Integrand size = 11, antiderivative size = 98 \[ \int \frac {x^2}{3+x^4} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )}{2 \sqrt {2} \sqrt [4]{3}}+\frac {\arctan \left (1+\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{3} x}{\sqrt {3}+x^2}\right )}{2 \sqrt {2} \sqrt [4]{3}} \] Output:
1/12*arctan(-1+1/3*2^(1/2)*x*3^(3/4))*2^(1/2)*3^(3/4)+1/12*arctan(1+1/3*2^ (1/2)*x*3^(3/4))*2^(1/2)*3^(3/4)-1/12*arctanh(2^(1/2)*3^(1/4)*x/(3^(1/2)+x ^2))*2^(1/2)*3^(3/4)
Time = 0.04 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.03 \[ \int \frac {x^2}{3+x^4} \, dx=\frac {-2 \arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )+2 \arctan \left (1+\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )+\log \left (3-\sqrt {2} 3^{3/4} x+\sqrt {3} x^2\right )-\log \left (3+\sqrt {2} 3^{3/4} x+\sqrt {3} x^2\right )}{4 \sqrt {2} \sqrt [4]{3}} \] Input:
Integrate[x^2/(3 + x^4),x]
Output:
(-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 + Sqr t[3]*x^2])/(4*Sqrt[2]*3^(1/4))
Time = 0.51 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.41, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {826, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^2}{x^4+3} \, dx\) |
\(\Big \downarrow \) 826 |
\(\displaystyle \frac {1}{2} \int \frac {x^2+\sqrt {3}}{x^4+3}dx-\frac {1}{2} \int \frac {\sqrt {3}-x^2}{x^4+3}dx\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1}{x^2-\sqrt {2} \sqrt [4]{3} x+\sqrt {3}}dx+\frac {1}{2} \int \frac {1}{x^2+\sqrt {2} \sqrt [4]{3} x+\sqrt {3}}dx\right )-\frac {1}{2} \int \frac {\sqrt {3}-x^2}{x^4+3}dx\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )^2-1}d\left (1-\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} x}{\sqrt [4]{3}}+1\right )^2-1}d\left (\frac {\sqrt {2} x}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}\right )-\frac {1}{2} \int \frac {\sqrt {3}-x^2}{x^4+3}dx\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}\right )-\frac {1}{2} \int \frac {\sqrt {3}-x^2}{x^4+3}dx\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt [4]{3}-2 x}{x^2-\sqrt {2} \sqrt [4]{3} x+\sqrt {3}}dx}{2 \sqrt {2} \sqrt [4]{3}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} x+\sqrt [4]{3}\right )}{x^2+\sqrt {2} \sqrt [4]{3} x+\sqrt {3}}dx}{2 \sqrt {2} \sqrt [4]{3}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt [4]{3}-2 x}{x^2-\sqrt {2} \sqrt [4]{3} x+\sqrt {3}}dx}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} x+\sqrt [4]{3}\right )}{x^2+\sqrt {2} \sqrt [4]{3} x+\sqrt {3}}dx}{2 \sqrt {2} \sqrt [4]{3}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt [4]{3}-2 x}{x^2-\sqrt {2} \sqrt [4]{3} x+\sqrt {3}}dx}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\int \frac {\sqrt {2} x+\sqrt [4]{3}}{x^2+\sqrt {2} \sqrt [4]{3} x+\sqrt {3}}dx}{2 \sqrt [4]{3}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {1}{2} \left (\frac {\arctan \left (\frac {\sqrt {2} x}{\sqrt [4]{3}}+1\right )}{\sqrt {2} \sqrt [4]{3}}-\frac {\arctan \left (1-\frac {\sqrt {2} x}{\sqrt [4]{3}}\right )}{\sqrt {2} \sqrt [4]{3}}\right )+\frac {1}{2} \left (\frac {\log \left (x^2-\sqrt {2} \sqrt [4]{3} x+\sqrt {3}\right )}{2 \sqrt {2} \sqrt [4]{3}}-\frac {\log \left (x^2+\sqrt {2} \sqrt [4]{3} x+\sqrt {3}\right )}{2 \sqrt {2} \sqrt [4]{3}}\right )\) |
Input:
Int[x^2/(3 + x^4),x]
Output:
(-(ArcTan[1 - (Sqrt[2]*x)/3^(1/4)]/(Sqrt[2]*3^(1/4))) + ArcTan[1 + (Sqrt[2 ]*x)/3^(1/4)]/(Sqrt[2]*3^(1/4)))/2 + (Log[Sqrt[3] - Sqrt[2]*3^(1/4)*x + x^ 2]/(2*Sqrt[2]*3^(1/4)) - Log[Sqrt[3] + Sqrt[2]*3^(1/4)*x + x^2]/(2*Sqrt[2] *3^(1/4)))/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.22
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4}+3\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{4}\) | \(22\) |
default | \(\frac {3^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-3^{\frac {1}{4}} x \sqrt {2}+\sqrt {3}}{x^{2}+3^{\frac {1}{4}} x \sqrt {2}+\sqrt {3}}\right )+2 \arctan \left (1+\frac {\sqrt {2}\, x 3^{\frac {3}{4}}}{3}\right )+2 \arctan \left (-1+\frac {\sqrt {2}\, x 3^{\frac {3}{4}}}{3}\right )\right )}{24}\) | \(73\) |
meijerg | \(\frac {3^{\frac {3}{4}} \left (\frac {x^{3} \sqrt {2}\, \ln \left (1-\frac {\sqrt {2}\, 3^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{3}+\frac {\sqrt {3}\, \sqrt {x^{4}}}{3}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{6-\sqrt {2}\, 3^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}-\frac {x^{3} \sqrt {2}\, \ln \left (1+\frac {\sqrt {2}\, 3^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{3}+\frac {\sqrt {3}\, \sqrt {x^{4}}}{3}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, 3^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{6+\sqrt {2}\, 3^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}\right )}{12}\) | \(171\) |
Input:
int(x^2/(x^4+3),x,method=_RETURNVERBOSE)
Output:
1/4*sum(1/_R*ln(x-_R),_R=RootOf(_Z^4+3))
Time = 0.08 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77 \[ \int \frac {x^2}{3+x^4} \, dx=\frac {1}{24} \cdot 12^{\frac {3}{4}} \arctan \left (\frac {1}{6} \cdot 12^{\frac {3}{4}} x + 1\right ) + \frac {1}{24} \cdot 12^{\frac {3}{4}} \arctan \left (\frac {1}{6} \cdot 12^{\frac {3}{4}} x - 1\right ) - \frac {1}{48} \cdot 12^{\frac {3}{4}} \log \left (2 \, x^{2} + 2 \cdot 12^{\frac {1}{4}} x + 2 \, \sqrt {3}\right ) + \frac {1}{48} \cdot 12^{\frac {3}{4}} \log \left (2 \, x^{2} - 2 \cdot 12^{\frac {1}{4}} x + 2 \, \sqrt {3}\right ) \] Input:
integrate(x^2/(x^4+3),x, algorithm="fricas")
Output:
1/24*12^(3/4)*arctan(1/6*12^(3/4)*x + 1) + 1/24*12^(3/4)*arctan(1/6*12^(3/ 4)*x - 1) - 1/48*12^(3/4)*log(2*x^2 + 2*12^(1/4)*x + 2*sqrt(3)) + 1/48*12^ (3/4)*log(2*x^2 - 2*12^(1/4)*x + 2*sqrt(3))
Time = 0.20 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.27 \[ \int \frac {x^2}{3+x^4} \, dx=\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (x^{2} - \sqrt {2} \cdot \sqrt [4]{3} x + \sqrt {3} \right )}}{24} - \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \log {\left (x^{2} + \sqrt {2} \cdot \sqrt [4]{3} x + \sqrt {3} \right )}}{24} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} x}{3} - 1 \right )}}{12} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot 3^{\frac {3}{4}} x}{3} + 1 \right )}}{12} \] Input:
integrate(x**2/(x**4+3),x)
Output:
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
Time = 0.11 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{3+x^4} \, dx=\frac {1}{12} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{12} \cdot 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, x - 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) - \frac {1}{24} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (x^{2} + 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {3}\right ) + \frac {1}{24} \cdot 3^{\frac {3}{4}} \sqrt {2} \log \left (x^{2} - 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {3}\right ) \] Input:
integrate(x^2/(x^4+3),x, algorithm="maxima")
Output:
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))
Time = 0.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.97 \[ \int \frac {x^2}{3+x^4} \, dx=\frac {1}{12} \cdot 108^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{12} \cdot 108^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, x - 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) - \frac {1}{24} \cdot 108^{\frac {1}{4}} \log \left (x^{2} + 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {3}\right ) + \frac {1}{24} \cdot 108^{\frac {1}{4}} \log \left (x^{2} - 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {3}\right ) \] Input:
integrate(x^2/(x^4+3),x, algorithm="giac")
Output:
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))
Time = 0.13 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.46 \[ \int \frac {x^2}{3+x^4} \, dx=\sqrt {2}\,3^{3/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,x\,\left (\frac {1}{6}-\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (\frac {1}{12}-\frac {1}{12}{}\mathrm {i}\right )+\sqrt {2}\,3^{3/4}\,\mathrm {atan}\left (\sqrt {2}\,3^{3/4}\,x\,\left (\frac {1}{6}+\frac {1}{6}{}\mathrm {i}\right )\right )\,\left (\frac {1}{12}+\frac {1}{12}{}\mathrm {i}\right ) \] Input:
int(x^2/(x^4 + 3),x)
Output:
2^(1/2)*3^(3/4)*atan(2^(1/2)*3^(3/4)*x*(1/6 - 1i/6))*(1/12 - 1i/12) + 2^(1 /2)*3^(3/4)*atan(2^(1/2)*3^(3/4)*x*(1/6 + 1i/6))*(1/12 + 1i/12)
Time = 0.21 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.85 \[ \int \frac {x^2}{3+x^4} \, dx=\frac {\sqrt {6}\, 3^{\frac {1}{4}} \left (-2 \mathit {atan} \left (\frac {\left (\sqrt {2}\, 3^{\frac {1}{4}}-2 x \right ) 3^{\frac {3}{4}}}{3 \sqrt {2}}\right )+2 \mathit {atan} \left (\frac {\left (\sqrt {2}\, 3^{\frac {1}{4}}+2 x \right ) 3^{\frac {3}{4}}}{3 \sqrt {2}}\right )+\mathrm {log}\left (-\sqrt {2}\, 3^{\frac {1}{4}} x +\sqrt {3}+x^{2}\right )-\mathrm {log}\left (\sqrt {2}\, 3^{\frac {1}{4}} x +\sqrt {3}+x^{2}\right )\right )}{24} \] Input:
int(x^2/(x^4+3),x)
Output:
(sqrt(6)*3**(1/4)*( - 2*atan((sqrt(2)*3**(1/4) - 2*x)/(sqrt(2)*3**(1/4))) + 2*atan((sqrt(2)*3**(1/4) + 2*x)/(sqrt(2)*3**(1/4))) + log( - sqrt(2)*3** (1/4)*x + sqrt(3) + x**2) - log(sqrt(2)*3**(1/4)*x + sqrt(3) + x**2)))/24