3.4.49 \(\int \frac {x^9}{1-x^4+x^8} \, dx\) [349]

Optimal. Leaf size=57 \[ \frac {x^2}{2}+\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}} \]

[Out]

1/2*x^2+1/12*ln(1+x^4-3^(1/2)*x^2)*3^(1/2)-1/12*ln(1+x^4+3^(1/2)*x^2)*3^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1373, 1136, 1178, 642} \begin {gather*} \frac {x^2}{2}+\frac {\log \left (x^4-\sqrt {3} x^2+1\right )}{4 \sqrt {3}}-\frac {\log \left (x^4+\sqrt {3} x^2+1\right )}{4 \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^9/(1 - x^4 + x^8),x]

[Out]

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

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1136

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

Rule 1178

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

Rule 1373

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]

Rubi steps

\begin {align*} \int \frac {x^9}{1-x^4+x^8} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^4}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=\frac {x^2}{2}-\frac {1}{2} \text {Subst}\left (\int \frac {1-x^2}{1-x^2+x^4} \, dx,x,x^2\right )\\ &=\frac {x^2}{2}+\frac {\text {Subst}\left (\int \frac {\sqrt {3}+2 x}{-1-\sqrt {3} x-x^2} \, dx,x,x^2\right )}{4 \sqrt {3}}+\frac {\text {Subst}\left (\int \frac {\sqrt {3}-2 x}{-1+\sqrt {3} x-x^2} \, dx,x,x^2\right )}{4 \sqrt {3}}\\ &=\frac {x^2}{2}+\frac {\log \left (1-\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}}-\frac {\log \left (1+\sqrt {3} x^2+x^4\right )}{4 \sqrt {3}}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 55, normalized size = 0.96 \begin {gather*} \frac {1}{12} \left (6 x^2+\sqrt {3} \log \left (-1+\sqrt {3} x^2-x^4\right )-\sqrt {3} \log \left (1+\sqrt {3} x^2+x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^9/(1 - x^4 + x^8),x]

[Out]

(6*x^2 + Sqrt[3]*Log[-1 + Sqrt[3]*x^2 - x^4] - Sqrt[3]*Log[1 + Sqrt[3]*x^2 + x^4])/12

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Maple [A]
time = 0.02, size = 44, normalized size = 0.77

method	result	size
default	\(\frac {x^{2}}{2}+\frac {\ln \left (1+x^{4}-x^{2} \sqrt {3}\right ) \sqrt {3}}{12}-\frac {\ln \left (1+x^{4}+x^{2} \sqrt {3}\right ) \sqrt {3}}{12}\)	\(44\)
risch	\(\frac {x^{2}}{2}+\frac {\ln \left (1+x^{4}-x^{2} \sqrt {3}\right ) \sqrt {3}}{12}-\frac {\ln \left (1+x^{4}+x^{2} \sqrt {3}\right ) \sqrt {3}}{12}\)	\(44\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(x^8-x^4+1),x,method=_RETURNVERBOSE)

[Out]

1/2*x^2+1/12*ln(1+x^4-x^2*3^(1/2))*3^(1/2)-1/12*ln(1+x^4+x^2*3^(1/2))*3^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(x^8-x^4+1),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.35, size = 47, normalized size = 0.82 \begin {gather*} \frac {1}{2} \, x^{2} + \frac {1}{12} \, \sqrt {3} \log \left (\frac {x^{8} + 5 \, x^{4} - 2 \, \sqrt {3} {\left (x^{6} + x^{2}\right )} + 1}{x^{8} - x^{4} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(x^8-x^4+1),x, algorithm="fricas")

[Out]

1/2*x^2 + 1/12*sqrt(3)*log((x^8 + 5*x^4 - 2*sqrt(3)*(x^6 + x^2) + 1)/(x^8 - x^4 + 1))

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Sympy [A]
time = 0.04, size = 48, normalized size = 0.84 \begin {gather*} \frac {x^{2}}{2} + \frac {\sqrt {3} \log {\left (x^{4} - \sqrt {3} x^{2} + 1 \right )}}{12} - \frac {\sqrt {3} \log {\left (x^{4} + \sqrt {3} x^{2} + 1 \right )}}{12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(x**8-x**4+1),x)

[Out]

x**2/2 + sqrt(3)*log(x**4 - sqrt(3)*x**2 + 1)/12 - sqrt(3)*log(x**4 + sqrt(3)*x**2 + 1)/12

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (43) = 86\).
time = 3.93, size = 99, normalized size = 1.74 \begin {gather*} \frac {1}{2} \, x^{2} + \frac {1}{4} \, {\left (x^{4} - 1\right )} \arctan \left (2 \, x^{2} + \sqrt {3}\right ) + \frac {1}{4} \, {\left (x^{4} - 1\right )} \arctan \left (2 \, x^{2} - \sqrt {3}\right ) + \frac {1}{24} \, {\left (\sqrt {3} x^{4} - \sqrt {3}\right )} \log \left (x^{4} + \sqrt {3} x^{2} + 1\right ) - \frac {1}{24} \, {\left (\sqrt {3} x^{4} - \sqrt {3}\right )} \log \left (x^{4} - \sqrt {3} x^{2} + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(x^8-x^4+1),x, algorithm="giac")

[Out]

1/2*x^2 + 1/4*(x^4 - 1)*arctan(2*x^2 + sqrt(3)) + 1/4*(x^4 - 1)*arctan(2*x^2 - sqrt(3)) + 1/24*(sqrt(3)*x^4 -
sqrt(3))*log(x^4 + sqrt(3)*x^2 + 1) - 1/24*(sqrt(3)*x^4 - sqrt(3))*log(x^4 - sqrt(3)*x^2 + 1)

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Mupad [B]
time = 1.31, size = 29, normalized size = 0.51 \begin {gather*} \frac {x^2}{2}-\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {2\,\sqrt {3}\,x^2}{9\,\left (\frac {2\,x^4}{9}+\frac {2}{9}\right )}\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(x^8 - x^4 + 1),x)

[Out]

x^2/2 - (3^(1/2)*atanh((2*3^(1/2)*x^2)/(9*((2*x^4)/9 + 2/9))))/6

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