3.1326 \(\int \frac{1}{a+b x^6} \, dx\)

Optimal. Leaf size=215 \[ -\frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} a^{5/6} \sqrt [6]{b}}+\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} a^{5/6} \sqrt [6]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}} \]

[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))

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Rubi [A]  time = 0.398242, antiderivative size = 215, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {209, 634, 618, 204, 628, 205} \[ -\frac{\log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} a^{5/6} \sqrt [6]{b}}+\frac{\log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt{3} a^{5/6} \sqrt [6]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} \sqrt [6]{b}} \]

Antiderivative was successfully verified.

[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 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}{a+b x^6} \, dx &=\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{\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 )}{6 \sqrt{3} a^{5/6} \sqrt [6]{b}}-\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 )}{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]  time = 0.0179342, size = 154, normalized size = 0.72 \[ \frac{-\sqrt{3} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+\sqrt{3} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+4 \tan ^{-1}\left (\frac{\sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \tan ^{-1}\left (\frac{2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{12 a^{5/6} \sqrt [6]{b}} \]

Antiderivative was successfully verified.

[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))

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Maple [A]  time = 0.029, size = 159, normalized size = 0.7 \begin{align*}{\frac{\sqrt{3}}{12\,a}\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}{6\,a}\sqrt [6]{{\frac{a}{b}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ) }+{\frac{1}{3\,a}\sqrt [6]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ) }-{\frac{\sqrt{3}}{12\,a}\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}{6\,a}\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),x)

[Out]

1/12/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))+1/6/a*(1/b*a)^(1/6)*arctan(2*x/(1/b
*a)^(1/6)+3^(1/2))+1/3/a*(1/b*a)^(1/6)*arctan(x/(1/b*a)^(1/6))-1/12/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))+1/6/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(1/(b*x^6+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.93207, size = 902, normalized size = 4.2 \begin{align*} \frac{1}{3} \, \sqrt{3} \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} \arctan \left (-\frac{2}{3} \, \sqrt{3} a^{4} b x \left (-\frac{1}{a^{5} b}\right )^{\frac{5}{6}} + \frac{2}{3} \, \sqrt{3} \sqrt{a^{2} \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{3}} + a x \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} + x^{2}} a^{4} b \left (-\frac{1}{a^{5} b}\right )^{\frac{5}{6}} + \frac{1}{3} \, \sqrt{3}\right ) + \frac{1}{3} \, \sqrt{3} \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} \arctan \left (-\frac{2}{3} \, \sqrt{3} a^{4} b x \left (-\frac{1}{a^{5} b}\right )^{\frac{5}{6}} + \frac{2}{3} \, \sqrt{3} \sqrt{a^{2} \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{3}} - a x \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} + x^{2}} a^{4} b \left (-\frac{1}{a^{5} b}\right )^{\frac{5}{6}} - \frac{1}{3} \, \sqrt{3}\right ) + \frac{1}{12} \, \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} \log \left (a^{2} \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{3}} + a x \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} + x^{2}\right ) - \frac{1}{12} \, \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} \log \left (a^{2} \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{3}} - a x \left (-\frac{1}{a^{5} b}\right )^{\frac{1}{6}} + x^{2}\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.1552, size = 20, normalized size = 0.09 \begin{align*} \operatorname{RootSum}{\left (46656 t^{6} a^{5} b + 1, \left ( t \mapsto t \log{\left (6 t a + x \right )} \right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)))

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Giac [A]  time = 1.22092, size = 257, normalized size = 1.2 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)