Integrand size = 13, antiderivative size = 220 \[ \int \frac {x^6}{a+b x^6} \, dx=\frac {x}{b}-\frac {\sqrt [6]{a} \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 b^{7/6}}+\frac {\sqrt [6]{a} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}-\frac {\sqrt [6]{a} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 b^{7/6}}+\frac {\sqrt [6]{a} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} b^{7/6}}-\frac {\sqrt [6]{a} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} b^{7/6}} \]
x/b-1/3*a^(1/6)*arctan(b^(1/6)*x/a^(1/6))/b^(7/6)+1/6*a^(1/6)*arctan((-2*b ^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6))/b^(7/6)-1/6*a^(1/6)*arctan((2*b^(1/6)*x +a^(1/6)*3^(1/2))/a^(1/6))/b^(7/6)+1/12*a^(1/6)*ln(a^(1/3)+b^(1/3)*x^2-a^( 1/6)*b^(1/6)*x*3^(1/2))/b^(7/6)*3^(1/2)-1/12*a^(1/6)*ln(a^(1/3)+b^(1/3)*x^ 2+a^(1/6)*b^(1/6)*x*3^(1/2))/b^(7/6)*3^(1/2)
Time = 0.04 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.83 \[ \int \frac {x^6}{a+b x^6} \, dx=\frac {12 \sqrt [6]{b} x-4 \sqrt [6]{a} \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt [6]{a} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \sqrt [6]{a} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+\sqrt {3} \sqrt [6]{a} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )-\sqrt {3} \sqrt [6]{a} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{12 b^{7/6}} \]
(12*b^(1/6)*x - 4*a^(1/6)*ArcTan[(b^(1/6)*x)/a^(1/6)] + 2*a^(1/6)*ArcTan[S qrt[3] - (2*b^(1/6)*x)/a^(1/6)] - 2*a^(1/6)*ArcTan[Sqrt[3] + (2*b^(1/6)*x) /a^(1/6)] + Sqrt[3]*a^(1/6)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1 /3)*x^2] - Sqrt[3]*a^(1/6)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/ 3)*x^2])/(12*b^(7/6))
Time = 0.46 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {843, 753, 27, 218, 1142, 25, 27, 1082, 217, 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.
| \(\displaystyle \int \frac {x^6}{a+b x^6} \, dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {x}{b}-\frac {a \int \frac {1}{b x^6+a}dx}{b}\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (\frac {\int \frac {1}{\sqrt [3]{b} x^2+\sqrt [3]{a}}dx}{3 a^{2/3}}+\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} x}{2 \left (\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}\right )}dx}{3 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{b} x+2 \sqrt [6]{a}}{2 \left (\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}\right )}dx}{3 a^{5/6}}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (\frac {\int \frac {1}{\sqrt [3]{b} x^2+\sqrt [3]{a}}dx}{3 a^{2/3}}+\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{b} x+2 \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 a^{5/6}}\right )}{b}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{b} x+2 \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{b}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x\right )}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x\right )}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{b}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {\int \frac {1}{-\left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {\int \frac {1}{-\left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{b}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{b}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x}{b}-\frac {a \left (\frac {-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{b}\) |
x/b - (a*(ArcTan[(b^(1/6)*x)/a^(1/6)]/(3*a^(5/6)*b^(1/6)) + (-(ArcTan[Sqrt [3]*(1 - (2*b^(1/6)*x)/(Sqrt[3]*a^(1/6)))]/b^(1/6)) - (Sqrt[3]*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(2*b^(1/6)))/(6*a^(5/6)) + (A rcTan[Sqrt[3]*(1 + (2*b^(1/6)*x)/(Sqrt[3]*a^(1/6)))]/b^(1/6) + (Sqrt[3]*Lo g[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(2*b^(1/6)))/(6*a^(5 /6))))/b
3.14.20.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> Module[{r = Numerator[Rt[a/ b, n]], s = Denominator[Rt[a/b, n]], k, u, v}, Simp[u = Int[(r - s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[ (r + s*Cos[(2*k - 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2* x^2), x]; 2*(r^2/(a*n)) Int[1/(r^2 + s^2*x^2), x] + 2*(r/(a*n)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] && PosQ[a /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_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.42 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.15
| method | result | size |
| risch | \(\frac {x}{b}-\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{6 b^{2}}\) | \(34\) |
| default | \(\frac {x}{b}-\frac {\left (\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}\right ) a}{b}\) | \(171\) |
Time = 0.30 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.96 \[ \int \frac {x^6}{a+b x^6} \, dx=-\frac {{\left (\sqrt {-3} b + b\right )} \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} b + b\right )} \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} + x\right ) - {\left (\sqrt {-3} b + b\right )} \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} b + b\right )} \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} + x\right ) + {\left (\sqrt {-3} b - b\right )} \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} b - b\right )} \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} + x\right ) - {\left (\sqrt {-3} b - b\right )} \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} b - b\right )} \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} + x\right ) + 2 \, b \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} \log \left (b \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} + x\right ) - 2 \, b \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} \log \left (-b \left (-\frac {a}{b^{7}}\right )^{\frac {1}{6}} + x\right ) - 12 \, x}{12 \, b} \]
-1/12*((sqrt(-3)*b + b)*(-a/b^7)^(1/6)*log(1/2*(sqrt(-3)*b + b)*(-a/b^7)^( 1/6) + x) - (sqrt(-3)*b + b)*(-a/b^7)^(1/6)*log(-1/2*(sqrt(-3)*b + b)*(-a/ b^7)^(1/6) + x) + (sqrt(-3)*b - b)*(-a/b^7)^(1/6)*log(1/2*(sqrt(-3)*b - b) *(-a/b^7)^(1/6) + x) - (sqrt(-3)*b - b)*(-a/b^7)^(1/6)*log(-1/2*(sqrt(-3)* b - b)*(-a/b^7)^(1/6) + x) + 2*b*(-a/b^7)^(1/6)*log(b*(-a/b^7)^(1/6) + x) - 2*b*(-a/b^7)^(1/6)*log(-b*(-a/b^7)^(1/6) + x) - 12*x)/b
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.10 \[ \int \frac {x^6}{a+b x^6} \, dx=\operatorname {RootSum} {\left (46656 t^{6} b^{7} + a, \left ( t \mapsto t \log {\left (- 6 t b + x \right )} \right )\right )} + \frac {x}{b} \]
Time = 0.28 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.88 \[ \int \frac {x^6}{a+b x^6} \, dx=-\frac {\frac {\sqrt {3} a^{\frac {1}{6}} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{b^{\frac {1}{6}}} - \frac {\sqrt {3} a^{\frac {1}{6}} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{b^{\frac {1}{6}}} + \frac {4 \, a^{\frac {1}{3}} \arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, a^{\frac {1}{3}} \arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, a^{\frac {1}{3}} \arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}}{12 \, b} + \frac {x}{b} \]
-1/12*(sqrt(3)*a^(1/6)*log(b^(1/3)*x^2 + sqrt(3)*a^(1/6)*b^(1/6)*x + a^(1/ 3))/b^(1/6) - sqrt(3)*a^(1/6)*log(b^(1/3)*x^2 - sqrt(3)*a^(1/6)*b^(1/6)*x + a^(1/3))/b^(1/6) + 4*a^(1/3)*arctan(b^(1/3)*x/sqrt(a^(1/3)*b^(1/3)))/sqr t(a^(1/3)*b^(1/3)) + 2*a^(1/3)*arctan((2*b^(1/3)*x + sqrt(3)*a^(1/6)*b^(1/ 6))/sqrt(a^(1/3)*b^(1/3)))/sqrt(a^(1/3)*b^(1/3)) + 2*a^(1/3)*arctan((2*b^( 1/3)*x - sqrt(3)*a^(1/6)*b^(1/6))/sqrt(a^(1/3)*b^(1/3)))/sqrt(a^(1/3)*b^(1 /3)))/b + x/b
Time = 0.27 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.82 \[ \int \frac {x^6}{a+b x^6} \, dx=\frac {x}{b} - \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, b^{2}} + \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, b^{2}} - \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, b^{2}} - \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, b^{2}} - \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, b^{2}} \]
x/b - 1/12*sqrt(3)*(a*b^5)^(1/6)*log(x^2 + sqrt(3)*x*(a/b)^(1/6) + (a/b)^( 1/3))/b^2 + 1/12*sqrt(3)*(a*b^5)^(1/6)*log(x^2 - sqrt(3)*x*(a/b)^(1/6) + ( a/b)^(1/3))/b^2 - 1/6*(a*b^5)^(1/6)*arctan((2*x + sqrt(3)*(a/b)^(1/6))/(a/ b)^(1/6))/b^2 - 1/6*(a*b^5)^(1/6)*arctan((2*x - sqrt(3)*(a/b)^(1/6))/(a/b) ^(1/6))/b^2 - 1/3*(a*b^5)^(1/6)*arctan(x/(a/b)^(1/6))/b^2
Time = 5.55 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.03 \[ \int \frac {x^6}{a+b x^6} \, dx=\frac {x}{b}+\frac {{\left (-a\right )}^{1/6}\,\mathrm {atan}\left (\frac {b^{1/6}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/6}}\right )\,1{}\mathrm {i}}{3\,b^{7/6}}+\frac {{\left (-a\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-a\right )}^{25/6}\,x\,1{}\mathrm {i}}{b^{1/6}\,\left (\frac {{\left (-a\right )}^{13/3}}{b^{1/3}}+\frac {\sqrt {3}\,{\left (-a\right )}^{13/3}\,1{}\mathrm {i}}{b^{1/3}}\right )}+\frac {\sqrt {3}\,{\left (-a\right )}^{25/6}\,x}{b^{1/6}\,\left (\frac {{\left (-a\right )}^{13/3}}{b^{1/3}}+\frac {\sqrt {3}\,{\left (-a\right )}^{13/3}\,1{}\mathrm {i}}{b^{1/3}}\right )}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3\,b^{7/6}}-\frac {{\left (-a\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-a\right )}^{25/6}\,x\,1{}\mathrm {i}}{b^{1/6}\,\left (\frac {{\left (-a\right )}^{13/3}}{b^{1/3}}-\frac {\sqrt {3}\,{\left (-a\right )}^{13/3}\,1{}\mathrm {i}}{b^{1/3}}\right )}-\frac {\sqrt {3}\,{\left (-a\right )}^{25/6}\,x}{b^{1/6}\,\left (\frac {{\left (-a\right )}^{13/3}}{b^{1/3}}-\frac {\sqrt {3}\,{\left (-a\right )}^{13/3}\,1{}\mathrm {i}}{b^{1/3}}\right )}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3\,b^{7/6}} \]
x/b + ((-a)^(1/6)*atan((b^(1/6)*x*1i)/(-a)^(1/6))*1i)/(3*b^(7/6)) + ((-a)^ (1/6)*atan(((-a)^(25/6)*x*1i)/(b^(1/6)*((-a)^(13/3)/b^(1/3) + (3^(1/2)*(-a )^(13/3)*1i)/b^(1/3))) + (3^(1/2)*(-a)^(25/6)*x)/(b^(1/6)*((-a)^(13/3)/b^( 1/3) + (3^(1/2)*(-a)^(13/3)*1i)/b^(1/3))))*((3^(1/2)*1i)/2 - 1/2)*1i)/(3*b ^(7/6)) - ((-a)^(1/6)*atan(((-a)^(25/6)*x*1i)/(b^(1/6)*((-a)^(13/3)/b^(1/3 ) - (3^(1/2)*(-a)^(13/3)*1i)/b^(1/3))) - (3^(1/2)*(-a)^(25/6)*x)/(b^(1/6)* ((-a)^(13/3)/b^(1/3) - (3^(1/2)*(-a)^(13/3)*1i)/b^(1/3))))*((3^(1/2)*1i)/2 + 1/2)*1i)/(3*b^(7/6))