Integrand size = 13, antiderivative size = 104 \[ \int \frac {x^6}{2+3 x^4} \, dx=\frac {x^3}{9}+\frac {\arctan \left (1-\sqrt [4]{6} x\right )}{3\ 6^{3/4}}-\frac {\arctan \left (1+\sqrt [4]{6} x\right )}{3\ 6^{3/4}}-\frac {\log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{6\ 6^{3/4}}+\frac {\log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{6\ 6^{3/4}} \]
1/9*x^3-1/18*arctan(-1+6^(1/4)*x)*6^(1/4)-1/18*arctan(1+6^(1/4)*x)*6^(1/4) -1/36*ln(-6^(3/4)*x+3*x^2+6^(1/2))*6^(1/4)+1/36*ln(6^(3/4)*x+3*x^2+6^(1/2) )*6^(1/4)
Time = 0.02 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.94 \[ \int \frac {x^6}{2+3 x^4} \, dx=\frac {1}{36} \left (4 x^3+2 \sqrt [4]{6} \arctan \left (1-\sqrt [4]{6} x\right )-2 \sqrt [4]{6} \arctan \left (1+\sqrt [4]{6} x\right )-\sqrt [4]{6} \log \left (2-2 \sqrt [4]{6} x+\sqrt {6} x^2\right )+\sqrt [4]{6} \log \left (2+2 \sqrt [4]{6} x+\sqrt {6} x^2\right )\right ) \]
(4*x^3 + 2*6^(1/4)*ArcTan[1 - 6^(1/4)*x] - 2*6^(1/4)*ArcTan[1 + 6^(1/4)*x] - 6^(1/4)*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + 6^(1/4)*Log[2 + 2*6^(1/4)* x + Sqrt[6]*x^2])/36
Time = 0.36 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.38, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {843, 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.
| \(\displaystyle \int \frac {x^6}{3 x^4+2} \, dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {x^3}{9}-\frac {2}{3} \int \frac {x^2}{3 x^4+2}dx\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {x^3}{9}-\frac {2}{3} \left (\frac {\int \frac {\sqrt {3} x^2+\sqrt {2}}{3 x^4+2}dx}{2 \sqrt {3}}-\frac {\int \frac {\sqrt {2}-\sqrt {3} x^2}{3 x^4+2}dx}{2 \sqrt {3}}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {x^3}{9}-\frac {2}{3} \left (\frac {\frac {\int \frac {1}{x^2-\frac {2^{3/4} x}{\sqrt [4]{3}}+\sqrt {\frac {2}{3}}}dx}{2 \sqrt {3}}+\frac {\int \frac {1}{x^2+\frac {2^{3/4} x}{\sqrt [4]{3}}+\sqrt {\frac {2}{3}}}dx}{2 \sqrt {3}}}{2 \sqrt {3}}-\frac {\int \frac {\sqrt {2}-\sqrt {3} x^2}{3 x^4+2}dx}{2 \sqrt {3}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x^3}{9}-\frac {2}{3} \left (\frac {\frac {\int \frac {1}{-\left (1-\sqrt [4]{6} x\right )^2-1}d\left (1-\sqrt [4]{6} x\right )}{2^{3/4} \sqrt [4]{3}}-\frac {\int \frac {1}{-\left (\sqrt [4]{6} x+1\right )^2-1}d\left (\sqrt [4]{6} x+1\right )}{2^{3/4} \sqrt [4]{3}}}{2 \sqrt {3}}-\frac {\int \frac {\sqrt {2}-\sqrt {3} x^2}{3 x^4+2}dx}{2 \sqrt {3}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x^3}{9}-\frac {2}{3} \left (\frac {\frac {\arctan \left (\sqrt [4]{6} x+1\right )}{2^{3/4} \sqrt [4]{3}}-\frac {\arctan \left (1-\sqrt [4]{6} x\right )}{2^{3/4} \sqrt [4]{3}}}{2 \sqrt {3}}-\frac {\int \frac {\sqrt {2}-\sqrt {3} x^2}{3 x^4+2}dx}{2 \sqrt {3}}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {x^3}{9}-\frac {2}{3} \left (\frac {\frac {\arctan \left (\sqrt [4]{6} x+1\right )}{2^{3/4} \sqrt [4]{3}}-\frac {\arctan \left (1-\sqrt [4]{6} x\right )}{2^{3/4} \sqrt [4]{3}}}{2 \sqrt {3}}-\frac {-\frac {\int -\frac {6^{3/4}-6 x}{3 x^2-6^{3/4} x+\sqrt {6}}dx}{2\ 2^{3/4} \sqrt [4]{3}}-\frac {\int -\frac {6^{3/4} \left (\sqrt [4]{6} x+1\right )}{3 x^2+6^{3/4} x+\sqrt {6}}dx}{2\ 2^{3/4} \sqrt [4]{3}}}{2 \sqrt {3}}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x^3}{9}-\frac {2}{3} \left (\frac {\frac {\arctan \left (\sqrt [4]{6} x+1\right )}{2^{3/4} \sqrt [4]{3}}-\frac {\arctan \left (1-\sqrt [4]{6} x\right )}{2^{3/4} \sqrt [4]{3}}}{2 \sqrt {3}}-\frac {\frac {\int \frac {6^{3/4}-6 x}{3 x^2-6^{3/4} x+\sqrt {6}}dx}{2\ 2^{3/4} \sqrt [4]{3}}+\frac {\int \frac {6^{3/4} \left (\sqrt [4]{6} x+1\right )}{3 x^2+6^{3/4} x+\sqrt {6}}dx}{2\ 2^{3/4} \sqrt [4]{3}}}{2 \sqrt {3}}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^3}{9}-\frac {2}{3} \left (\frac {\frac {\arctan \left (\sqrt [4]{6} x+1\right )}{2^{3/4} \sqrt [4]{3}}-\frac {\arctan \left (1-\sqrt [4]{6} x\right )}{2^{3/4} \sqrt [4]{3}}}{2 \sqrt {3}}-\frac {\frac {\int \frac {6^{3/4}-6 x}{3 x^2-6^{3/4} x+\sqrt {6}}dx}{2\ 2^{3/4} \sqrt [4]{3}}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt [4]{6} x+1}{3 x^2+6^{3/4} x+\sqrt {6}}dx}{2 \sqrt {3}}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x^3}{9}-\frac {2}{3} \left (\frac {\frac {\arctan \left (\sqrt [4]{6} x+1\right )}{2^{3/4} \sqrt [4]{3}}-\frac {\arctan \left (1-\sqrt [4]{6} x\right )}{2^{3/4} \sqrt [4]{3}}}{2 \sqrt {3}}-\frac {\frac {\log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{2\ 2^{3/4} \sqrt [4]{3}}-\frac {\log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{2\ 2^{3/4} \sqrt [4]{3}}}{2 \sqrt {3}}\right )\) |
x^3/9 - (2*((-(ArcTan[1 - 6^(1/4)*x]/(2^(3/4)*3^(1/4))) + ArcTan[1 + 6^(1/ 4)*x]/(2^(3/4)*3^(1/4)))/(2*Sqrt[3]) - (-1/2*Log[Sqrt[6] - 6^(3/4)*x + 3*x ^2]/(2^(3/4)*3^(1/4)) + Log[Sqrt[6] + 6^(3/4)*x + 3*x^2]/(2*2^(3/4)*3^(1/4 )))/(2*Sqrt[3])))/3
3.7.94.3.1 Defintions of rubi rules used
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n)^(p + 1)/(b*(m + n*p + 1))), x] - Simp[ a*c^n*((m - n + 1)/(b*(m + n*p + 1))) Int[(c*x)^(m - n)*(a + b*x^n)^p, x] , x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n* p + 1, 0] && IntBinomialQ[a, b, c, n, m, p, x]
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 = 3.97 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.29
| method | result | size |
| risch | \(\frac {x^{3}}{9}-\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}\right )}{18}\) | \(30\) |
| default | \(\frac {x^{3}}{9}-\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{216}\) | \(99\) |
| meijerg | \(\frac {54^{\frac {3}{4}} \left (\frac {2 x^{3} 3^{\frac {3}{4}} 2^{\frac {1}{4}}}{3}-\frac {x^{3} 3^{\frac {3}{4}} 2^{\frac {1}{4}} \left (\frac {2^{\frac {1}{4}} 27^{\frac {3}{4}} \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{27 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {2 \,2^{\frac {1}{4}} 27^{\frac {3}{4}} \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{27 \left (x^{4}\right )^{\frac {3}{4}}}-\frac {2^{\frac {1}{4}} 27^{\frac {3}{4}} \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{27 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {2 \,2^{\frac {1}{4}} 27^{\frac {3}{4}} \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{27 \left (x^{4}\right )^{\frac {3}{4}}}\right )}{2}\right )}{324}\) | \(195\) |
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87 \[ \int \frac {x^6}{2+3 x^4} \, dx=\frac {1}{9} \, x^{3} - \left (\frac {1}{648} i - \frac {1}{648}\right ) \cdot 54^{\frac {3}{4}} \sqrt {2} \log \left (6 \, x + \left (i + 1\right ) \cdot 54^{\frac {1}{4}} \sqrt {2}\right ) + \left (\frac {1}{648} i + \frac {1}{648}\right ) \cdot 54^{\frac {3}{4}} \sqrt {2} \log \left (6 \, x - \left (i - 1\right ) \cdot 54^{\frac {1}{4}} \sqrt {2}\right ) - \left (\frac {1}{648} i + \frac {1}{648}\right ) \cdot 54^{\frac {3}{4}} \sqrt {2} \log \left (6 \, x + \left (i - 1\right ) \cdot 54^{\frac {1}{4}} \sqrt {2}\right ) + \left (\frac {1}{648} i - \frac {1}{648}\right ) \cdot 54^{\frac {3}{4}} \sqrt {2} \log \left (6 \, x - \left (i + 1\right ) \cdot 54^{\frac {1}{4}} \sqrt {2}\right ) \]
1/9*x^3 - (1/648*I - 1/648)*54^(3/4)*sqrt(2)*log(6*x + (I + 1)*54^(1/4)*sq rt(2)) + (1/648*I + 1/648)*54^(3/4)*sqrt(2)*log(6*x - (I - 1)*54^(1/4)*sqr t(2)) - (1/648*I + 1/648)*54^(3/4)*sqrt(2)*log(6*x + (I - 1)*54^(1/4)*sqrt (2)) + (1/648*I - 1/648)*54^(3/4)*sqrt(2)*log(6*x - (I + 1)*54^(1/4)*sqrt( 2))
Time = 0.18 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {x^6}{2+3 x^4} \, dx=\frac {x^{3}}{9} - \frac {\sqrt [4]{6} \log {\left (x^{2} - \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{36} + \frac {\sqrt [4]{6} \log {\left (x^{2} + \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{36} - \frac {\sqrt [4]{6} \operatorname {atan}{\left (\sqrt [4]{6} x - 1 \right )}}{18} - \frac {\sqrt [4]{6} \operatorname {atan}{\left (\sqrt [4]{6} x + 1 \right )}}{18} \]
x**3/9 - 6**(1/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3)/36 + 6**(1/4)*log(x **2 + 6**(3/4)*x/3 + sqrt(6)/3)/36 - 6**(1/4)*atan(6**(1/4)*x - 1)/18 - 6* *(1/4)*atan(6**(1/4)*x + 1)/18
Time = 0.29 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.21 \[ \int \frac {x^6}{2+3 x^4} \, dx=\frac {1}{9} \, x^{3} - \frac {1}{18} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) - \frac {1}{18} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{36} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) - \frac {1}{36} \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) \]
1/9*x^3 - 1/18*3^(1/4)*2^(1/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3 ^(1/4)*2^(3/4))) - 1/18*3^(1/4)*2^(1/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt (3)*x - 3^(1/4)*2^(3/4))) + 1/36*3^(1/4)*2^(1/4)*log(sqrt(3)*x^2 + 3^(1/4) *2^(3/4)*x + sqrt(2)) - 1/36*3^(1/4)*2^(1/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^( 3/4)*x + sqrt(2))
Time = 0.30 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.96 \[ \int \frac {x^6}{2+3 x^4} \, dx=\frac {1}{9} \, x^{3} - \frac {1}{18} \cdot 6^{\frac {1}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) - \frac {1}{18} \cdot 6^{\frac {1}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{36} \cdot 6^{\frac {1}{4}} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {1}{36} \cdot 6^{\frac {1}{4}} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]
1/9*x^3 - 1/18*6^(1/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3) ^(1/4))) - 1/18*6^(1/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3 )^(1/4))) + 1/36*6^(1/4)*log(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/ 36*6^(1/4)*log(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3))
Time = 0.14 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.37 \[ \int \frac {x^6}{2+3 x^4} \, dx=\frac {x^3}{9}+6^{1/4}\,\mathrm {atan}\left (6^{1/4}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{18}+\frac {1}{18}{}\mathrm {i}\right )+6^{1/4}\,\mathrm {atan}\left (6^{1/4}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )\right )\,\left (-\frac {1}{18}-\frac {1}{18}{}\mathrm {i}\right ) \]