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

Optimal. Leaf size=23 \[ -\frac {\tanh ^{-1}\left (\frac {2 x^4+3}{\sqrt {5}}\right )}{2 \sqrt {5}} \]

[Out]

-1/10*arctanh(1/5*(2*x^4+3)*5^(1/2))*5^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {1352, 618, 206} \[ -\frac {\tanh ^{-1}\left (\frac {2 x^4+3}{\sqrt {5}}\right )}{2 \sqrt {5}} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(1 + 3*x^4 + x^8),x]

[Out]

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

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

Rule 1352

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 38, normalized size = 1.65 \[ \frac {\log \left (-2 x^4+\sqrt {5}-3\right )-\log \left (2 x^4+\sqrt {5}+3\right )}{4 \sqrt {5}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(1 + 3*x^4 + x^8),x]

[Out]

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

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fricas [B]  time = 0.84, size = 43, normalized size = 1.87 \[ \frac {1}{20} \, \sqrt {5} \log \left (\frac {2 \, x^{8} + 6 \, x^{4} - \sqrt {5} {\left (2 \, x^{4} + 3\right )} + 7}{x^{8} + 3 \, x^{4} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*sqrt(5)*log((2*x^8 + 6*x^4 - sqrt(5)*(2*x^4 + 3) + 7)/(x^8 + 3*x^4 + 1))

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giac [A]  time = 0.60, size = 31, normalized size = 1.35 \[ \frac {1}{20} \, \sqrt {5} \log \left (\frac {2 \, x^{4} - \sqrt {5} + 3}{2 \, x^{4} + \sqrt {5} + 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*sqrt(5)*log((2*x^4 - sqrt(5) + 3)/(2*x^4 + sqrt(5) + 3))

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maple [A]  time = 0.00, size = 19, normalized size = 0.83 \[ -\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 x^{4}+3\right ) \sqrt {5}}{5}\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*arctanh(1/5*(2*x^4+3)*5^(1/2))*5^(1/2)

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maxima [A]  time = 1.48, size = 31, normalized size = 1.35 \[ \frac {1}{20} \, \sqrt {5} \log \left (\frac {2 \, x^{4} - \sqrt {5} + 3}{2 \, x^{4} + \sqrt {5} + 3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/20*sqrt(5)*log((2*x^4 - sqrt(5) + 3)/(2*x^4 + sqrt(5) + 3))

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mupad [B]  time = 1.33, size = 30, normalized size = 1.30 \[ \frac {\sqrt {5}\,\mathrm {atanh}\left (\frac {8\,\sqrt {5}\,x^4+3\,\sqrt {5}}{18\,x^4+7}\right )}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(3*x^4 + x^8 + 1),x)

[Out]

(5^(1/2)*atanh((3*5^(1/2) + 8*5^(1/2)*x^4)/(18*x^4 + 7)))/10

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sympy [A]  time = 0.12, size = 42, normalized size = 1.83 \[ \frac {\sqrt {5} \log {\left (x^{4} - \frac {\sqrt {5}}{2} + \frac {3}{2} \right )}}{20} - \frac {\sqrt {5} \log {\left (x^{4} + \frac {\sqrt {5}}{2} + \frac {3}{2} \right )}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(5)*log(x**4 - sqrt(5)/2 + 3/2)/20 - sqrt(5)*log(x**4 + sqrt(5)/2 + 3/2)/20

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