\(\int \frac {1}{a+b x^6} \, dx\) [1326]

Optimal result

Integrand size = 9, antiderivative size = 215 \[ \int \frac {1}{a+b x^6} \, dx=\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}} \]

[Out]

1/3*arctan(b^(1/6)*x/a^(1/6))/a^(5/6)/b^(1/6)-1/6*arctan((-2*b^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6))/a^(5/6)/b^(1/
6)+1/6*arctan((2*b^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6))/a^(5/6)/b^(1/6)-1/12*ln(a^(1/3)+b^(1/3)*x^2-a^(1/6)*b^(1/
6)*x*3^(1/2))/a^(5/6)/b^(1/6)*3^(1/2)+1/12*ln(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*3^(1/2))/a^(5/6)/b^(1/6)*3
^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {215, 648, 632, 210, 642, 211} \[ \int \frac {1}{a+b x^6} \, dx=\frac {\arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}-\frac {\log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {\log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}} \]

[In]

Int[(a + b*x^6)^(-1),x]

[Out]

ArcTan[(b^(1/6)*x)/a^(1/6)]/(3*a^(5/6)*b^(1/6)) - ArcTan[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*x)/a^(1/6)]/(6*a^(5/6)*b
^(1/6)) + ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)]/(6*a^(5/6)*b^(1/6)) - Log[a^(1/3) - Sqrt[3]*a^(1/6)*
b^(1/6)*x + b^(1/3)*x^2]/(4*Sqrt[3]*a^(5/6)*b^(1/6)) + Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2]/
(4*Sqrt[3]*a^(5/6)*b^(1/6))

Rule 210

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])

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 215

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] + Dist[2*(r/(a*n)), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)
/4, 0] && PosQ[a/b]

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sqrt [6]{a}-\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{3 a^{5/6}}+\frac {\int \frac {\sqrt [6]{a}+\frac {1}{2} \sqrt {3} \sqrt [6]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{3 a^{5/6}}+\frac {\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{3 a^{2/3}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}+\frac {\int \frac {1}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{12 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{12 a^{2/3}}-\frac {\int \frac {-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {\int \frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{b} x}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{6 \sqrt {3} a^{5/6} \sqrt [6]{b}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [6]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{6 \sqrt {3} a^{5/6} \sqrt [6]{b}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}-\frac {\tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}+\frac {\tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}-\frac {\log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}}+\frac {\log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{5/6} \sqrt [6]{b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.72 \[ \int \frac {1}{a+b x^6} \, dx=\frac {4 \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )-\sqrt {3} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )+\sqrt {3} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{12 a^{5/6} \sqrt [6]{b}} \]

[In]

Integrate[(a + b*x^6)^(-1),x]

[Out]

(4*ArcTan[(b^(1/6)*x)/a^(1/6)] - 2*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/6)] + 2*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/
a^(1/6)] - Sqrt[3]*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2] + Sqrt[3]*Log[a^(1/3) + Sqrt[3]*a^(1
/6)*b^(1/6)*x + b^(1/3)*x^2])/(12*a^(5/6)*b^(1/6))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 4.38 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.13

method	result	size
risch	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}}{6 b}\)	\(27\)
default	\(\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 a}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}\)	\(159\)

[In]

int(1/(b*x^6+a),x,method=_RETURNVERBOSE)

[Out]

1/6/b*sum(1/_R^5*ln(x-_R),_R=RootOf(_Z^6*b+a))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00 \[ \int \frac {1}{a+b x^6} \, dx=\frac {1}{12} \, {\left (\sqrt {-3} + 1\right )} \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a + a\right )} \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} + x\right ) - \frac {1}{12} \, {\left (\sqrt {-3} + 1\right )} \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a + a\right )} \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} + x\right ) + \frac {1}{12} \, {\left (\sqrt {-3} - 1\right )} \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a - a\right )} \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} + x\right ) - \frac {1}{12} \, {\left (\sqrt {-3} - 1\right )} \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a - a\right )} \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} + x\right ) + \frac {1}{6} \, \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} \log \left (a \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} + x\right ) - \frac {1}{6} \, \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} \log \left (-a \left (-\frac {1}{a^{5} b}\right )^{\frac {1}{6}} + x\right ) \]

[In]

integrate(1/(b*x^6+a),x, algorithm="fricas")

[Out]

1/12*(sqrt(-3) + 1)*(-1/(a^5*b))^(1/6)*log(1/2*(sqrt(-3)*a + a)*(-1/(a^5*b))^(1/6) + x) - 1/12*(sqrt(-3) + 1)*
(-1/(a^5*b))^(1/6)*log(-1/2*(sqrt(-3)*a + a)*(-1/(a^5*b))^(1/6) + x) + 1/12*(sqrt(-3) - 1)*(-1/(a^5*b))^(1/6)*
log(1/2*(sqrt(-3)*a - a)*(-1/(a^5*b))^(1/6) + x) - 1/12*(sqrt(-3) - 1)*(-1/(a^5*b))^(1/6)*log(-1/2*(sqrt(-3)*a
- a)*(-1/(a^5*b))^(1/6) + x) + 1/6*(-1/(a^5*b))^(1/6)*log(a*(-1/(a^5*b))^(1/6) + x) - 1/6*(-1/(a^5*b))^(1/6)*
log(-a*(-1/(a^5*b))^(1/6) + x)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.09 \[ \int \frac {1}{a+b x^6} \, dx=\operatorname {RootSum} {\left (46656 t^{6} a^{5} b + 1, \left ( t \mapsto t \log {\left (6 t a + x \right )} \right )\right )} \]

[In]

integrate(1/(b*x**6+a),x)

[Out]

RootSum(46656*_t**6*a**5*b + 1, Lambda(_t, _t*log(6*_t*a + x)))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.86 \[ \int \frac {1}{a+b x^6} \, dx=\frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{12 \, a^{\frac {5}{6}} b^{\frac {1}{6}}} - \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{12 \, a^{\frac {5}{6}} b^{\frac {1}{6}}} + \frac {\arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{3 \, a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {\arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{6 \, a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {\arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{6 \, a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} \]

[In]

integrate(1/(b*x^6+a),x, algorithm="maxima")

[Out]

1/12*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)) - 1/12*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)) + 1/3*arctan(b^(1/3)*x/sqrt(a^(1/3)*b^(1/3)))/
(a^(2/3)*sqrt(a^(1/3)*b^(1/3))) + 1/6*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))) + 1/6*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)))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.88 \[ \int \frac {1}{a+b x^6} \, dx=\frac {\sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, a b} - \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, a b} + \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, a b} + \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, a b} + \frac {\left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, a b} \]

[In]

integrate(1/(b*x^6+a),x, algorithm="giac")

[Out]

1/12*sqrt(3)*(a*b^5)^(1/6)*log(x^2 + sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/(a*b) - 1/12*sqrt(3)*(a*b^5)^(1/6)*l
og(x^2 - sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/(a*b) + 1/6*(a*b^5)^(1/6)*arctan((2*x + sqrt(3)*(a/b)^(1/6))/(a/
b)^(1/6))/(a*b) + 1/6*(a*b^5)^(1/6)*arctan((2*x - sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a*b) + 1/3*(a*b^5)^(1/6)*
arctan(x/(a/b)^(1/6))/(a*b)

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.02 \[ \int \frac {1}{a+b x^6} \, dx=-\frac {\mathrm {atanh}\left (\frac {b^{1/6}\,x}{{\left (-a\right )}^{1/6}}\right )}{3\,{\left (-a\right )}^{5/6}\,b^{1/6}}+\frac {\mathrm {atan}\left (\frac {b^{29/6}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{5/6}\,\left (\frac {b^{14/3}}{{\left (-a\right )}^{2/3}}+\frac {\sqrt {3}\,b^{14/3}\,1{}\mathrm {i}}{{\left (-a\right )}^{2/3}}\right )}+\frac {\sqrt {3}\,b^{29/6}\,x}{{\left (-a\right )}^{5/6}\,\left (\frac {b^{14/3}}{{\left (-a\right )}^{2/3}}+\frac {\sqrt {3}\,b^{14/3}\,1{}\mathrm {i}}{{\left (-a\right )}^{2/3}}\right )}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{6\,{\left (-a\right )}^{5/6}\,b^{1/6}}-\frac {\mathrm {atan}\left (\frac {b^{29/6}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{5/6}\,\left (\frac {b^{14/3}}{{\left (-a\right )}^{2/3}}-\frac {\sqrt {3}\,b^{14/3}\,1{}\mathrm {i}}{{\left (-a\right )}^{2/3}}\right )}-\frac {\sqrt {3}\,b^{29/6}\,x}{{\left (-a\right )}^{5/6}\,\left (\frac {b^{14/3}}{{\left (-a\right )}^{2/3}}-\frac {\sqrt {3}\,b^{14/3}\,1{}\mathrm {i}}{{\left (-a\right )}^{2/3}}\right )}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{6\,{\left (-a\right )}^{5/6}\,b^{1/6}} \]

[In]

int(1/(a + b*x^6),x)

[Out]

(atan((b^(29/6)*x*1i)/((-a)^(5/6)*(b^(14/3)/(-a)^(2/3) + (3^(1/2)*b^(14/3)*1i)/(-a)^(2/3))) + (3^(1/2)*b^(29/6
)*x)/((-a)^(5/6)*(b^(14/3)/(-a)^(2/3) + (3^(1/2)*b^(14/3)*1i)/(-a)^(2/3))))*(3^(1/2)*1i - 1)*1i)/(6*(-a)^(5/6)
*b^(1/6)) - atanh((b^(1/6)*x)/(-a)^(1/6))/(3*(-a)^(5/6)*b^(1/6)) - (atan((b^(29/6)*x*1i)/((-a)^(5/6)*(b^(14/3)
/(-a)^(2/3) - (3^(1/2)*b^(14/3)*1i)/(-a)^(2/3))) - (3^(1/2)*b^(29/6)*x)/((-a)^(5/6)*(b^(14/3)/(-a)^(2/3) - (3^
(1/2)*b^(14/3)*1i)/(-a)^(2/3))))*(3^(1/2)*1i + 1)*1i)/(6*(-a)^(5/6)*b^(1/6))