Integrand size = 9, antiderivative size = 215 \[ \int \frac {1}{b+a x^6} \, dx=\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{6 \sqrt [6]{a} b^{5/6}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{b}+2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{6 \sqrt [6]{a} b^{5/6}}-\frac {\log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{4 \sqrt {3} \sqrt [6]{a} b^{5/6}}+\frac {\log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{4 \sqrt {3} \sqrt [6]{a} b^{5/6}} \] Output:
1/3*arctan(a^(1/6)*x/b^(1/6))/a^(1/6)/b^(5/6)-1/6*arctan((3^(1/2)*b^(1/6)- 2*a^(1/6)*x)/b^(1/6))/a^(1/6)/b^(5/6)+1/6*arctan((3^(1/2)*b^(1/6)+2*a^(1/6 )*x)/b^(1/6))/a^(1/6)/b^(5/6)-1/12*ln(b^(1/3)-3^(1/2)*a^(1/6)*b^(1/6)*x+a^ (1/3)*x^2)*3^(1/2)/a^(1/6)/b^(5/6)+1/12*ln(b^(1/3)+3^(1/2)*a^(1/6)*b^(1/6) *x+a^(1/3)*x^2)*3^(1/2)/a^(1/6)/b^(5/6)
Time = 0.03 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.72 \[ \int \frac {1}{b+a x^6} \, dx=\frac {4 \arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )-2 \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )+2 \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )-\sqrt {3} \log \left (\sqrt [3]{b}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )+\sqrt {3} \log \left (\sqrt [3]{b}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{12 \sqrt [6]{a} b^{5/6}} \] Input:
Integrate[(b + a*x^6)^(-1),x]
Output:
(4*ArcTan[(a^(1/6)*x)/b^(1/6)] - 2*ArcTan[Sqrt[3] - (2*a^(1/6)*x)/b^(1/6)] + 2*ArcTan[Sqrt[3] + (2*a^(1/6)*x)/b^(1/6)] - Sqrt[3]*Log[b^(1/3) - Sqrt[ 3]*a^(1/6)*b^(1/6)*x + a^(1/3)*x^2] + Sqrt[3]*Log[b^(1/3) + Sqrt[3]*a^(1/6 )*b^(1/6)*x + a^(1/3)*x^2])/(12*a^(1/6)*b^(5/6))
Time = 0.72 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.98, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {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.
| \(\displaystyle \int \frac {1}{a x^6+b} \, dx\) |
| \(\Big \downarrow \) 753 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt [3]{a} x^2+\sqrt [3]{b}}dx}{3 b^{2/3}}+\frac {\int \frac {2 \sqrt [6]{b}-\sqrt {3} \sqrt [6]{a} x}{2 \left (\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}\right )}dx}{3 b^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a} x+2 \sqrt [6]{b}}{2 \left (\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}\right )}dx}{3 b^{5/6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {1}{\sqrt [3]{a} x^2+\sqrt [3]{b}}dx}{3 b^{2/3}}+\frac {\int \frac {2 \sqrt [6]{b}-\sqrt {3} \sqrt [6]{a} x}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 b^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a} x+2 \sqrt [6]{b}}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 b^{5/6}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\int \frac {2 \sqrt [6]{b}-\sqrt {3} \sqrt [6]{a} x}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 b^{5/6}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a} x+2 \sqrt [6]{b}}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 b^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx-\frac {\sqrt {3} \int -\frac {\sqrt [6]{a} \left (\sqrt {3} \sqrt [6]{b}-2 \sqrt [6]{a} x\right )}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{2 \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{a} \left (2 \sqrt [6]{a} x+\sqrt {3} \sqrt [6]{b}\right )}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{2 \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{a} \left (\sqrt {3} \sqrt [6]{b}-2 \sqrt [6]{a} x\right )}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{2 \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {\sqrt {3} \int \frac {\sqrt [6]{a} \left (2 \sqrt [6]{a} x+\sqrt {3} \sqrt [6]{b}\right )}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{2 \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 b^{5/6}}+\frac {\frac {1}{2} \sqrt [6]{b} \int \frac {1}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{a} x+\sqrt {3} \sqrt [6]{b}}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx}{6 b^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {\int \frac {1}{-\left (1-\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}\right )^2-\frac {1}{3}}d\left (1-\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}\right )}{\sqrt {3} \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{a} x+\sqrt {3} \sqrt [6]{b}}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx-\frac {\int \frac {1}{-\left (\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}+1\right )^2-\frac {1}{3}}d\left (\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}+1\right )}{\sqrt {3} \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\frac {1}{2} \sqrt {3} \int \frac {\sqrt {3} \sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [3]{a} x^2-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}\right )\right )}{\sqrt [6]{a}}}{6 b^{5/6}}+\frac {\frac {1}{2} \sqrt {3} \int \frac {2 \sqrt [6]{a} x+\sqrt {3} \sqrt [6]{b}}{\sqrt [3]{a} x^2+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b}}dx+\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}+1\right )\right )}{\sqrt [6]{a}}}{6 b^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {-\frac {\arctan \left (\sqrt {3} \left (1-\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}\right )\right )}{\sqrt [6]{a}}-\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{2 \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\frac {\arctan \left (\sqrt {3} \left (\frac {2 \sqrt [6]{a} x}{\sqrt {3} \sqrt [6]{b}}+1\right )\right )}{\sqrt [6]{a}}+\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{2 \sqrt [6]{a}}}{6 b^{5/6}}+\frac {\arctan \left (\frac {\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 \sqrt [6]{a} b^{5/6}}\) |
Input:
Int[(b + a*x^6)^(-1),x]
Output:
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)) + (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))
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_) + (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 = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.13
| method | result | size |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} a +b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{6 a}\) | \(27\) |
| default | \(\frac {\left (\frac {b}{a}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{3 b}-\frac {\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 b}+\frac {\left (\frac {b}{a}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b}+\frac {\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 b}+\frac {\left (\frac {b}{a}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b}\) | \(159\) |
Input:
int(1/(a*x^6+b),x,method=_RETURNVERBOSE)
Output:
1/6/a*sum(1/_R^5*ln(x-_R),_R=RootOf(_Z^6*a+b))
Time = 0.07 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00 \[ \int \frac {1}{b+a x^6} \, dx=\frac {1}{12} \, {\left (\sqrt {-3} + 1\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} b + b\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} + x\right ) - \frac {1}{12} \, {\left (\sqrt {-3} + 1\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} b + b\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} + x\right ) + \frac {1}{12} \, {\left (\sqrt {-3} - 1\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} b - b\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} + x\right ) - \frac {1}{12} \, {\left (\sqrt {-3} - 1\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} b - b\right )} \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} + x\right ) + \frac {1}{6} \, \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (b \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} + x\right ) - \frac {1}{6} \, \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} \log \left (-b \left (-\frac {1}{a b^{5}}\right )^{\frac {1}{6}} + x\right ) \] Input:
integrate(1/(a*x^6+b),x, algorithm="fricas")
Output:
1/12*(sqrt(-3) + 1)*(-1/(a*b^5))^(1/6)*log(1/2*(sqrt(-3)*b + b)*(-1/(a*b^5 ))^(1/6) + x) - 1/12*(sqrt(-3) + 1)*(-1/(a*b^5))^(1/6)*log(-1/2*(sqrt(-3)* b + b)*(-1/(a*b^5))^(1/6) + x) + 1/12*(sqrt(-3) - 1)*(-1/(a*b^5))^(1/6)*lo g(1/2*(sqrt(-3)*b - b)*(-1/(a*b^5))^(1/6) + x) - 1/12*(sqrt(-3) - 1)*(-1/( a*b^5))^(1/6)*log(-1/2*(sqrt(-3)*b - b)*(-1/(a*b^5))^(1/6) + x) + 1/6*(-1/ (a*b^5))^(1/6)*log(b*(-1/(a*b^5))^(1/6) + x) - 1/6*(-1/(a*b^5))^(1/6)*log( -b*(-1/(a*b^5))^(1/6) + x)
Time = 0.07 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.09 \[ \int \frac {1}{b+a x^6} \, dx=\operatorname {RootSum} {\left (46656 t^{6} a b^{5} + 1, \left ( t \mapsto t \log {\left (6 t b + x \right )} \right )\right )} \] Input:
integrate(1/(a*x**6+b),x)
Output:
RootSum(46656*_t**6*a*b**5 + 1, Lambda(_t, _t*log(6*_t*b + x)))
Time = 0.10 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.86 \[ \int \frac {1}{b+a x^6} \, dx=\frac {\sqrt {3} \log \left (a^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + b^{\frac {1}{3}}\right )}{12 \, a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {\sqrt {3} \log \left (a^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + b^{\frac {1}{3}}\right )}{12 \, a^{\frac {1}{6}} b^{\frac {5}{6}}} + \frac {\arctan \left (\frac {a^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{3 \, b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {\arctan \left (\frac {2 \, a^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{6 \, b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {\arctan \left (\frac {2 \, a^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{6 \, b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} \] Input:
integrate(1/(a*x^6+b),x, algorithm="maxima")
Output:
1/12*sqrt(3)*log(a^(1/3)*x^2 + sqrt(3)*a^(1/6)*b^(1/6)*x + b^(1/3))/(a^(1/ 6)*b^(5/6)) - 1/12*sqrt(3)*log(a^(1/3)*x^2 - sqrt(3)*a^(1/6)*b^(1/6)*x + b ^(1/3))/(a^(1/6)*b^(5/6)) + 1/3*arctan(a^(1/3)*x/sqrt(a^(1/3)*b^(1/3)))/(b ^(2/3)*sqrt(a^(1/3)*b^(1/3))) + 1/6*arctan((2*a^(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))) + 1/6*arct an((2*a^(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)))
Time = 0.11 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.88 \[ \int \frac {1}{b+a x^6} \, dx=\frac {\sqrt {3} \left (a^{5} b\right )^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{6}} + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 \, a b} - \frac {\sqrt {3} \left (a^{5} b\right )^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{6}} + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 \, a b} + \frac {\left (a^{5} b\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {b}{a}\right )^{\frac {1}{6}}}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{6 \, a b} + \frac {\left (a^{5} b\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {b}{a}\right )^{\frac {1}{6}}}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{6 \, a b} + \frac {\left (a^{5} b\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{3 \, a b} \] Input:
integrate(1/(a*x^6+b),x, algorithm="giac")
Output:
1/12*sqrt(3)*(a^5*b)^(1/6)*log(x^2 + sqrt(3)*x*(b/a)^(1/6) + (b/a)^(1/3))/ (a*b) - 1/12*sqrt(3)*(a^5*b)^(1/6)*log(x^2 - sqrt(3)*x*(b/a)^(1/6) + (b/a) ^(1/3))/(a*b) + 1/6*(a^5*b)^(1/6)*arctan((2*x + sqrt(3)*(b/a)^(1/6))/(b/a) ^(1/6))/(a*b) + 1/6*(a^5*b)^(1/6)*arctan((2*x - sqrt(3)*(b/a)^(1/6))/(b/a) ^(1/6))/(a*b) + 1/3*(a^5*b)^(1/6)*arctan(x/(b/a)^(1/6))/(a*b)
Time = 9.90 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.02 \[ \int \frac {1}{b+a x^6} \, dx=\frac {\mathrm {atanh}\left (\frac {{\left (-a\right )}^{1/6}\,x}{b^{1/6}}\right )}{3\,{\left (-a\right )}^{1/6}\,b^{5/6}}-\frac {\mathrm {atan}\left (\frac {{\left (-a\right )}^{29/6}\,x\,1{}\mathrm {i}}{b^{5/6}\,\left (\frac {{\left (-a\right )}^{14/3}}{b^{2/3}}+\frac {\sqrt {3}\,{\left (-a\right )}^{14/3}\,1{}\mathrm {i}}{b^{2/3}}\right )}+\frac {\sqrt {3}\,{\left (-a\right )}^{29/6}\,x}{b^{5/6}\,\left (\frac {{\left (-a\right )}^{14/3}}{b^{2/3}}+\frac {\sqrt {3}\,{\left (-a\right )}^{14/3}\,1{}\mathrm {i}}{b^{2/3}}\right )}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{6\,{\left (-a\right )}^{1/6}\,b^{5/6}}+\frac {\mathrm {atan}\left (\frac {{\left (-a\right )}^{29/6}\,x\,1{}\mathrm {i}}{b^{5/6}\,\left (\frac {{\left (-a\right )}^{14/3}}{b^{2/3}}-\frac {\sqrt {3}\,{\left (-a\right )}^{14/3}\,1{}\mathrm {i}}{b^{2/3}}\right )}-\frac {\sqrt {3}\,{\left (-a\right )}^{29/6}\,x}{b^{5/6}\,\left (\frac {{\left (-a\right )}^{14/3}}{b^{2/3}}-\frac {\sqrt {3}\,{\left (-a\right )}^{14/3}\,1{}\mathrm {i}}{b^{2/3}}\right )}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{6\,{\left (-a\right )}^{1/6}\,b^{5/6}} \] Input:
int(1/(b + a*x^6),x)
Output:
atanh(((-a)^(1/6)*x)/b^(1/6))/(3*(-a)^(1/6)*b^(5/6)) - (atan(((-a)^(29/6)* x*1i)/(b^(5/6)*((-a)^(14/3)/b^(2/3) + (3^(1/2)*(-a)^(14/3)*1i)/b^(2/3))) + (3^(1/2)*(-a)^(29/6)*x)/(b^(5/6)*((-a)^(14/3)/b^(2/3) + (3^(1/2)*(-a)^(14 /3)*1i)/b^(2/3))))*(3^(1/2)*1i - 1)*1i)/(6*(-a)^(1/6)*b^(5/6)) + (atan(((- a)^(29/6)*x*1i)/(b^(5/6)*((-a)^(14/3)/b^(2/3) - (3^(1/2)*(-a)^(14/3)*1i)/b ^(2/3))) - (3^(1/2)*(-a)^(29/6)*x)/(b^(5/6)*((-a)^(14/3)/b^(2/3) - (3^(1/2 )*(-a)^(14/3)*1i)/b^(2/3))))*(3^(1/2)*1i + 1)*1i)/(6*(-a)^(1/6)*b^(5/6))
Time = 0.25 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.58 \[ \int \frac {1}{b+a x^6} \, dx=\frac {-2 \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}-2 a^{\frac {1}{3}} x}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )+2 \mathit {atan} \left (\frac {b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}+2 a^{\frac {1}{3}} x}{b^{\frac {1}{6}} a^{\frac {1}{6}}}\right )+4 \mathit {atan} \left (\frac {a^{\frac {1}{6}} x}{b^{\frac {1}{6}}}\right )-\sqrt {3}\, \mathrm {log}\left (-b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}} x^{2}+b^{\frac {1}{3}}\right )+\sqrt {3}\, \mathrm {log}\left (b^{\frac {1}{6}} a^{\frac {1}{6}} \sqrt {3}\, x +a^{\frac {1}{3}} x^{2}+b^{\frac {1}{3}}\right )}{12 b^{\frac {5}{6}} a^{\frac {1}{6}}} \] Input:
int(1/(a*x^6+b),x)
Output:
(b**(1/6)*a**(1/6)*( - 2*atan((b**(1/6)*a**(1/6)*sqrt(3) - 2*a**(1/3)*x)/( b**(1/6)*a**(1/6))) + 2*atan((b**(1/6)*a**(1/6)*sqrt(3) + 2*a**(1/3)*x)/(b **(1/6)*a**(1/6))) + 4*atan((a**(1/3)*x)/(b**(1/6)*a**(1/6))) - sqrt(3)*lo g( - b**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3)*x**2 + b**(1/3)) + sqrt(3)*log (b**(1/6)*a**(1/6)*sqrt(3)*x + a**(1/3)*x**2 + b**(1/3))))/(12*a**(1/3)*b)