac {\sqrt {5}}{2}-\frac {3}{2}\right )}{40}-\frac {3\,\sqrt {5}\,\ln \left (x^4+\frac {\sqrt {5}}{2}-\frac {3}{2}\right )}{40} \]

[In]

int(x^7/(x^8 - 3*x^4 + 1),x)

[Out]

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