Integrand size = 9, antiderivative size = 167 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {x}{6 a \left (a+b x^6\right )}+\frac {5 \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}-\frac {5 \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac {5 \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac {5 \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x}{\sqrt [3]{a}+\sqrt [3]{b} x^2}\right )}{12 \sqrt {3} a^{11/6} \sqrt [6]{b}} \] Output:
1/6*x/a/(b*x^6+a)+5/18*arctan(b^(1/6)*x/a^(1/6))/a^(11/6)/b^(1/6)+5/36*arc tan(-3^(1/2)+2*b^(1/6)*x/a^(1/6))/a^(11/6)/b^(1/6)+5/36*arctan(3^(1/2)+2*b ^(1/6)*x/a^(1/6))/a^(11/6)/b^(1/6)+5/36*arctanh(3^(1/2)*a^(1/6)*b^(1/6)*x/ (a^(1/3)+b^(1/3)*x^2))*3^(1/2)/a^(11/6)/b^(1/6)
Time = 0.08 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {\frac {12 a^{5/6} x}{a+b x^6}+\frac {20 \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}-\frac {10 \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}+\frac {10 \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}-\frac {5 \sqrt {3} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{\sqrt [6]{b}}+\frac {5 \sqrt {3} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{\sqrt [6]{b}}}{72 a^{11/6}} \] Input:
Integrate[(a + b*x^6)^(-2),x]
Output:
((12*a^(5/6)*x)/(a + b*x^6) + (20*ArcTan[(b^(1/6)*x)/a^(1/6)])/b^(1/6) - ( 10*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)])/b^(1/6) + (10*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)])/b^(1/6) - (5*Sqrt[3]*Log[a^(1/3) - Sqrt[3]*a^(1/6 )*b^(1/6)*x + b^(1/3)*x^2])/b^(1/6) + (5*Sqrt[3]*Log[a^(1/3) + Sqrt[3]*a^( 1/6)*b^(1/6)*x + b^(1/3)*x^2])/b^(1/6))/(72*a^(11/6))
Time = 0.65 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.41, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.111, Rules used = {749, 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.
| \(\displaystyle \int \frac {1}{\left (a+b x^6\right )^2} \, dx\) |
| \(\Big \downarrow \) 749 |
| \(\displaystyle \frac {5 \int \frac {1}{b x^6+a}dx}{6 a}+\frac {x}{6 a \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle \frac {5 \left (\frac {\int \frac {1}{\sqrt [3]{b} x^2+\sqrt [3]{a}}dx}{3 a^{2/3}}+\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} x}{2 \left (\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}\right )}dx}{3 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{b} x+2 \sqrt [6]{a}}{2 \left (\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}\right )}dx}{3 a^{5/6}}\right )}{6 a}+\frac {x}{6 a \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {\int \frac {1}{\sqrt [3]{b} x^2+\sqrt [3]{a}}dx}{3 a^{2/3}}+\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{b} x+2 \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 a^{5/6}}\right )}{6 a}+\frac {x}{6 a \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {5 \left (\frac {\int \frac {2 \sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{b} x+2 \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{6 a}+\frac {x}{6 a \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {5 \left (\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x\right )}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{6 a}+\frac {x}{6 a \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \left (\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x\right )}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{6 a}+\frac {x}{6 a \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5 \left (\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{6 a}+\frac {x}{6 a \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {5 \left (\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {\int \frac {1}{-\left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {\int \frac {1}{-\left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{6 a}+\frac {x}{6 a \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {5 \left (\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{6 a}+\frac {x}{6 a \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {5 \left (\frac {-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{2 \sqrt [6]{b}}}{6 a^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}\right )}{6 a}+\frac {x}{6 a \left (a+b x^6\right )}\) |
Input:
Int[(a + b*x^6)^(-2),x]
Output:
x/(6*a*(a + b*x^6)) + (5*(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)) - (Sqr t[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)) + (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))))/(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_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (Inte gerQ[2*p] || Denominator[p + 1/n] < Denominator[p])
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_) + (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 = 0.58 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.28
| method | result | size |
| risch | \(\frac {x}{6 a \left (b \,x^{6}+a \right )}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{36 a b}\) | \(46\) |
| default | \(-\frac {\left (\frac {a}{b}\right )^{\frac {1}{3}} x}{36 a^{2} \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}-\frac {\sqrt {\frac {a}{b}}\, \sqrt {3}}{36 a^{2} \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}+\frac {5 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a^{2}}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{36 a^{2}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{3}} x}{36 a^{2} \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}+\frac {\sqrt {\frac {a}{b}}\, \sqrt {3}}{36 a^{2} \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}-\frac {5 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a^{2}}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{36 a^{2}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{3}} x}{18 a^{2} \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 a^{2}}\) | \(343\) |
Input:
int(1/(b*x^6+a)^2,x,method=_RETURNVERBOSE)
Output:
1/6*x/a/(b*x^6+a)+5/36/a/b*sum(1/_R^5*ln(x-_R),_R=RootOf(_Z^6*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (115) = 230\).
Time = 0.07 (sec) , antiderivative size = 351, normalized size of antiderivative = 2.10 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {10 \, {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (a^{2} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x\right ) - 10 \, {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (-a^{2} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x\right ) + 5 \, {\left (a b x^{6} + a^{2} + \sqrt {-3} {\left (a b x^{6} + a^{2}\right )}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a^{2} + a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x\right ) - 5 \, {\left (a b x^{6} + a^{2} + \sqrt {-3} {\left (a b x^{6} + a^{2}\right )}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a^{2} + a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x\right ) - 5 \, {\left (a b x^{6} + a^{2} - \sqrt {-3} {\left (a b x^{6} + a^{2}\right )}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a^{2} - a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x\right ) + 5 \, {\left (a b x^{6} + a^{2} - \sqrt {-3} {\left (a b x^{6} + a^{2}\right )}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a^{2} - a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x\right ) + 12 \, x}{72 \, {\left (a b x^{6} + a^{2}\right )}} \] Input:
integrate(1/(b*x^6+a)^2,x, algorithm="fricas")
Output:
1/72*(10*(a*b*x^6 + a^2)*(-1/(a^11*b))^(1/6)*log(a^2*(-1/(a^11*b))^(1/6) + x) - 10*(a*b*x^6 + a^2)*(-1/(a^11*b))^(1/6)*log(-a^2*(-1/(a^11*b))^(1/6) + x) + 5*(a*b*x^6 + a^2 + sqrt(-3)*(a*b*x^6 + a^2))*(-1/(a^11*b))^(1/6)*lo g(1/2*(sqrt(-3)*a^2 + a^2)*(-1/(a^11*b))^(1/6) + x) - 5*(a*b*x^6 + a^2 + s qrt(-3)*(a*b*x^6 + a^2))*(-1/(a^11*b))^(1/6)*log(-1/2*(sqrt(-3)*a^2 + a^2) *(-1/(a^11*b))^(1/6) + x) - 5*(a*b*x^6 + a^2 - sqrt(-3)*(a*b*x^6 + a^2))*( -1/(a^11*b))^(1/6)*log(1/2*(sqrt(-3)*a^2 - a^2)*(-1/(a^11*b))^(1/6) + x) + 5*(a*b*x^6 + a^2 - sqrt(-3)*(a*b*x^6 + a^2))*(-1/(a^11*b))^(1/6)*log(-1/2 *(sqrt(-3)*a^2 - a^2)*(-1/(a^11*b))^(1/6) + x) + 12*x)/(a*b*x^6 + a^2)
Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.23 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {x}{6 a^{2} + 6 a b x^{6}} + \operatorname {RootSum} {\left (2176782336 t^{6} a^{11} b + 15625, \left ( t \mapsto t \log {\left (\frac {36 t a^{2}}{5} + x \right )} \right )\right )} \] Input:
integrate(1/(b*x**6+a)**2,x)
Output:
x/(6*a**2 + 6*a*b*x**6) + RootSum(2176782336*_t**6*a**11*b + 15625, Lambda (_t, _t*log(36*_t*a**2/5 + x)))
Time = 0.13 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {x}{6 \, {\left (a b x^{6} + a^{2}\right )}} + \frac {5 \, {\left (\frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} - \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} + \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{72 \, a} \] Input:
integrate(1/(b*x^6+a)^2,x, algorithm="maxima")
Output:
1/6*x/(a*b*x^6 + a^2) + 5/72*(sqrt(3)*log(b^(1/3)*x^2 + sqrt(3)*a^(1/6)*b^ (1/6)*x + a^(1/3))/(a^(5/6)*b^(1/6)) - sqrt(3)*log(b^(1/3)*x^2 - sqrt(3)*a ^(1/6)*b^(1/6)*x + a^(1/3))/(a^(5/6)*b^(1/6)) + 4*arctan(b^(1/3)*x/sqrt(a^ (1/3)*b^(1/3)))/(a^(2/3)*sqrt(a^(1/3)*b^(1/3))) + 2*arctan((2*b^(1/3)*x + sqrt(3)*a^(1/6)*b^(1/6))/sqrt(a^(1/3)*b^(1/3)))/(a^(2/3)*sqrt(a^(1/3)*b^(1 /3))) + 2*arctan((2*b^(1/3)*x - sqrt(3)*a^(1/6)*b^(1/6))/sqrt(a^(1/3)*b^(1 /3)))/(a^(2/3)*sqrt(a^(1/3)*b^(1/3))))/a
Time = 0.12 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {x}{6 \, {\left (b x^{6} + a\right )} a} + \frac {5 \, \sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a^{2} b} - \frac {5 \, \sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a^{2} b} + \frac {5 \, \left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a^{2} b} + \frac {5 \, \left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a^{2} b} + \frac {5 \, \left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, a^{2} b} \] Input:
integrate(1/(b*x^6+a)^2,x, algorithm="giac")
Output:
1/6*x/((b*x^6 + a)*a) + 5/72*sqrt(3)*(a*b^5)^(1/6)*log(x^2 + sqrt(3)*x*(a/ b)^(1/6) + (a/b)^(1/3))/(a^2*b) - 5/72*sqrt(3)*(a*b^5)^(1/6)*log(x^2 - sqr t(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/(a^2*b) + 5/36*(a*b^5)^(1/6)*arctan((2*x + sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a^2*b) + 5/36*(a*b^5)^(1/6)*arctan(( 2*x - sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a^2*b) + 5/18*(a*b^5)^(1/6)*arcta n(x/(a/b)^(1/6))/(a^2*b)
Time = 0.19 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.45 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {x}{6\,a\,\left (b\,x^6+a\right )}-\frac {\mathrm {atan}\left (\frac {b^{1/6}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/6}}\right )\,5{}\mathrm {i}}{18\,{\left (-a\right )}^{11/6}\,b^{1/6}}+\frac {\mathrm {atan}\left (\frac {b^{29/6}\,x\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{35/6}\,\left (\frac {3125\,b^{14/3}}{7776\,{\left (-a\right )}^{17/3}}-\frac {\sqrt {3}\,b^{14/3}\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{17/3}}\right )}-\frac {3125\,\sqrt {3}\,b^{29/6}\,x}{7776\,{\left (-a\right )}^{35/6}\,\left (\frac {3125\,b^{14/3}}{7776\,{\left (-a\right )}^{17/3}}-\frac {\sqrt {3}\,b^{14/3}\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{17/3}}\right )}\right )\,\left (5+\sqrt {3}\,5{}\mathrm {i}\right )\,1{}\mathrm {i}}{36\,{\left (-a\right )}^{11/6}\,b^{1/6}}-\frac {\mathrm {atan}\left (\frac {b^{29/6}\,x\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{35/6}\,\left (\frac {3125\,b^{14/3}}{7776\,{\left (-a\right )}^{17/3}}+\frac {\sqrt {3}\,b^{14/3}\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{17/3}}\right )}+\frac {3125\,\sqrt {3}\,b^{29/6}\,x}{7776\,{\left (-a\right )}^{35/6}\,\left (\frac {3125\,b^{14/3}}{7776\,{\left (-a\right )}^{17/3}}+\frac {\sqrt {3}\,b^{14/3}\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{17/3}}\right )}\right )\,\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )\,1{}\mathrm {i}}{36\,{\left (-a\right )}^{11/6}\,b^{1/6}} \] Input:
int(1/(a + b*x^6)^2,x)
Output:
x/(6*a*(a + b*x^6)) - (atan((b^(1/6)*x*1i)/(-a)^(1/6))*5i)/(18*(-a)^(11/6) *b^(1/6)) + (atan((b^(29/6)*x*3125i)/(7776*(-a)^(35/6)*((3125*b^(14/3))/(7 776*(-a)^(17/3)) - (3^(1/2)*b^(14/3)*3125i)/(7776*(-a)^(17/3)))) - (3125*3 ^(1/2)*b^(29/6)*x)/(7776*(-a)^(35/6)*((3125*b^(14/3))/(7776*(-a)^(17/3)) - (3^(1/2)*b^(14/3)*3125i)/(7776*(-a)^(17/3)))))*(3^(1/2)*5i + 5)*1i)/(36*( -a)^(11/6)*b^(1/6)) - (atan((b^(29/6)*x*3125i)/(7776*(-a)^(35/6)*((3125*b^ (14/3))/(7776*(-a)^(17/3)) + (3^(1/2)*b^(14/3)*3125i)/(7776*(-a)^(17/3)))) + (3125*3^(1/2)*b^(29/6)*x)/(7776*(-a)^(35/6)*((3125*b^(14/3))/(7776*(-a) ^(17/3)) + (3^(1/2)*b^(14/3)*3125i)/(7776*(-a)^(17/3)))))*(3^(1/2)*5i - 5) *1i)/(36*(-a)^(11/6)*b^(1/6))
Time = 0.26 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.98 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {-10 b^{\frac {5}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 b^{\frac {1}{3}} x}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )-10 b^{\frac {11}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 b^{\frac {1}{3}} x}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) x^{6}+10 b^{\frac {5}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 b^{\frac {1}{3}} x}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )+10 b^{\frac {11}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 b^{\frac {1}{3}} x}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right ) x^{6}+20 b^{\frac {5}{6}} a^{\frac {7}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} x}{a^{\frac {1}{6}}}\right )+20 b^{\frac {11}{6}} a^{\frac {1}{6}} \mathit {atan} \left (\frac {b^{\frac {1}{6}} x}{a^{\frac {1}{6}}}\right ) x^{6}-5 b^{\frac {5}{6}} a^{\frac {7}{6}} \sqrt {3}\, \mathrm {log}\left (-b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+b^{\frac {1}{3}} x^{2}\right )-5 b^{\frac {11}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (-b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+b^{\frac {1}{3}} x^{2}\right ) x^{6}+5 b^{\frac {5}{6}} a^{\frac {7}{6}} \sqrt {3}\, \mathrm {log}\left (b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+b^{\frac {1}{3}} x^{2}\right )+5 b^{\frac {11}{6}} a^{\frac {1}{6}} \sqrt {3}\, \mathrm {log}\left (b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}}+b^{\frac {1}{3}} x^{2}\right ) x^{6}+12 a b x}{72 a^{2} b \left (b \,x^{6}+a \right )} \] Input:
int(1/(b*x^6+a)^2,x)
Output:
( - 10*b**(5/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*b**(1/3)*x)/( b**(1/6)*a**(1/6)))*a - 10*b**(5/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt( 3) - 2*b**(1/3)*x)/(b**(1/6)*a**(1/6)))*b*x**6 + 10*b**(5/6)*a**(1/6)*atan ((b**(1/6)*a**(1/6)*sqrt(3) + 2*b**(1/3)*x)/(b**(1/6)*a**(1/6)))*a + 10*b* *(5/6)*a**(1/6)*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*b**(1/3)*x)/(b**(1/6)* a**(1/6)))*b*x**6 + 20*b**(5/6)*a**(1/6)*atan((b**(1/3)*x)/(b**(1/6)*a**(1 /6)))*a + 20*b**(5/6)*a**(1/6)*atan((b**(1/3)*x)/(b**(1/6)*a**(1/6)))*b*x* *6 - 5*b**(5/6)*a**(1/6)*sqrt(3)*log( - b**(1/6)*a**(1/6)*sqrt(3)*x + a**( 1/3) + b**(1/3)*x**2)*a - 5*b**(5/6)*a**(1/6)*sqrt(3)*log( - b**(1/6)*a**( 1/6)*sqrt(3)*x + a**(1/3) + b**(1/3)*x**2)*b*x**6 + 5*b**(5/6)*a**(1/6)*sq rt(3)*log(b**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + b**(1/3)*x**2)*a + 5*b* *(5/6)*a**(1/6)*sqrt(3)*log(b**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3) + b**(1 /3)*x**2)*b*x**6 + 12*a*b*x)/(72*a**2*b*(a + b*x**6))