3.1339 \(\int \frac{1}{(a+b x^6)^2} \, dx\)

Optimal. Leaf size=232 \[ -\frac{5 \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{11/6} \sqrt [6]{b}}+\frac{5 \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{11/6} \sqrt [6]{b}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac{x}{6 a \left (a+b x^6\right )} \]

[Out]

x/(6*a*(a + b*x^6)) + (5*ArcTan[(b^(1/6)*x)/a^(1/6)])/(18*a^(11/6)*b^(1/6)) - (5*ArcTan[(Sqrt[3]*a^(1/6) - 2*b
^(1/6)*x)/a^(1/6)])/(36*a^(11/6)*b^(1/6)) + (5*ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)])/(36*a^(11/6)*b
^(1/6)) - (5*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sqrt[3]*a^(11/6)*b^(1/6)) + (5*Log[a^
(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sqrt[3]*a^(11/6)*b^(1/6))

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Rubi [A]  time = 0.396952, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.778, Rules used = {199, 209, 634, 618, 204, 628, 205} \[ -\frac{5 \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{11/6} \sqrt [6]{b}}+\frac{5 \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{11/6} \sqrt [6]{b}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}-\frac{5 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac{x}{6 a \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^6)^(-2),x]

[Out]

x/(6*a*(a + b*x^6)) + (5*ArcTan[(b^(1/6)*x)/a^(1/6)])/(18*a^(11/6)*b^(1/6)) - (5*ArcTan[(Sqrt[3]*a^(1/6) - 2*b
^(1/6)*x)/a^(1/6)])/(36*a^(11/6)*b^(1/6)) + (5*ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)])/(36*a^(11/6)*b
^(1/6)) - (5*Log[a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sqrt[3]*a^(11/6)*b^(1/6)) + (5*Log[a^
(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/(24*Sqrt[3]*a^(11/6)*b^(1/6))

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(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] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rule 209

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*Int[1/(r^2 +
s^2*x^2), x])/(a*n) + 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 634

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]

Rule 618

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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

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 205

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

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b x^6\right )^2} \, dx &=\frac{x}{6 a \left (a+b x^6\right )}+\frac{5 \int \frac{1}{a+b x^6} \, dx}{6 a}\\ &=\frac{x}{6 a \left (a+b x^6\right )}+\frac{5 \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}{18 a^{11/6}}+\frac{5 \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}{18 a^{11/6}}+\frac{5 \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{18 a^{5/3}}\\ &=\frac{x}{6 a \left (a+b x^6\right )}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}+\frac{5 \int \frac{1}{\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a^{5/3}}+\frac{5 \int \frac{1}{\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a^{5/3}}-\frac{5 \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}{24 \sqrt{3} a^{11/6} \sqrt [6]{b}}+\frac{5 \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}{24 \sqrt{3} a^{11/6} \sqrt [6]{b}}\\ &=\frac{x}{6 a \left (a+b x^6\right )}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}-\frac{5 \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{11/6} \sqrt [6]{b}}+\frac{5 \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{11/6} \sqrt [6]{b}}+\frac{5 \operatorname{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 )}{36 \sqrt{3} a^{11/6} \sqrt [6]{b}}-\frac{5 \operatorname{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 )}{36 \sqrt{3} a^{11/6} \sqrt [6]{b}}\\ &=\frac{x}{6 a \left (a+b x^6\right )}+\frac{5 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}-\frac{5 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac{5 \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}-\frac{5 \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{11/6} \sqrt [6]{b}}+\frac{5 \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{11/6} \sqrt [6]{b}}\\ \end{align*}

Mathematica [A]  time = 0.108227, size = 192, normalized size = 0.83 \[ \frac{\frac{12 a^{5/6} x}{a+b x^6}-\frac{5 \sqrt{3} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\sqrt [6]{b}}+\frac{5 \sqrt{3} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{\sqrt [6]{b}}+\frac{20 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}-\frac{10 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}+\frac{10 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{\sqrt [6]{b}}}{72 a^{11/6}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^6)^(-2),x]

[Out]

((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))

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Maple [B]  time = 0.236, size = 343, normalized size = 1.5 \begin{align*} -{\frac{x}{36\,{a}^{2}}\sqrt [3]{{\frac{a}{b}}} \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}-{\frac{\sqrt{3}}{36\,{a}^{2}}\sqrt{{\frac{a}{b}}} \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{5\,\sqrt{3}}{72\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{5}{36\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) }+{\frac{x}{18\,{a}^{2}}\sqrt [3]{{\frac{a}{b}}} \left ({x}^{2}+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{5}{18\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{x}{36\,{a}^{2}}\sqrt [3]{{\frac{a}{b}}} \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}+{\frac{\sqrt{3}}{36\,{a}^{2}}\sqrt{{\frac{a}{b}}} \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) ^{-1}}-{\frac{5\,\sqrt{3}}{72\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{5}{36\,{a}^{2}}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-\sqrt{3} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/36/a^2/(x^2+3^(1/2)*(1/b*a)^(1/6)*x+(1/b*a)^(1/3))*(1/b*a)^(1/3)*x-1/36/a^2/(x^2+3^(1/2)*(1/b*a)^(1/6)*x+(1
/b*a)^(1/3))*(1/b*a)^(1/2)*3^(1/2)+5/72/a^2*3^(1/2)*(1/b*a)^(1/6)*ln(x^2+3^(1/2)*(1/b*a)^(1/6)*x+(1/b*a)^(1/3)
)+5/36/a^2*(1/b*a)^(1/6)*arctan(2*x/(1/b*a)^(1/6)+3^(1/2))+1/18*(1/b*a)^(1/3)/a^2*x/(x^2+(1/b*a)^(1/3))+5/18*(
1/b*a)^(1/6)/a^2*arctan(x/(1/b*a)^(1/6))-1/36/a^2/(x^2-3^(1/2)*(1/b*a)^(1/6)*x+(1/b*a)^(1/3))*(1/b*a)^(1/3)*x+
1/36/a^2/(x^2-3^(1/2)*(1/b*a)^(1/6)*x+(1/b*a)^(1/3))*(1/b*a)^(1/2)*3^(1/2)-5/72/a^2*3^(1/2)*(1/b*a)^(1/6)*ln(x
^2-3^(1/2)*(1/b*a)^(1/6)*x+(1/b*a)^(1/3))+5/36/a^2*(1/b*a)^(1/6)*arctan(2*x/(1/b*a)^(1/6)-3^(1/2))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.62036, size = 1102, normalized size = 4.75 \begin{align*} \frac{20 \, \sqrt{3}{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} \arctan \left (-\frac{2}{3} \, \sqrt{3} a^{9} b x \left (-\frac{1}{a^{11} b}\right )^{\frac{5}{6}} + \frac{2}{3} \, \sqrt{3} \sqrt{a^{4} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{3}} + a^{2} x \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} + x^{2}} a^{9} b \left (-\frac{1}{a^{11} b}\right )^{\frac{5}{6}} + \frac{1}{3} \, \sqrt{3}\right ) + 20 \, \sqrt{3}{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} \arctan \left (-\frac{2}{3} \, \sqrt{3} a^{9} b x \left (-\frac{1}{a^{11} b}\right )^{\frac{5}{6}} + \frac{2}{3} \, \sqrt{3} \sqrt{a^{4} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{3}} - a^{2} x \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} + x^{2}} a^{9} b \left (-\frac{1}{a^{11} b}\right )^{\frac{5}{6}} - \frac{1}{3} \, \sqrt{3}\right ) + 5 \,{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} \log \left (a^{4} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{3}} + a^{2} x \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} + x^{2}\right ) - 5 \,{\left (a b x^{6} + a^{2}\right )} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} \log \left (a^{4} \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{3}} - a^{2} x \left (-\frac{1}{a^{11} b}\right )^{\frac{1}{6}} + x^{2}\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 ) - 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 ) + 12 \, x}{72 \,{\left (a b x^{6} + a^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(20*sqrt(3)*(a*b*x^6 + a^2)*(-1/(a^11*b))^(1/6)*arctan(-2/3*sqrt(3)*a^9*b*x*(-1/(a^11*b))^(5/6) + 2/3*sqr
t(3)*sqrt(a^4*(-1/(a^11*b))^(1/3) + a^2*x*(-1/(a^11*b))^(1/6) + x^2)*a^9*b*(-1/(a^11*b))^(5/6) + 1/3*sqrt(3))
+ 20*sqrt(3)*(a*b*x^6 + a^2)*(-1/(a^11*b))^(1/6)*arctan(-2/3*sqrt(3)*a^9*b*x*(-1/(a^11*b))^(5/6) + 2/3*sqrt(3)
*sqrt(a^4*(-1/(a^11*b))^(1/3) - a^2*x*(-1/(a^11*b))^(1/6) + x^2)*a^9*b*(-1/(a^11*b))^(5/6) - 1/3*sqrt(3)) + 5*
(a*b*x^6 + a^2)*(-1/(a^11*b))^(1/6)*log(a^4*(-1/(a^11*b))^(1/3) + a^2*x*(-1/(a^11*b))^(1/6) + x^2) - 5*(a*b*x^
6 + a^2)*(-1/(a^11*b))^(1/6)*log(a^4*(-1/(a^11*b))^(1/3) - a^2*x*(-1/(a^11*b))^(1/6) + x^2) + 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) + 12*x)/(a*b*x^6 + a^2)

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Sympy [A]  time = 0.917377, size = 39, normalized size = 0.17 \begin{align*} \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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)))

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Giac [A]  time = 1.12728, size = 277, normalized size = 1.19 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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