3.1333 \(\int \frac{x^6}{(a+b x^6)^2} \, dx\)

Optimal. Leaf size=232 \[ -\frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{5/6} b^{7/6}}+\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{5/6} b^{7/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{7/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}-\frac{x}{6 b \left (a+b x^6\right )} \]

[Out]

-x/(6*b*(a + b*x^6)) + ArcTan[(b^(1/6)*x)/a^(1/6)]/(18*a^(5/6)*b^(7/6)) - ArcTan[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*
x)/a^(1/6)]/(36*a^(5/6)*b^(7/6)) + ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)]/(36*a^(5/6)*b^(7/6)) - Log[
a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2]/(24*Sqrt[3]*a^(5/6)*b^(7/6)) + Log[a^(1/3) + Sqrt[3]*a^(1/6
)*b^(1/6)*x + b^(1/3)*x^2]/(24*Sqrt[3]*a^(5/6)*b^(7/6))

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Rubi [A]  time = 0.419271, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538, Rules used = {288, 209, 634, 618, 204, 628, 205} \[ -\frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{5/6} b^{7/6}}+\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{5/6} b^{7/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{7/6}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}-\frac{x}{6 b \left (a+b x^6\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^6/(a + b*x^6)^2,x]

[Out]

-x/(6*b*(a + b*x^6)) + ArcTan[(b^(1/6)*x)/a^(1/6)]/(18*a^(5/6)*b^(7/6)) - ArcTan[(Sqrt[3]*a^(1/6) - 2*b^(1/6)*
x)/a^(1/6)]/(36*a^(5/6)*b^(7/6)) + ArcTan[(Sqrt[3]*a^(1/6) + 2*b^(1/6)*x)/a^(1/6)]/(36*a^(5/6)*b^(7/6)) - Log[
a^(1/3) - Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2]/(24*Sqrt[3]*a^(5/6)*b^(7/6)) + Log[a^(1/3) + Sqrt[3]*a^(1/6
)*b^(1/6)*x + b^(1/3)*x^2]/(24*Sqrt[3]*a^(5/6)*b^(7/6))

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

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{x^6}{\left (a+b x^6\right )^2} \, dx &=-\frac{x}{6 b \left (a+b x^6\right )}+\frac{\int \frac{1}{a+b x^6} \, dx}{6 b}\\ &=-\frac{x}{6 b \left (a+b x^6\right )}+\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}{18 a^{5/6} b}+\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}{18 a^{5/6} b}+\frac{\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{18 a^{2/3} b}\\ &=-\frac{x}{6 b \left (a+b x^6\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{7/6}}-\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}{24 \sqrt{3} a^{5/6} b^{7/6}}+\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}{24 \sqrt{3} a^{5/6} b^{7/6}}+\frac{\int \frac{1}{\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a^{2/3} b}+\frac{\int \frac{1}{\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{72 a^{2/3} b}\\ &=-\frac{x}{6 b \left (a+b x^6\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{7/6}}-\frac{\log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{5/6} b^{7/6}}+\frac{\log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{5/6} b^{7/6}}+\frac{\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^{5/6} b^{7/6}}-\frac{\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^{5/6} b^{7/6}}\\ &=-\frac{x}{6 b \left (a+b x^6\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{5/6} b^{7/6}}-\frac{\tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}+\frac{\tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{5/6} b^{7/6}}-\frac{\log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{5/6} b^{7/6}}+\frac{\log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt{3} a^{5/6} b^{7/6}}\\ \end{align*}

Mathematica [A]  time = 0.111091, size = 191, normalized size = 0.82 \[ \frac{-\frac{\sqrt{3} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{a^{5/6}}+\frac{\sqrt{3} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{a^{5/6}}+\frac{4 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{a^{5/6}}-\frac{2 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{a^{5/6}}+\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{a^{5/6}}-\frac{12 \sqrt [6]{b} x}{a+b x^6}}{72 b^{7/6}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6/(a + b*x^6)^2,x]

[Out]

((-12*b^(1/6)*x)/(a + b*x^6) + (4*ArcTan[(b^(1/6)*x)/a^(1/6)])/a^(5/6) - (2*ArcTan[Sqrt[3] - (2*b^(1/6)*x)/a^(
1/6)])/a^(5/6) + (2*ArcTan[Sqrt[3] + (2*b^(1/6)*x)/a^(1/6)])/a^(5/6) - (Sqrt[3]*Log[a^(1/3) - Sqrt[3]*a^(1/6)*
b^(1/6)*x + b^(1/3)*x^2])/a^(5/6) + (Sqrt[3]*Log[a^(1/3) + Sqrt[3]*a^(1/6)*b^(1/6)*x + b^(1/3)*x^2])/a^(5/6))/
(72*b^(7/6))

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Maple [A]  time = 0.033, size = 189, normalized size = 0.8 \begin{align*} -{\frac{x}{6\,b \left ( b{x}^{6}+a \right ) }}+{\frac{\sqrt{3}}{72\,ab}\sqrt [6]{{\frac{a}{b}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{36\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) }+{\frac{1}{18\,ab}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{\sqrt{3}}{72\,ab}\sqrt [6]{{\frac{a}{b}}}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }+{\frac{1}{36\,ab}\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(x^6/(b*x^6+a)^2,x)

[Out]

-1/6*x/b/(b*x^6+a)+1/72/b/a*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))+1/36/b/a*(1/b*
a)^(1/6)*arctan(2*x/(1/b*a)^(1/6)+3^(1/2))+1/18/b/a*(1/b*a)^(1/6)*arctan(x/(1/b*a)^(1/6))-1/72/b/a*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))+1/36/b/a*(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(x^6/(b*x^6+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.56689, size = 1150, normalized size = 4.96 \begin{align*} \frac{4 \, \sqrt{3}{\left (b^{2} x^{6} + a b\right )} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2}{3} \, \sqrt{3} a^{4} b^{6} x \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{5}{6}} + \frac{2}{3} \, \sqrt{3} \sqrt{a^{2} b^{2} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{3}} + a b x \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} + x^{2}} a^{4} b^{6} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{5}{6}} + \frac{1}{3} \, \sqrt{3}\right ) + 4 \, \sqrt{3}{\left (b^{2} x^{6} + a b\right )} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2}{3} \, \sqrt{3} a^{4} b^{6} x \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{5}{6}} + \frac{2}{3} \, \sqrt{3} \sqrt{a^{2} b^{2} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{3}} - a b x \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} + x^{2}} a^{4} b^{6} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{5}{6}} - \frac{1}{3} \, \sqrt{3}\right ) +{\left (b^{2} x^{6} + a b\right )} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} \log \left (a^{2} b^{2} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{3}} + a b x \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} + x^{2}\right ) -{\left (b^{2} x^{6} + a b\right )} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} \log \left (a^{2} b^{2} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{3}} - a b x \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} + x^{2}\right ) + 2 \,{\left (b^{2} x^{6} + a b\right )} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} \log \left (a b \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} + x\right ) - 2 \,{\left (b^{2} x^{6} + a b\right )} \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} \log \left (-a b \left (-\frac{1}{a^{5} b^{7}}\right )^{\frac{1}{6}} + x\right ) - 12 \, x}{72 \,{\left (b^{2} x^{6} + a b\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(4*sqrt(3)*(b^2*x^6 + a*b)*(-1/(a^5*b^7))^(1/6)*arctan(-2/3*sqrt(3)*a^4*b^6*x*(-1/(a^5*b^7))^(5/6) + 2/3*
sqrt(3)*sqrt(a^2*b^2*(-1/(a^5*b^7))^(1/3) + a*b*x*(-1/(a^5*b^7))^(1/6) + x^2)*a^4*b^6*(-1/(a^5*b^7))^(5/6) + 1
/3*sqrt(3)) + 4*sqrt(3)*(b^2*x^6 + a*b)*(-1/(a^5*b^7))^(1/6)*arctan(-2/3*sqrt(3)*a^4*b^6*x*(-1/(a^5*b^7))^(5/6
) + 2/3*sqrt(3)*sqrt(a^2*b^2*(-1/(a^5*b^7))^(1/3) - a*b*x*(-1/(a^5*b^7))^(1/6) + x^2)*a^4*b^6*(-1/(a^5*b^7))^(
5/6) - 1/3*sqrt(3)) + (b^2*x^6 + a*b)*(-1/(a^5*b^7))^(1/6)*log(a^2*b^2*(-1/(a^5*b^7))^(1/3) + a*b*x*(-1/(a^5*b
^7))^(1/6) + x^2) - (b^2*x^6 + a*b)*(-1/(a^5*b^7))^(1/6)*log(a^2*b^2*(-1/(a^5*b^7))^(1/3) - a*b*x*(-1/(a^5*b^7
))^(1/6) + x^2) + 2*(b^2*x^6 + a*b)*(-1/(a^5*b^7))^(1/6)*log(a*b*(-1/(a^5*b^7))^(1/6) + x) - 2*(b^2*x^6 + a*b)
*(-1/(a^5*b^7))^(1/6)*log(-a*b*(-1/(a^5*b^7))^(1/6) + x) - 12*x)/(b^2*x^6 + a*b)

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Sympy [A]  time = 0.917527, size = 39, normalized size = 0.17 \begin{align*} - \frac{x}{6 a b + 6 b^{2} x^{6}} + \operatorname{RootSum}{\left (2176782336 t^{6} a^{5} b^{7} + 1, \left ( t \mapsto t \log{\left (36 t a b + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/(6*a*b + 6*b**2*x**6) + RootSum(2176782336*_t**6*a**5*b**7 + 1, Lambda(_t, _t*log(36*_t*a*b + x)))

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Giac [A]  time = 1.42111, size = 277, normalized size = 1.19 \begin{align*} -\frac{x}{6 \,{\left (b x^{6} + a\right )} 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 )}{72 \, a b^{2}} - \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 )}{72 \, a b^{2}} + \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 )}{36 \, a b^{2}} + \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 )}{36 \, a b^{2}} + \frac{\left (a b^{5}\right )^{\frac{1}{6}} \arctan \left (\frac{x}{\left (\frac{a}{b}\right )^{\frac{1}{6}}}\right )}{18 \, a b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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