\(\int \frac {1}{x^2 (a+b x^6)} \, dx\) [1328]

Optimal result

Integrand size = 13, antiderivative size = 223 \[ \int \frac {1}{x^2 \left (a+b x^6\right )} \, dx=-\frac {1}{a x}-\frac {\sqrt [6]{b} \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{7/6}}+\frac {\sqrt [6]{b} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}}+\frac {\sqrt [6]{b} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}} \]

[Out]

-1/a/x-1/3*b^(1/6)*arctan(b^(1/6)*x/a^(1/6))/a^(7/6)+1/6*b^(1/6)*arctan((-2*b^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6)
)/a^(7/6)-1/6*b^(1/6)*arctan((2*b^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6))/a^(7/6)-1/12*b^(1/6)*ln(a^(1/3)+b^(1/3)*x^
2-a^(1/6)*b^(1/6)*x*3^(1/2))/a^(7/6)*3^(1/2)+1/12*b^(1/6)*ln(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*3^(1/2))/a^
(7/6)*3^(1/2)

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {331, 301, 648, 632, 210, 642, 211} \[ \int \frac {1}{x^2 \left (a+b x^6\right )} \, dx=-\frac {\sqrt [6]{b} \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{7/6}}+\frac {\sqrt [6]{b} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}}+\frac {\sqrt [6]{b} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}}-\frac {1}{a x} \]

[In]

Int[1/(x^2*(a + b*x^6)),x]

[Out]

-(1/(a*x)) - (b^(1/6)*ArcTan[(b^(1/6)*x)/a^(1/6)])/(3*a^(7/6)) + (b^(1/6)*ArcTan[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*
x)/a^(1/6)])/(6*a^(7/6)) - (b^(1/6)*ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)])/(6*a^(7/6)) - (b^(1/6)*Lo
g[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(4*Sqrt[3]*a^(7/6)) + (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^(7/6))

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 301

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

Rule 331

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] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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 {1}{a x}-\frac {b \int \frac {x^4}{a+b x^6} \, dx}{a} \\ & = -\frac {1}{a x}-\frac {\sqrt [3]{b} \int \frac {-\frac {\sqrt [6]{a}}{2}+\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^{7/6}}-\frac {\sqrt [3]{b} \int \frac {-\frac {\sqrt [6]{a}}{2}-\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^{7/6}}-\frac {\sqrt [3]{b} \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{3 a} \\ & = -\frac {1}{a x}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{7/6}}-\frac {\sqrt [6]{b} \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^{7/6}}+\frac {\sqrt [6]{b} \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^{7/6}}-\frac {\sqrt [3]{b} \int \frac {1}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{12 a}-\frac {\sqrt [3]{b} \int \frac {1}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{12 a} \\ & = -\frac {1}{a x}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{7/6}}-\frac {\sqrt [6]{b} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}}+\frac {\sqrt [6]{b} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}}-\frac {\sqrt [6]{b} \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^{7/6}}+\frac {\sqrt [6]{b} \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^{7/6}} \\ & = -\frac {1}{a x}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{7/6}}+\frac {\sqrt [6]{b} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}}+\frac {\sqrt [6]{b} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^2 \left (a+b x^6\right )} \, dx=-\frac {12 \sqrt [6]{a}+4 \sqrt [6]{b} x \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \sqrt [6]{b} x \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt [6]{b} x \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+\sqrt {3} \sqrt [6]{b} x \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )-\sqrt {3} \sqrt [6]{b} x \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{12 a^{7/6} x} \]

[In]

Integrate[1/(x^2*(a + b*x^6)),x]

[Out]

-1/12*(12*a^(1/6) + 4*b^(1/6)*x*ArcTan[(b^(1/6)*x)/a^(1/6)] - 2*b^(1/6)*x*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(1/
6)] + 2*b^(1/6)*x*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)] + Sqrt[3]*b^(1/6)*x*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^
(1/6)*x + b^(1/3)*x^2] - Sqrt[3]*b^(1/6)*x*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(a^(7/6)*x)

Maple [C] (verified)

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

Time = 4.63 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.22

method	result	size
risch	\(-\frac {1}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{7} \textit {\_Z}^{6}+b \right )}{\sum }\textit {\_R} \ln \left (\left (7 \textit {\_R}^{6} a^{7}+6 b \right ) x +a^{6} \textit {\_R}^{5}\right )\right )}{6}\)	\(50\)
default	\(-\frac {\left (\frac {\arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a}+\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{6}}}\right ) b}{a}-\frac {1}{a x}\)	\(174\)

[In]

int(1/x^2/(b*x^6+a),x,method=_RETURNVERBOSE)

[Out]

-1/a/x+1/6*sum(_R*ln((7*_R^6*a^7+6*b)*x+a^6*_R^5),_R=RootOf(_Z^6*a^7+b))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x^2 \left (a+b x^6\right )} \, dx=-\frac {2 \, a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (a^{6} \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} + b x\right ) - 2 \, a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (-a^{6} \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} + b x\right ) - {\left (\sqrt {-3} a x - a x\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (b x + \frac {1}{2} \, {\left (\sqrt {-3} a^{6} + a^{6}\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}}\right ) + {\left (\sqrt {-3} a x - a x\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (b x - \frac {1}{2} \, {\left (\sqrt {-3} a^{6} + a^{6}\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}}\right ) - {\left (\sqrt {-3} a x + a x\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (b x + \frac {1}{2} \, {\left (\sqrt {-3} a^{6} - a^{6}\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}}\right ) + {\left (\sqrt {-3} a x + a x\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (b x - \frac {1}{2} \, {\left (\sqrt {-3} a^{6} - a^{6}\right )} \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}}\right ) + 12}{12 \, a x} \]

[In]

integrate(1/x^2/(b*x^6+a),x, algorithm="fricas")

[Out]

-1/12*(2*a*x*(-b/a^7)^(1/6)*log(a^6*(-b/a^7)^(5/6) + b*x) - 2*a*x*(-b/a^7)^(1/6)*log(-a^6*(-b/a^7)^(5/6) + b*x
) - (sqrt(-3)*a*x - a*x)*(-b/a^7)^(1/6)*log(b*x + 1/2*(sqrt(-3)*a^6 + a^6)*(-b/a^7)^(5/6)) + (sqrt(-3)*a*x - a
*x)*(-b/a^7)^(1/6)*log(b*x - 1/2*(sqrt(-3)*a^6 + a^6)*(-b/a^7)^(5/6)) - (sqrt(-3)*a*x + a*x)*(-b/a^7)^(1/6)*lo
g(b*x + 1/2*(sqrt(-3)*a^6 - a^6)*(-b/a^7)^(5/6)) + (sqrt(-3)*a*x + a*x)*(-b/a^7)^(1/6)*log(b*x - 1/2*(sqrt(-3)
*a^6 - a^6)*(-b/a^7)^(5/6)) + 12)/(a*x)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.13 \[ \int \frac {1}{x^2 \left (a+b x^6\right )} \, dx=\operatorname {RootSum} {\left (46656 t^{6} a^{7} + b, \left ( t \mapsto t \log {\left (- \frac {7776 t^{5} a^{6}}{b} + x \right )} \right )\right )} - \frac {1}{a x} \]

[In]

integrate(1/x**2/(b*x**6+a),x)

[Out]

RootSum(46656*_t**6*a**7 + b, Lambda(_t, _t*log(-7776*_t**5*a**6/b + x))) - 1/(a*x)

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^2 \left (a+b x^6\right )} \, dx=\frac {b {\left (\frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{12 \, a} - \frac {1}{a x} \]

[In]

integrate(1/x^2/(b*x^6+a),x, algorithm="maxima")

[Out]

1/12*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*arctan(b^(1/3)*x/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))) - 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)*sqr
t(a^(1/3)*b^(1/3))))/a - 1/(a*x)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^2 \left (a+b x^6\right )} \, dx=-\frac {b \left (\frac {a}{b}\right )^{\frac {5}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, a^{2}} - \frac {1}{a x} + \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {5}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, a^{2} b^{4}} - \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {5}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, a^{2} b^{4}} - \frac {\left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, a^{2} b^{4}} - \frac {\left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, a^{2} b^{4}} \]

[In]

integrate(1/x^2/(b*x^6+a),x, algorithm="giac")

[Out]

-1/3*b*(a/b)^(5/6)*arctan(x/(a/b)^(1/6))/a^2 - 1/(a*x) + 1/12*sqrt(3)*(a*b^5)^(5/6)*log(x^2 + sqrt(3)*x*(a/b)^
(1/6) + (a/b)^(1/3))/(a^2*b^4) - 1/12*sqrt(3)*(a*b^5)^(5/6)*log(x^2 - sqrt(3)*x*(a/b)^(1/6) + (a/b)^(1/3))/(a^
2*b^4) - 1/6*(a*b^5)^(5/6)*arctan((2*x + sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a^2*b^4) - 1/6*(a*b^5)^(5/6)*arcta
n((2*x - sqrt(3)*(a/b)^(1/6))/(a/b)^(1/6))/(a^2*b^4)

Mupad [B] (verification not implemented)

Time = 5.61 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.67 \[ \int \frac {1}{x^2 \left (a+b x^6\right )} \, dx=-\frac {1}{a\,x}-\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/6}\,x\,1{}\mathrm {i}}{a^{1/6}}\right )\,1{}\mathrm {i}}{3\,a^{7/6}}-\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {a^{13/2}\,{\left (-b\right )}^{13/2}\,x\,2{}\mathrm {i}}{a^{20/3}\,{\left (-b\right )}^{19/3}-\sqrt {3}\,a^{20/3}\,{\left (-b\right )}^{19/3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3\,a^{7/6}}+\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {a^{13/2}\,{\left (-b\right )}^{13/2}\,x\,2{}\mathrm {i}}{a^{20/3}\,{\left (-b\right )}^{19/3}+\sqrt {3}\,a^{20/3}\,{\left (-b\right )}^{19/3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3\,a^{7/6}} \]

[In]

int(1/(x^2*(a + b*x^6)),x)

[Out]

((-b)^(1/6)*atan((a^(13/2)*(-b)^(13/2)*x*2i)/(a^(20/3)*(-b)^(19/3) + 3^(1/2)*a^(20/3)*(-b)^(19/3)*1i))*((3^(1/
2)*1i)/2 - 1/2)*1i)/(3*a^(7/6)) - ((-b)^(1/6)*atan(((-b)^(1/6)*x*1i)/a^(1/6))*1i)/(3*a^(7/6)) - ((-b)^(1/6)*at
an((a^(13/2)*(-b)^(13/2)*x*2i)/(a^(20/3)*(-b)^(19/3) - 3^(1/2)*a^(20/3)*(-b)^(19/3)*1i))*((3^(1/2)*1i)/2 + 1/2
)*1i)/(3*a^(7/6)) - 1/(a*x)