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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0256436, antiderivative size = 23, normalized size of antiderivative = 1., 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 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]

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 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])

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*}

Mathematica [A]  time = 0.0100449, 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])

________________________________________________________________________________________

Maple [A]  time = 0., size = 19, normalized size = 0.8 \begin{align*} -{\frac{\sqrt{5}}{10}{\it Artanh} \left ({\frac{ \left ( 2\,{x}^{4}+3 \right ) \sqrt{5}}{5}} \right ) } \end{align*}

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)

________________________________________________________________________________________

Maxima [A]  time = 1.49359, size = 42, normalized size = 1.83 \begin{align*} \frac{1}{20} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) \end{align*}

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))

________________________________________________________________________________________

Fricas [B]  time = 1.45554, size = 107, normalized size = 4.65 \begin{align*} \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 ) \end{align*}

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))

________________________________________________________________________________________

Sympy [A]  time = 0.121867, size = 42, normalized size = 1.83 \begin{align*} \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} \end{align*}

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

________________________________________________________________________________________

Giac [A]  time = 1.29294, size = 42, normalized size = 1.83 \begin{align*} \frac{1}{20} \, \sqrt{5} \log \left (\frac{2 \, x^{4} - \sqrt{5} + 3}{2 \, x^{4} + \sqrt{5} + 3}\right ) \end{align*}

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))