Integrand size = 13, antiderivative size = 244 \[ \int \frac {1}{x^2 \left (a+b x^6\right )^2} \, dx=-\frac {7}{6 a^2 x}+\frac {1}{6 a x \left (a+b x^6\right )}-\frac {7 \sqrt [6]{b} \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{13/6}}+\frac {7 \sqrt [6]{b} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac {7 \sqrt [6]{b} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{13/6}}-\frac {7 \sqrt [6]{b} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{13/6}}+\frac {7 \sqrt [6]{b} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{13/6}} \]
-7/6/a^2/x+1/6/a/x/(b*x^6+a)-7/18*b^(1/6)*arctan(b^(1/6)*x/a^(1/6))/a^(13/ 6)+7/36*b^(1/6)*arctan((-2*b^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6))/a^(13/6)-7/ 36*b^(1/6)*arctan((2*b^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6))/a^(13/6)-7/72*b^( 1/6)*ln(a^(1/3)+b^(1/3)*x^2-a^(1/6)*b^(1/6)*x*3^(1/2))/a^(13/6)*3^(1/2)+7/ 72*b^(1/6)*ln(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*3^(1/2))/a^(13/6)*3^(1 /2)
Time = 0.17 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.84 \[ \int \frac {1}{x^2 \left (a+b x^6\right )^2} \, dx=\frac {-\frac {72 \sqrt [6]{a}}{x}-\frac {12 \sqrt [6]{a} b x^5}{a+b x^6}-28 \sqrt [6]{b} \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )+14 \sqrt [6]{b} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-14 \sqrt [6]{b} \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-7 \sqrt {3} \sqrt [6]{b} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )+7 \sqrt {3} \sqrt [6]{b} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{72 a^{13/6}} \]
((-72*a^(1/6))/x - (12*a^(1/6)*b*x^5)/(a + b*x^6) - 28*b^(1/6)*ArcTan[(b^( 1/6)*x)/a^(1/6)] + 14*b^(1/6)*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] - 14 *b^(1/6)*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] - 7*Sqrt[3]*b^(1/6)*Log[a ^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2] + 7*Sqrt[3]*b^(1/6)*Log[ a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(72*a^(13/6))
Time = 0.47 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {819, 847, 824, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (a+b x^6\right )^2} \, dx\) |
| \(\Big \downarrow \) 819 |
| \(\displaystyle \frac {7 \int \frac {1}{x^2 \left (b x^6+a\right )}dx}{6 a}+\frac {1}{6 a x \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 847 |
| \(\displaystyle \frac {7 \left (-\frac {b \int \frac {x^4}{b x^6+a}dx}{a}-\frac {1}{a x}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 824 |
| \(\displaystyle \frac {7 \left (-\frac {b \left (\frac {\int \frac {1}{\sqrt [3]{b} x^2+\sqrt [3]{a}}dx}{3 b^{2/3}}+\frac {\int -\frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} x}{2 \left (\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}\right )}dx}{3 \sqrt [6]{a} b^{2/3}}+\frac {\int -\frac {\sqrt {3} \sqrt [6]{b} x+\sqrt [6]{a}}{2 \left (\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}\right )}dx}{3 \sqrt [6]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 \left (-\frac {b \left (\frac {\int \frac {1}{\sqrt [3]{b} x^2+\sqrt [3]{a}}dx}{3 b^{2/3}}-\frac {\int \frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}-\frac {\int \frac {\sqrt {3} \sqrt [6]{b} x+\sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}\right )}{a}-\frac {1}{a x}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {7 \left (-\frac {b \left (-\frac {\int \frac {\sqrt [6]{a}-\sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}-\frac {\int \frac {\sqrt {3} \sqrt [6]{b} x+\sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {1}{a x}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {7 \left (-\frac {b \left (-\frac {-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {\sqrt {3} \int -\frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x\right )}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {1}{a x}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {7 \left (-\frac {b \left (-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x\right )}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \int \frac {\sqrt [6]{b} \left (2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}\right )}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{2 \sqrt [6]{b}}-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {1}{a x}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 \left (-\frac {b \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {1}{2} \sqrt [6]{a} \int \frac {1}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {1}{a x}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {7 \left (-\frac {b \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {\int \frac {1}{-\left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {\int \frac {1}{-\left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )}{\sqrt {3} \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {1}{a x}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {7 \left (-\frac {b \left (-\frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [3]{b} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx+\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{b} x+\sqrt {3} \sqrt [6]{a}}{\sqrt [3]{b} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}}dx-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {1}{a x}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^6\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {7 \left (-\frac {b \left (-\frac {\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )\right )}{\sqrt [6]{b}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{2 \sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}-\frac {\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{2 \sqrt [6]{b}}-\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}+1\right )\right )}{\sqrt [6]{b}}}{6 \sqrt [6]{a} b^{2/3}}+\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 \sqrt [6]{a} b^{5/6}}\right )}{a}-\frac {1}{a x}\right )}{6 a}+\frac {1}{6 a x \left (a+b x^6\right )}\) |
1/(6*a*x*(a + b*x^6)) + (7*(-(1/(a*x)) - (b*(ArcTan[(b^(1/6)*x)/a^(1/6)]/( 3*a^(1/6)*b^(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^(1/6)*b^(2/3)) - (-(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^(1/6)*b^(2/3))))/a))/(6*a)
3.14.41.3.1 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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-( c*x)^(m + 1))*((a + b*x^n)^(p + 1)/(a*c*n*(p + 1))), x] + Simp[(m + n*(p + 1) + 1)/(a*n*(p + 1)) Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a , b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p , x]
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator [Rt[a/b, n]], s = Denominator[Rt[a/b, n]], k, u}, Simp[u = Int[(r*Cos[(2*k - 1)*m*(Pi/n)] - s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 - 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] + Int[(r*Cos[(2*k - 1)*m*(Pi/n)] + s*Cos[(2*k - 1)*(m + 1)*(Pi/n)]*x)/(r^2 + 2*r*s*Cos[(2*k - 1)*(Pi/n)]*x + s^2*x^2), x] ; 2*(-1)^(m/2)*(r^(m + 2)/(a*n*s^m)) Int[1/(r^2 + s^2*x^2), x] + 2*(r^(m + 1)/(a*n*s^m)) Sum[u, {k, 1, (n - 2)/4}], x]] /; FreeQ[{a, b}, x] && IGt Q[(n - 2)/4, 0] && IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x )^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))) Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a , b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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 = 4.40 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.29
| method | result | size |
| risch | \(\frac {-\frac {7 b \,x^{6}}{6 a^{2}}-\frac {1}{a}}{x \left (b \,x^{6}+a \right )}+\frac {7 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{13} \textit {\_Z}^{6}+b \right )}{\sum }\textit {\_R} \ln \left (\left (7 \textit {\_R}^{6} a^{13}+6 b \right ) x +a^{11} \textit {\_R}^{5}\right )\right )}{36}\) | \(70\) |
| default | \(-\frac {b \left (\frac {x^{5}}{6 b \,x^{6}+6 a}+\frac {7 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {7 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a}+\frac {7 \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{36 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {7 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a}+\frac {7 \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{36 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{a^{2}}-\frac {1}{a^{2} x}\) | \(188\) |
(-7/6*b/a^2*x^6-1/a)/x/(b*x^6+a)+7/36*sum(_R*ln((7*_R^6*a^13+6*b)*x+a^11*_ R^5),_R=RootOf(_Z^6*a^13+b))
Leaf count of result is larger than twice the leaf count of optimal. 396 vs. \(2 (168) = 336\).
Time = 0.28 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.62 \[ \int \frac {1}{x^2 \left (a+b x^6\right )^2} \, dx=-\frac {84 \, b x^{6} + 14 \, {\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} \log \left (16807 \, a^{11} \left (-\frac {b}{a^{13}}\right )^{\frac {5}{6}} + 16807 \, b x\right ) - 14 \, {\left (a^{2} b x^{7} + a^{3} x\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} \log \left (-16807 \, a^{11} \left (-\frac {b}{a^{13}}\right )^{\frac {5}{6}} + 16807 \, b x\right ) + 7 \, {\left (a^{2} b x^{7} + a^{3} x - \sqrt {-3} {\left (a^{2} b x^{7} + a^{3} x\right )}\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} \log \left (16807 \, b x + \frac {16807}{2} \, {\left (\sqrt {-3} a^{11} + a^{11}\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {5}{6}}\right ) - 7 \, {\left (a^{2} b x^{7} + a^{3} x - \sqrt {-3} {\left (a^{2} b x^{7} + a^{3} x\right )}\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} \log \left (16807 \, b x - \frac {16807}{2} \, {\left (\sqrt {-3} a^{11} + a^{11}\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {5}{6}}\right ) - 7 \, {\left (a^{2} b x^{7} + a^{3} x + \sqrt {-3} {\left (a^{2} b x^{7} + a^{3} x\right )}\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} \log \left (16807 \, b x + \frac {16807}{2} \, {\left (\sqrt {-3} a^{11} - a^{11}\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {5}{6}}\right ) + 7 \, {\left (a^{2} b x^{7} + a^{3} x + \sqrt {-3} {\left (a^{2} b x^{7} + a^{3} x\right )}\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {1}{6}} \log \left (16807 \, b x - \frac {16807}{2} \, {\left (\sqrt {-3} a^{11} - a^{11}\right )} \left (-\frac {b}{a^{13}}\right )^{\frac {5}{6}}\right ) + 72 \, a}{72 \, {\left (a^{2} b x^{7} + a^{3} x\right )}} \]
-1/72*(84*b*x^6 + 14*(a^2*b*x^7 + a^3*x)*(-b/a^13)^(1/6)*log(16807*a^11*(- b/a^13)^(5/6) + 16807*b*x) - 14*(a^2*b*x^7 + a^3*x)*(-b/a^13)^(1/6)*log(-1 6807*a^11*(-b/a^13)^(5/6) + 16807*b*x) + 7*(a^2*b*x^7 + a^3*x - sqrt(-3)*( a^2*b*x^7 + a^3*x))*(-b/a^13)^(1/6)*log(16807*b*x + 16807/2*(sqrt(-3)*a^11 + a^11)*(-b/a^13)^(5/6)) - 7*(a^2*b*x^7 + a^3*x - sqrt(-3)*(a^2*b*x^7 + a ^3*x))*(-b/a^13)^(1/6)*log(16807*b*x - 16807/2*(sqrt(-3)*a^11 + a^11)*(-b/ a^13)^(5/6)) - 7*(a^2*b*x^7 + a^3*x + sqrt(-3)*(a^2*b*x^7 + a^3*x))*(-b/a^ 13)^(1/6)*log(16807*b*x + 16807/2*(sqrt(-3)*a^11 - a^11)*(-b/a^13)^(5/6)) + 7*(a^2*b*x^7 + a^3*x + sqrt(-3)*(a^2*b*x^7 + a^3*x))*(-b/a^13)^(1/6)*log (16807*b*x - 16807/2*(sqrt(-3)*a^11 - a^11)*(-b/a^13)^(5/6)) + 72*a)/(a^2* b*x^7 + a^3*x)
Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.23 \[ \int \frac {1}{x^2 \left (a+b x^6\right )^2} \, dx=\frac {- 6 a - 7 b x^{6}}{6 a^{3} x + 6 a^{2} b x^{7}} + \operatorname {RootSum} {\left (2176782336 t^{6} a^{13} + 117649 b, \left ( t \mapsto t \log {\left (- \frac {60466176 t^{5} a^{11}}{16807 b} + x \right )} \right )\right )} \]
(-6*a - 7*b*x**6)/(6*a**3*x + 6*a**2*b*x**7) + RootSum(2176782336*_t**6*a* *13 + 117649*b, Lambda(_t, _t*log(-60466176*_t**5*a**11/(16807*b) + x)))
Time = 0.28 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^2 \left (a+b x^6\right )^2} \, dx=-\frac {7 \, b x^{6} + 6 \, a}{6 \, {\left (a^{2} b x^{7} + a^{3} x\right )}} + \frac {7 \, b {\left (\frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{72 \, a^{2}} \]
-1/6*(7*b*x^6 + 6*a)/(a^2*b*x^7 + a^3*x) + 7/72*b*(sqrt(3)*log(b^(1/3)*x^2 + sqrt(3)*a^(1/6)*b^(1/6)*x + a^(1/3))/(a^(1/6)*b^(5/6)) - sqrt(3)*log(b^ (1/3)*x^2 - sqrt(3)*a^(1/6)*b^(1/6)*x + a^(1/3))/(a^(1/6)*b^(5/6)) - 4*arc tan(b^(1/3)*x/sqrt(a^(1/3)*b^(1/3)))/(b^(2/3)*sqrt(a^(1/3)*b^(1/3))) - 2*a rctan((2*b^(1/3)*x + sqrt(3)*a^(1/6)*b^(1/6))/sqrt(a^(1/3)*b^(1/3)))/(b^(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)))/(b^(2/3)*sqrt(a^(1/3)*b^(1/3))))/a^2
Time = 0.34 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 \left (a+b x^6\right )^2} \, dx=-\frac {7 \, b \left (\frac {a}{b}\right )^{\frac {5}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, a^{3}} - \frac {7 \, b x^{6} + 6 \, a}{6 \, {\left (b x^{7} + a x\right )} a^{2}} + \frac {7 \, \sqrt {3} \left (a b^{5}\right )^{\frac {5}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a^{3} b^{4}} - \frac {7 \, \sqrt {3} \left (a b^{5}\right )^{\frac {5}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a^{3} b^{4}} - \frac {7 \, \left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a^{3} b^{4}} - \frac {7 \, \left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a^{3} b^{4}} \]
-7/18*b*(a/b)^(5/6)*arctan(x/(a/b)^(1/6))/a^3 - 1/6*(7*b*x^6 + 6*a)/((b*x^ 7 + a*x)*a^2) + 7/72*sqrt(3)*(a*b^5)^(5/6)*log(x^2 + sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/(a^3*b^4) - 7/72*sqrt(3)*(a*b^5)^(5/6)*log(x^2 - sqrt(3)*x *(a/b)^(1/6) + (a/b)^(1/3))/(a^3*b^4) - 7/36*(a*b^5)^(5/6)*arctan((2*x + s qrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a^3*b^4) - 7/36*(a*b^5)^(5/6)*arctan((2* x - sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a^3*b^4)
Time = 5.54 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.69 \[ \int \frac {1}{x^2 \left (a+b x^6\right )^2} \, dx=-\frac {\frac {1}{a}+\frac {7\,b\,x^6}{6\,a^2}}{b\,x^7+a\,x}-\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/6}\,x\,1{}\mathrm {i}}{a^{1/6}}\right )\,7{}\mathrm {i}}{18\,a^{13/6}}-\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {a^{21/2}\,{\left (-b\right )}^{13/2}\,x\,43563744{}\mathrm {i}}{21781872\,a^{32/3}\,{\left (-b\right )}^{19/3}-\sqrt {3}\,a^{32/3}\,{\left (-b\right )}^{19/3}\,21781872{}\mathrm {i}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,7{}\mathrm {i}}{18\,a^{13/6}}+\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {a^{21/2}\,{\left (-b\right )}^{13/2}\,x\,43563744{}\mathrm {i}}{21781872\,a^{32/3}\,{\left (-b\right )}^{19/3}+\sqrt {3}\,a^{32/3}\,{\left (-b\right )}^{19/3}\,21781872{}\mathrm {i}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,7{}\mathrm {i}}{18\,a^{13/6}} \]
((-b)^(1/6)*atan((a^(21/2)*(-b)^(13/2)*x*43563744i)/(21781872*a^(32/3)*(-b )^(19/3) + 3^(1/2)*a^(32/3)*(-b)^(19/3)*21781872i))*((3^(1/2)*1i)/2 - 1/2) *7i)/(18*a^(13/6)) - ((-b)^(1/6)*atan(((-b)^(1/6)*x*1i)/a^(1/6))*7i)/(18*a ^(13/6)) - ((-b)^(1/6)*atan((a^(21/2)*(-b)^(13/2)*x*43563744i)/(21781872*a ^(32/3)*(-b)^(19/3) - 3^(1/2)*a^(32/3)*(-b)^(19/3)*21781872i))*((3^(1/2)*1 i)/2 + 1/2)*7i)/(18*a^(13/6)) - (1/a + (7*b*x^6)/(6*a^2))/(a*x + b*x^7)