\(\int \frac {1}{(a+b x^6)^2} \, dx\) [1339]

Optimal result

Integrand size = 9, antiderivative size = 232 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {x}{6 a \left (a+b x^6\right )}+\frac {5 \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}-\frac {5 \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac {5 \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}-\frac {5 \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{11/6} \sqrt [6]{b}}+\frac {5 \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{11/6} \sqrt [6]{b}} \]

[Out]

1/6*x/a/(b*x^6+a)+5/18*arctan(b^(1/6)*x/a^(1/6))/a^(11/6)/b^(1/6)-5/36*arctan((-2*b^(1/6)*x+a^(1/6)*3^(1/2))/a
^(1/6))/a^(11/6)/b^(1/6)+5/36*arctan((2*b^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6))/a^(11/6)/b^(1/6)-5/72*ln(a^(1/3)+b
^(1/3)*x^2-a^(1/6)*b^(1/6)*x*3^(1/2))/a^(11/6)/b^(1/6)*3^(1/2)+5/72*ln(a^(1/3)+b^(1/3)*x^2+a^(1/6)*b^(1/6)*x*3
^(1/2))/a^(11/6)/b^(1/6)*3^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {205, 215, 648, 632, 210, 642, 211} \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {5 \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{18 a^{11/6} \sqrt [6]{b}}-\frac {5 \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}+\frac {5 \arctan \left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{36 a^{11/6} \sqrt [6]{b}}-\frac {5 \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{11/6} \sqrt [6]{b}}+\frac {5 \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{24 \sqrt {3} a^{11/6} \sqrt [6]{b}}+\frac {x}{6 a \left (a+b x^6\right )} \]

[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 205

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] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

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 {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 \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 )}{36 \sqrt {3} a^{11/6} \sqrt [6]{b}}-\frac {5 \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 )}{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] (verified)

Time = 0.13 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {\frac {12 a^{5/6} x}{a+b x^6}+\frac {20 \arctan \left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}-\frac {10 \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}+\frac {10 \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{\sqrt [6]{b}}-\frac {5 \sqrt {3} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{\sqrt [6]{b}}+\frac {5 \sqrt {3} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{\sqrt [6]{b}}}{72 a^{11/6}} \]

[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))

Maple [C] (verified)

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

Time = 4.38 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.20

method	result	size
risch	\(\frac {x}{6 a \left (b \,x^{6}+a \right )}+\frac {5 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} b +a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{5}}\right )}{36 b a}\)	\(46\)
default	\(\frac {\left (\frac {a}{b}\right )^{\frac {1}{3}} x}{18 a^{2} \left (x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 a^{2}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{3}} x}{36 a^{2} \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}-\frac {\sqrt {\frac {a}{b}}\, \sqrt {3}}{36 a^{2} \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}+\frac {5 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a^{2}}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{36 a^{2}}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{3}} x}{36 a^{2} \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}+\frac {\sqrt {\frac {a}{b}}\, \sqrt {3}}{36 a^{2} \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}-\frac {5 \sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 a^{2}}+\frac {5 \left (\frac {a}{b}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{36 a^{2}}\)	\(343\)

[In]

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

[Out]

1/6*x/a/(b*x^6+a)+5/36/b/a*sum(1/_R^5*ln(x-_R),_R=RootOf(_Z^6*b+a))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 351 vs. \(2 (158) = 316\).

Time = 0.28 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.51 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {10 \, {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (a^{2} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x\right ) - 10 \, {\left (a b x^{6} + a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (-a^{2} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x\right ) + 5 \, {\left (a b x^{6} + a^{2} + \sqrt {-3} {\left (a b x^{6} + a^{2}\right )}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a^{2} + a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x\right ) - 5 \, {\left (a b x^{6} + a^{2} + \sqrt {-3} {\left (a b x^{6} + a^{2}\right )}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a^{2} + a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x\right ) - 5 \, {\left (a b x^{6} + a^{2} - \sqrt {-3} {\left (a b x^{6} + a^{2}\right )}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} a^{2} - a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x\right ) + 5 \, {\left (a b x^{6} + a^{2} - \sqrt {-3} {\left (a b x^{6} + a^{2}\right )}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} a^{2} - a^{2}\right )} \left (-\frac {1}{a^{11} b}\right )^{\frac {1}{6}} + x\right ) + 12 \, x}{72 \, {\left (a b x^{6} + a^{2}\right )}} \]

[In]

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

[Out]

1/72*(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) + 5*(a*b*x^6 + a^2 + sqrt(-3)*(a*b*x^6 + a^2))*(-1/(a^11*b))^(1/6)*
log(1/2*(sqrt(-3)*a^2 + a^2)*(-1/(a^11*b))^(1/6) + x) - 5*(a*b*x^6 + a^2 + sqrt(-3)*(a*b*x^6 + a^2))*(-1/(a^11
*b))^(1/6)*log(-1/2*(sqrt(-3)*a^2 + a^2)*(-1/(a^11*b))^(1/6) + x) - 5*(a*b*x^6 + a^2 - sqrt(-3)*(a*b*x^6 + a^2
))*(-1/(a^11*b))^(1/6)*log(1/2*(sqrt(-3)*a^2 - a^2)*(-1/(a^11*b))^(1/6) + x) + 5*(a*b*x^6 + a^2 - sqrt(-3)*(a*
b*x^6 + a^2))*(-1/(a^11*b))^(1/6)*log(-1/2*(sqrt(-3)*a^2 - a^2)*(-1/(a^11*b))^(1/6) + x) + 12*x)/(a*b*x^6 + a^
2)

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.17 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {x}{6 a^{2} + 6 a b x^{6}} + \operatorname {RootSum} {\left (2176782336 t^{6} a^{11} b + 15625, \left ( t \mapsto t \log {\left (\frac {36 t a^{2}}{5} + x \right )} \right )\right )} \]

[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)))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {x}{6 \, {\left (a b x^{6} + a^{2}\right )}} + \frac {5 \, {\left (\frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} - \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {5}{6}} b^{\frac {1}{6}}} + \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{a^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{72 \, a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {x}{6 \, {\left (b x^{6} + a\right )} a} + \frac {5 \, \sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a^{2} b} - \frac {5 \, \sqrt {3} \left (a b^{5}\right )^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{72 \, a^{2} b} + \frac {5 \, \left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a^{2} b} + \frac {5 \, \left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{36 \, a^{2} b} + \frac {5 \, \left (a b^{5}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{18 \, a^{2} b} \]

[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)

Mupad [B] (verification not implemented)

Time = 5.59 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (a+b x^6\right )^2} \, dx=\frac {x}{6\,a\,\left (b\,x^6+a\right )}-\frac {\mathrm {atan}\left (\frac {b^{1/6}\,x\,1{}\mathrm {i}}{{\left (-a\right )}^{1/6}}\right )\,5{}\mathrm {i}}{18\,{\left (-a\right )}^{11/6}\,b^{1/6}}+\frac {\mathrm {atan}\left (\frac {b^{29/6}\,x\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{35/6}\,\left (\frac {3125\,b^{14/3}}{7776\,{\left (-a\right )}^{17/3}}-\frac {\sqrt {3}\,b^{14/3}\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{17/3}}\right )}-\frac {3125\,\sqrt {3}\,b^{29/6}\,x}{7776\,{\left (-a\right )}^{35/6}\,\left (\frac {3125\,b^{14/3}}{7776\,{\left (-a\right )}^{17/3}}-\frac {\sqrt {3}\,b^{14/3}\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{17/3}}\right )}\right )\,\left (5+\sqrt {3}\,5{}\mathrm {i}\right )\,1{}\mathrm {i}}{36\,{\left (-a\right )}^{11/6}\,b^{1/6}}-\frac {\mathrm {atan}\left (\frac {b^{29/6}\,x\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{35/6}\,\left (\frac {3125\,b^{14/3}}{7776\,{\left (-a\right )}^{17/3}}+\frac {\sqrt {3}\,b^{14/3}\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{17/3}}\right )}+\frac {3125\,\sqrt {3}\,b^{29/6}\,x}{7776\,{\left (-a\right )}^{35/6}\,\left (\frac {3125\,b^{14/3}}{7776\,{\left (-a\right )}^{17/3}}+\frac {\sqrt {3}\,b^{14/3}\,3125{}\mathrm {i}}{7776\,{\left (-a\right )}^{17/3}}\right )}\right )\,\left (-5+\sqrt {3}\,5{}\mathrm {i}\right )\,1{}\mathrm {i}}{36\,{\left (-a\right )}^{11/6}\,b^{1/6}} \]

[In]

int(1/(a + b*x^6)^2,x)

[Out]

x/(6*a*(a + b*x^6)) - (atan((b^(1/6)*x*1i)/(-a)^(1/6))*5i)/(18*(-a)^(11/6)*b^(1/6)) + (atan((b^(29/6)*x*3125i)
/(7776*(-a)^(35/6)*((3125*b^(14/3))/(7776*(-a)^(17/3)) - (3^(1/2)*b^(14/3)*3125i)/(7776*(-a)^(17/3)))) - (3125
*3^(1/2)*b^(29/6)*x)/(7776*(-a)^(35/6)*((3125*b^(14/3))/(7776*(-a)^(17/3)) - (3^(1/2)*b^(14/3)*3125i)/(7776*(-
a)^(17/3)))))*(3^(1/2)*5i + 5)*1i)/(36*(-a)^(11/6)*b^(1/6)) - (atan((b^(29/6)*x*3125i)/(7776*(-a)^(35/6)*((312
5*b^(14/3))/(7776*(-a)^(17/3)) + (3^(1/2)*b^(14/3)*3125i)/(7776*(-a)^(17/3)))) + (3125*3^(1/2)*b^(29/6)*x)/(77
76*(-a)^(35/6)*((3125*b^(14/3))/(7776*(-a)^(17/3)) + (3^(1/2)*b^(14/3)*3125i)/(7776*(-a)^(17/3)))))*(3^(1/2)*5
i - 5)*1i)/(36*(-a)^(11/6)*b^(1/6))