3.707 \(\int \frac{x^2}{3+x^4} \, dx\)

Optimal. Leaf size=133 \[ \frac{\log \left (x^2-\sqrt{2} \sqrt [4]{3} x+\sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3}}-\frac{\log \left (x^2+\sqrt{2} \sqrt [4]{3} x+\sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{3}}\right )}{2 \sqrt{2} \sqrt [4]{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{3}}+1\right )}{2 \sqrt{2} \sqrt [4]{3}} \]

[Out]

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

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Rubi [A]  time = 0.0789787, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.546, Rules used = {297, 1162, 617, 204, 1165, 628} \[ \frac{\log \left (x^2-\sqrt{2} \sqrt [4]{3} x+\sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3}}-\frac{\log \left (x^2+\sqrt{2} \sqrt [4]{3} x+\sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3}}-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{3}}\right )}{2 \sqrt{2} \sqrt [4]{3}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{3}}+1\right )}{2 \sqrt{2} \sqrt [4]{3}} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(3 + x^4),x]

[Out]

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

Rule 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

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

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 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, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

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]

Rubi steps

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

Mathematica [A]  time = 0.0402054, size = 101, normalized size = 0.76 \[ \frac{\log \left (\sqrt{3} x^2-\sqrt{2} 3^{3/4} x+3\right )-\log \left (\sqrt{3} x^2+\sqrt{2} 3^{3/4} x+3\right )-2 \tan ^{-1}\left (1-\frac{\sqrt{2} x}{\sqrt [4]{3}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt [4]{3}}+1\right )}{4 \sqrt{2} \sqrt [4]{3}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(3 + x^4),x]

[Out]

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

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Maple [A]  time = 0.004, size = 85, normalized size = 0.6 \begin{align*}{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{12}\arctan \left ( 1+{\frac{x\sqrt{2}{3}^{{\frac{3}{4}}}}{3}} \right ) }+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{12}\arctan \left ( -1+{\frac{x\sqrt{2}{3}^{{\frac{3}{4}}}}{3}} \right ) }+{\frac{\sqrt{2}{3}^{{\frac{3}{4}}}}{24}\ln \left ({\frac{{x}^{2}-\sqrt [4]{3}x\sqrt{2}+\sqrt{3}}{{x}^{2}+\sqrt [4]{3}x\sqrt{2}+\sqrt{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(x^4+3),x)

[Out]

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

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Maxima [A]  time = 1.52931, size = 144, normalized size = 1.08 \begin{align*} \frac{1}{12} \cdot 3^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (2 \, x + 3^{\frac{1}{4}} \sqrt{2}\right )}\right ) + \frac{1}{12} \cdot 3^{\frac{3}{4}} \sqrt{2} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (2 \, x - 3^{\frac{1}{4}} \sqrt{2}\right )}\right ) - \frac{1}{24} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (x^{2} + 3^{\frac{1}{4}} \sqrt{2} x + \sqrt{3}\right ) + \frac{1}{24} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (x^{2} - 3^{\frac{1}{4}} \sqrt{2} x + \sqrt{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^4+3),x, algorithm="maxima")

[Out]

1/12*3^(3/4)*sqrt(2)*arctan(1/6*3^(3/4)*sqrt(2)*(2*x + 3^(1/4)*sqrt(2))) + 1/12*3^(3/4)*sqrt(2)*arctan(1/6*3^(
3/4)*sqrt(2)*(2*x - 3^(1/4)*sqrt(2))) - 1/24*3^(3/4)*sqrt(2)*log(x^2 + 3^(1/4)*sqrt(2)*x + sqrt(3)) + 1/24*3^(
3/4)*sqrt(2)*log(x^2 - 3^(1/4)*sqrt(2)*x + sqrt(3))

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Fricas [A]  time = 1.9325, size = 489, normalized size = 3.68 \begin{align*} -\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2} \arctan \left (-\frac{1}{3} \cdot 3^{\frac{3}{4}} \sqrt{2} x + \frac{1}{3} \cdot 3^{\frac{3}{4}} \sqrt{2} \sqrt{x^{2} + 3^{\frac{1}{4}} \sqrt{2} x + \sqrt{3}} - 1\right ) - \frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2} \arctan \left (-\frac{1}{3} \cdot 3^{\frac{3}{4}} \sqrt{2} x + \frac{1}{3} \cdot 3^{\frac{3}{4}} \sqrt{2} \sqrt{x^{2} - 3^{\frac{1}{4}} \sqrt{2} x + \sqrt{3}} + 1\right ) - \frac{1}{24} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (x^{2} + 3^{\frac{1}{4}} \sqrt{2} x + \sqrt{3}\right ) + \frac{1}{24} \cdot 3^{\frac{3}{4}} \sqrt{2} \log \left (x^{2} - 3^{\frac{1}{4}} \sqrt{2} x + \sqrt{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^4+3),x, algorithm="fricas")

[Out]

-1/6*3^(3/4)*sqrt(2)*arctan(-1/3*3^(3/4)*sqrt(2)*x + 1/3*3^(3/4)*sqrt(2)*sqrt(x^2 + 3^(1/4)*sqrt(2)*x + sqrt(3
)) - 1) - 1/6*3^(3/4)*sqrt(2)*arctan(-1/3*3^(3/4)*sqrt(2)*x + 1/3*3^(3/4)*sqrt(2)*sqrt(x^2 - 3^(1/4)*sqrt(2)*x
+ sqrt(3)) + 1) - 1/24*3^(3/4)*sqrt(2)*log(x^2 + 3^(1/4)*sqrt(2)*x + sqrt(3)) + 1/24*3^(3/4)*sqrt(2)*log(x^2
- 3^(1/4)*sqrt(2)*x + sqrt(3))

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Sympy [A]  time = 0.482094, size = 124, normalized size = 0.93 \begin{align*} \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (x^{2} - \sqrt{2} \sqrt [4]{3} x + \sqrt{3} \right )}}{24} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \log{\left (x^{2} + \sqrt{2} \sqrt [4]{3} x + \sqrt{3} \right )}}{24} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} x}{3} - 1 \right )}}{12} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt{2} \cdot 3^{\frac{3}{4}} x}{3} + 1 \right )}}{12} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2/(x**4+3),x)

[Out]

sqrt(2)*3**(3/4)*log(x**2 - sqrt(2)*3**(1/4)*x + sqrt(3))/24 - sqrt(2)*3**(3/4)*log(x**2 + sqrt(2)*3**(1/4)*x
+ sqrt(3))/24 + sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(3/4)*x/3 - 1)/12 + sqrt(2)*3**(3/4)*atan(sqrt(2)*3**(3/4)*x/
3 + 1)/12

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Giac [A]  time = 1.12474, size = 128, normalized size = 0.96 \begin{align*} \frac{1}{12} \cdot 108^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (2 \, x + 3^{\frac{1}{4}} \sqrt{2}\right )}\right ) + \frac{1}{12} \cdot 108^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 3^{\frac{3}{4}} \sqrt{2}{\left (2 \, x - 3^{\frac{1}{4}} \sqrt{2}\right )}\right ) - \frac{1}{24} \cdot 108^{\frac{1}{4}} \log \left (x^{2} + 3^{\frac{1}{4}} \sqrt{2} x + \sqrt{3}\right ) + \frac{1}{24} \cdot 108^{\frac{1}{4}} \log \left (x^{2} - 3^{\frac{1}{4}} \sqrt{2} x + \sqrt{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2/(x^4+3),x, algorithm="giac")

[Out]

1/12*108^(1/4)*arctan(1/6*3^(3/4)*sqrt(2)*(2*x + 3^(1/4)*sqrt(2))) + 1/12*108^(1/4)*arctan(1/6*3^(3/4)*sqrt(2)
*(2*x - 3^(1/4)*sqrt(2))) - 1/24*108^(1/4)*log(x^2 + 3^(1/4)*sqrt(2)*x + sqrt(3)) + 1/24*108^(1/4)*log(x^2 - 3
^(1/4)*sqrt(2)*x + sqrt(3))