Integrand size = 9, antiderivative size = 155 \[ \int \frac {1}{a+\frac {b}{x^6}} \, dx=\frac {x}{a}-\frac {\sqrt [6]{b} \arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{7/6}}+\frac {\sqrt [6]{b} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x}{\sqrt [3]{b}+\sqrt [3]{a} x^2}\right )}{2 \sqrt {3} a^{7/6}} \] Output:
x/a-1/3*b^(1/6)*arctan(a^(1/6)*x/b^(1/6))/a^(7/6)-1/6*b^(1/6)*arctan(-3^(1 /2)+2*a^(1/6)*x/b^(1/6))/a^(7/6)-1/6*b^(1/6)*arctan(3^(1/2)+2*a^(1/6)*x/b^ (1/6))/a^(7/6)-1/6*b^(1/6)*arctanh(3^(1/2)*a^(1/6)*b^(1/6)*x/(b^(1/3)+a^(1 /3)*x^2))*3^(1/2)/a^(7/6)
Time = 0.04 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.17 \[ \int \frac {1}{a+\frac {b}{x^6}} \, dx=\frac {12 \sqrt [6]{a} x-4 \sqrt [6]{b} \arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )+2 \sqrt [6]{b} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )-2 \sqrt [6]{b} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )+\sqrt {3} \sqrt [6]{b} \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )-\sqrt {3} \sqrt [6]{b} \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{12 a^{7/6}} \] Input:
Integrate[(a + b/x^6)^(-1),x]
Output:
(12*a^(1/6)*x - 4*b^(1/6)*ArcTan[(a^(1/6)*x)/b^(1/6)] + 2*b^(1/6)*ArcTan[S qrt[3] - (2*a^(1/6)*x)/b^(1/6)] - 2*b^(1/6)*ArcTan[Sqrt[3] + (2*a^(1/6)*x) /b^(1/6)] + Sqrt[3]*b^(1/6)*Log[b^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + a^(1 /3)*x^2] - Sqrt[3]*b^(1/6)*Log[b^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + a^(1/ 3)*x^2])/(12*a^(7/6))
Time = 0.77 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.43, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.222, Rules used = {772, 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.
| \(\displaystyle \int \frac {1}{a+\frac {b}{x^6}} \, dx\) |
| \(\Big \downarrow \) 772 |
| \(\displaystyle \int \frac {x^6}{a x^6+b}dx\) |
| \(\Big \downarrow \) 843 |
| \(\displaystyle \frac {x}{a}-\frac {b \int \frac {1}{a x^6+b}dx}{a}\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\int \frac {1}{\sqrt [3]{a} x^2+\sqrt [3]{b}}dx}{3 b^{2/3}}+\frac {\int \frac {2 \sqrt [6]{b}-\sqrt {3} \sqrt [6]{a} x}{2 \left (\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}\right )}dx}{3 b^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a} x+2 \sqrt [6]{b}}{2 \left (\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}\right )}dx}{3 b^{5/6}}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\int \frac {1}{\sqrt [3]{a} x^2+\sqrt [3]{b}}dx}{3 b^{2/3}}+\frac {\int \frac {2 \sqrt [6]{b}-\sqrt {3} \sqrt [6]{a} x}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 b^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a} x+2 \sqrt [6]{b}}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 b^{5/6}}\right )}{a}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\int \frac {2 \sqrt [6]{b}-\sqrt {3} \sqrt [6]{a} x}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 b^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a} x+2 \sqrt [6]{b}}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 b^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {1}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx-\frac {\sqrt {3} \int -\frac {\sqrt [6]{a} \left (\sqrt {3} \sqrt [6]{b}-2 \sqrt [6]{a} x\right )}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{2 \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{a} \left (2 \sqrt [6]{a} x+\sqrt {3} \sqrt [6]{b}\right )}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{2 \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {1}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{a} \left (\sqrt {3} \sqrt [6]{b}-2 \sqrt [6]{a} x\right )}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{2 \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{a} \left (2 \sqrt [6]{a} x+\sqrt {3} \sqrt [6]{b}\right )}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{2 \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {1}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 b^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{a} x+\sqrt {3} \sqrt [6]{b}}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 b^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {\int \frac {1}{-\left (1-\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}\right )}{\sqrt {3} \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{a} x+\sqrt {3} \sqrt [6]{b}}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx-\frac {\int \frac {1}{-\left (\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}+1\right )}{\sqrt {3} \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}\right )\right )}{\sqrt [6]{a}}}{6 b^{5/6}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{a} x+\sqrt {3} \sqrt [6]{b}}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}+1\right )\right )}{\sqrt [6]{a}}}{6 b^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {x}{a}-\frac {b \left (\frac {-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}\right )\right )}{\sqrt [6]{a}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{2 \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}+1\right )\right )}{\sqrt [6]{a}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{2 \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}\) |
Input:
Int[(a + b/x^6)^(-1),x]
Output:
x/a - (b*(ArcTan[(a^(1/6)*x)/b^(1/6)]/(3*a^(1/6)*b^(5/6)) + (-(ArcTan[Sqrt [3]*(1 - (2*a^(1/6)*x)/(Sqrt[3]*b^(1/6)))]/a^(1/6)) - (Sqrt[3]*Log[b^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + a^(1/3)*x^2])/(2*a^(1/6)))/(6*b^(5/6)) + (A rcTan[Sqrt[3]*(1 + (2*a^(1/6)*x)/(Sqrt[3]*b^(1/6)))]/a^(1/6) + (Sqrt[3]*Lo g[b^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + a^(1/3)*x^2])/(2*a^(1/6)))/(6*b^(5 /6))))/a
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[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && ILtQ[n, 0] && IntegerQ[p]
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 = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.22
| method | result | size |
| risch | \(\frac {x}{a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} a +b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{6 a^{2}}\) | \(34\) |
| default | \(\frac {x}{a}-\frac {\left (\frac {\left (\frac {b}{a}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{3 b}+\frac {\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 b}+\frac {\left (\frac {b}{a}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b}-\frac {\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 b}+\frac {\left (\frac {b}{a}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b}\right ) b}{a}\) | \(171\) |
Input:
int(1/(a+b/x^6),x,method=_RETURNVERBOSE)
Output:
x/a-1/6/a^2*b*sum(1/_R^5*ln(x-_R),_R=RootOf(_Z^6*a+b))
Leaf count of result is larger than twice the leaf count of optimal. 212 vs. \(2 (105) = 210\).
Time = 0.09 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.37 \[ \int \frac {1}{a+\frac {b}{x^6}} \, dx=-\frac {{\left (\sqrt {-3} a + a\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a + a\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x\right ) - {\left (\sqrt {-3} a + a\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a + a\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x\right ) + {\left (\sqrt {-3} a - a\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a - a\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x\right ) - {\left (\sqrt {-3} a - a\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a - a\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x\right ) + 2 \, a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x\right ) - 2 \, a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (-a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x\right ) - 12 \, x}{12 \, a} \] Input:
integrate(1/(a+b/x^6),x, algorithm="fricas")
Output:
-1/12*((sqrt(-3)*a + a)*(-b/a^7)^(1/6)*log(1/2*(sqrt(-3)*a + a)*(-b/a^7)^( 1/6) + x) - (sqrt(-3)*a + a)*(-b/a^7)^(1/6)*log(-1/2*(sqrt(-3)*a + a)*(-b/ a^7)^(1/6) + x) + (sqrt(-3)*a - a)*(-b/a^7)^(1/6)*log(1/2*(sqrt(-3)*a - a) *(-b/a^7)^(1/6) + x) - (sqrt(-3)*a - a)*(-b/a^7)^(1/6)*log(-1/2*(sqrt(-3)* a - a)*(-b/a^7)^(1/6) + x) + 2*a*(-b/a^7)^(1/6)*log(a*(-b/a^7)^(1/6) + x) - 2*a*(-b/a^7)^(1/6)*log(-a*(-b/a^7)^(1/6) + x) - 12*x)/a
Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.14 \[ \int \frac {1}{a+\frac {b}{x^6}} \, dx=\operatorname {RootSum} {\left (46656 t^{6} a^{7} + b, \left ( t \mapsto t \log {\left (- 6 t a + x \right )} \right )\right )} + \frac {x}{a} \] Input:
integrate(1/(a+b/x**6),x)
Output:
RootSum(46656*_t**6*a**7 + b, Lambda(_t, _t*log(-6*_t*a + x))) + x/a
Time = 0.11 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.25 \[ \int \frac {1}{a+\frac {b}{x^6}} \, dx=-\frac {\frac {\sqrt {3} b^{\frac {1}{6}} \log \left (a^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + b^{\frac {1}{3}}\right )}{a^{\frac {1}{6}}} - \frac {\sqrt {3} b^{\frac {1}{6}} \log \left (a^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + b^{\frac {1}{3}}\right )}{a^{\frac {1}{6}}} + \frac {4 \, b^{\frac {1}{3}} \arctan \left (\frac {a^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, b^{\frac {1}{3}} \arctan \left (\frac {2 \, a^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, b^{\frac {1}{3}} \arctan \left (\frac {2 \, a^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}}{12 \, a} + \frac {x}{a} \] Input:
integrate(1/(a+b/x^6),x, algorithm="maxima")
Output:
-1/12*(sqrt(3)*b^(1/6)*log(a^(1/3)*x^2 + sqrt(3)*a^(1/6)*b^(1/6)*x + b^(1/ 3))/a^(1/6) - sqrt(3)*b^(1/6)*log(a^(1/3)*x^2 - sqrt(3)*a^(1/6)*b^(1/6)*x + b^(1/3))/a^(1/6) + 4*b^(1/3)*arctan(a^(1/3)*x/sqrt(a^(1/3)*b^(1/3)))/sqr t(a^(1/3)*b^(1/3)) + 2*b^(1/3)*arctan((2*a^(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*b^(1/3)*arctan((2*a^( 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)))/a + x/a
Time = 0.12 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.16 \[ \int \frac {1}{a+\frac {b}{x^6}} \, dx=\frac {x}{a} - \frac {\sqrt {3} \left (a^{5} b\right )^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{6}} + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 \, a^{2}} + \frac {\sqrt {3} \left (a^{5} b\right )^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{6}} + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 \, a^{2}} - \frac {\left (a^{5} b\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {b}{a}\right )^{\frac {1}{6}}}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{6 \, a^{2}} - \frac {\left (a^{5} b\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {b}{a}\right )^{\frac {1}{6}}}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{6 \, a^{2}} - \frac {\left (a^{5} b\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{3 \, a^{2}} \] Input:
integrate(1/(a+b/x^6),x, algorithm="giac")
Output:
x/a - 1/12*sqrt(3)*(a^5*b)^(1/6)*log(x^2 + sqrt(3)*x*(b/a)^(1/6) + (b/a)^( 1/3))/a^2 + 1/12*sqrt(3)*(a^5*b)^(1/6)*log(x^2 - sqrt(3)*x*(b/a)^(1/6) + ( b/a)^(1/3))/a^2 - 1/6*(a^5*b)^(1/6)*arctan((2*x + sqrt(3)*(b/a)^(1/6))/(b/ a)^(1/6))/a^2 - 1/6*(a^5*b)^(1/6)*arctan((2*x - sqrt(3)*(b/a)^(1/6))/(b/a) ^(1/6))/a^2 - 1/3*(a^5*b)^(1/6)*arctan(x/(b/a)^(1/6))/a^2
Time = 0.45 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.46 \[ \int \frac {1}{a+\frac {b}{x^6}} \, dx=\frac {x}{a}+\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {a^{1/6}\,x\,1{}\mathrm {i}}{{\left (-b\right )}^{1/6}}\right )\,1{}\mathrm {i}}{3\,a^{7/6}}+\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{25/6}\,x\,1{}\mathrm {i}}{a^{1/6}\,\left (\frac {{\left (-b\right )}^{13/3}}{a^{1/3}}+\frac {\sqrt {3}\,{\left (-b\right )}^{13/3}\,1{}\mathrm {i}}{a^{1/3}}\right )}+\frac {\sqrt {3}\,{\left (-b\right )}^{25/6}\,x}{a^{1/6}\,\left (\frac {{\left (-b\right )}^{13/3}}{a^{1/3}}+\frac {\sqrt {3}\,{\left (-b\right )}^{13/3}\,1{}\mathrm {i}}{a^{1/3}}\right )}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3\,a^{7/6}}-\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{25/6}\,x\,1{}\mathrm {i}}{a^{1/6}\,\left (\frac {{\left (-b\right )}^{13/3}}{a^{1/3}}-\frac {\sqrt {3}\,{\left (-b\right )}^{13/3}\,1{}\mathrm {i}}{a^{1/3}}\right )}-\frac {\sqrt {3}\,{\left (-b\right )}^{25/6}\,x}{a^{1/6}\,\left (\frac {{\left (-b\right )}^{13/3}}{a^{1/3}}-\frac {\sqrt {3}\,{\left (-b\right )}^{13/3}\,1{}\mathrm {i}}{a^{1/3}}\right )}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3\,a^{7/6}} \] Input:
int(1/(a + b/x^6),x)
Output:
x/a + ((-b)^(1/6)*atan((a^(1/6)*x*1i)/(-b)^(1/6))*1i)/(3*a^(7/6)) + ((-b)^ (1/6)*atan(((-b)^(25/6)*x*1i)/(a^(1/6)*((-b)^(13/3)/a^(1/3) + (3^(1/2)*(-b )^(13/3)*1i)/a^(1/3))) + (3^(1/2)*(-b)^(25/6)*x)/(a^(1/6)*((-b)^(13/3)/a^( 1/3) + (3^(1/2)*(-b)^(13/3)*1i)/a^(1/3))))*((3^(1/2)*1i)/2 - 1/2)*1i)/(3*a ^(7/6)) - ((-b)^(1/6)*atan(((-b)^(25/6)*x*1i)/(a^(1/6)*((-b)^(13/3)/a^(1/3 ) - (3^(1/2)*(-b)^(13/3)*1i)/a^(1/3))) - (3^(1/2)*(-b)^(25/6)*x)/(a^(1/6)* ((-b)^(13/3)/a^(1/3) - (3^(1/2)*(-b)^(13/3)*1i)/a^(1/3))))*((3^(1/2)*1i)/2 + 1/2)*1i)/(3*a^(7/6))
Time = 0.22 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.01 \[ \int \frac {1}{a+\frac {b}{x^6}} \, dx=\frac {2 b^{\frac {1}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 a^{\frac {1}{3}} x}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )-2 b^{\frac {1}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 a^{\frac {1}{3}} x}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )-4 b^{\frac {1}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {a^{\frac {1}{6}} x}{b^{\frac {1}{6}}}\right )+b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (-b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}} x^{2}+b^{\frac {1}{3}}\right )-b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}} x^{2}+b^{\frac {1}{3}}\right )+12 a^{\frac {1}{3}} x}{12 a^{\frac {4}{3}}} \] Input:
int(1/(a+b/x^6),x)
Output:
(2*b**(1/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*a**(1/3)*x)/(b**( 1/6)*a**(1/6))) - 2*b**(1/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2* a**(1/3)*x)/(b**(1/6)*a**(1/6))) - 4*b**(1/6)*a**(1/6)*atan((a**(1/3)*x)/( b**(1/6)*a**(1/6))) + b**(1/6)*a**(1/6)*sqrt(3)*log( - b**(1/6)*a**(1/6)*s qrt(3)*x + a**(1/3)*x**2 + b**(1/3)) - b**(1/6)*a**(1/6)*sqrt(3)*log(b**(1 /6)*a**(1/6)*sqrt(3)*x + a**(1/3)*x**2 + b**(1/3)) + 12*a**(1/3)*x)/(12*a* *(1/3)*a)