3.2109 \(\int \frac{1}{a+\frac{b}{x^6}} \, dx\)

Optimal. Leaf size=220 \[ \frac{\sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{7/6}}+\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{6 a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{2 \sqrt [6]{a} x+\sqrt{3} \sqrt [6]{b}}{\sqrt [6]{b}}\right )}{6 a^{7/6}}+\frac{x}{a} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.43626, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889, Rules used = {193, 321, 209, 634, 618, 204, 628, 205} \[ \frac{\sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{7/6}}+\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{b}-2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{6 a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{2 \sqrt [6]{a} x+\sqrt{3} \sqrt [6]{b}}{\sqrt [6]{b}}\right )}{6 a^{7/6}}+\frac{x}{a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^6)^(-1),x]

[Out]

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

Rule 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 321

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*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 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{1}{a+\frac{b}{x^6}} \, dx &=\int \frac{x^6}{b+a x^6} \, dx\\ &=\frac{x}{a}-\frac{b \int \frac{1}{b+a x^6} \, dx}{a}\\ &=\frac{x}{a}-\frac{\sqrt [6]{b} \int \frac{\sqrt [6]{b}-\frac{1}{2} \sqrt{3} \sqrt [6]{a} x}{\sqrt [3]{b}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2} \, dx}{3 a}-\frac{\sqrt [6]{b} \int \frac{\sqrt [6]{b}+\frac{1}{2} \sqrt{3} \sqrt [6]{a} x}{\sqrt [3]{b}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2} \, dx}{3 a}-\frac{\sqrt [3]{b} \int \frac{1}{\sqrt [3]{b}+\sqrt [3]{a} x^2} \, dx}{3 a}\\ &=\frac{x}{a}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{7/6}}+\frac{\sqrt [6]{b} \int \frac{-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b}+2 \sqrt [3]{a} x}{\sqrt [3]{b}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} 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]{a} x}{\sqrt [3]{b}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2} \, dx}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [3]{b} \int \frac{1}{\sqrt [3]{b}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2} \, dx}{12 a}-\frac{\sqrt [3]{b} \int \frac{1}{\sqrt [3]{b}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2} \, dx}{12 a}\\ &=\frac{x}{a}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{7/6}}+\frac{\sqrt [6]{b} \log \left (\sqrt [3]{b}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [6]{b} \log \left (\sqrt [3]{b}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [6]{b} \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 \sqrt [6]{a} x}{\sqrt{3} \sqrt [6]{b}}\right )}{6 \sqrt{3} a^{7/6}}+\frac{\sqrt [6]{b} \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 \sqrt [6]{a} x}{\sqrt{3} \sqrt [6]{b}}\right )}{6 \sqrt{3} a^{7/6}}\\ &=\frac{x}{a}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{3 a^{7/6}}+\frac{\sqrt [6]{b} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{6 a^{7/6}}-\frac{\sqrt [6]{b} \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )}{6 a^{7/6}}+\frac{\sqrt [6]{b} \log \left (\sqrt [3]{b}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{4 \sqrt{3} a^{7/6}}-\frac{\sqrt [6]{b} \log \left (\sqrt [3]{b}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2\right )}{4 \sqrt{3} a^{7/6}}\\ \end{align*}

Mathematica [A]  time = 0.0560242, size = 182, normalized size = 0.83 \[ \frac{\sqrt{3} \sqrt [6]{b} \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )-\sqrt{3} \sqrt [6]{b} \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a} x^2+\sqrt [3]{b}\right )-4 \sqrt [6]{b} \tan ^{-1}\left (\frac{\sqrt [6]{a} x}{\sqrt [6]{b}}\right )+2 \sqrt [6]{b} \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{a} x}{\sqrt [6]{b}}\right )-2 \sqrt [6]{b} \tan ^{-1}\left (\frac{2 \sqrt [6]{a} x}{\sqrt [6]{b}}+\sqrt{3}\right )+12 \sqrt [6]{a} x}{12 a^{7/6}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^6)^(-1),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.06, size = 164, normalized size = 0.8 \begin{align*}{\frac{x}{a}}-{\frac{\sqrt{3}}{12\,a}\sqrt [6]{{\frac{b}{a}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{b}{a}}}x+\sqrt [3]{{\frac{b}{a}}} \right ) }-{\frac{1}{6\,a}\sqrt [6]{{\frac{b}{a}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{b}{a}}}}}}+\sqrt{3} \right ) }+{\frac{\sqrt{3}}{12\,a}\sqrt [6]{{\frac{b}{a}}}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{b}{a}}}x+\sqrt [3]{{\frac{b}{a}}} \right ) }-{\frac{1}{6\,a}\sqrt [6]{{\frac{b}{a}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{b}{a}}}}}}-\sqrt{3} \right ) }-{\frac{1}{3\,a}\sqrt [6]{{\frac{b}{a}}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{b}{a}}}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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/(a+b/x^6),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 1.58766, size = 803, normalized size = 3.65 \begin{align*} -\frac{4 \, \sqrt{3} a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2 \, \sqrt{3} a^{6} x \left (-\frac{b}{a^{7}}\right )^{\frac{5}{6}} - 2 \, \sqrt{3} \sqrt{a^{2} \left (-\frac{b}{a^{7}}\right )^{\frac{1}{3}} + a x \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} + x^{2}} a^{6} \left (-\frac{b}{a^{7}}\right )^{\frac{5}{6}} - \sqrt{3} b}{3 \, b}\right ) + 4 \, \sqrt{3} a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \arctan \left (-\frac{2 \, \sqrt{3} a^{6} x \left (-\frac{b}{a^{7}}\right )^{\frac{5}{6}} - 2 \, \sqrt{3} \sqrt{a^{2} \left (-\frac{b}{a^{7}}\right )^{\frac{1}{3}} - a x \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} + x^{2}} a^{6} \left (-\frac{b}{a^{7}}\right )^{\frac{5}{6}} + \sqrt{3} b}{3 \, b}\right ) + a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \log \left (a^{2} \left (-\frac{b}{a^{7}}\right )^{\frac{1}{3}} + a x \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} + x^{2}\right ) - a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \log \left (a^{2} \left (-\frac{b}{a^{7}}\right )^{\frac{1}{3}} - a x \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} + x^{2}\right ) + 2 \, a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \log \left (a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} + x\right ) - 2 \, a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} \log \left (-a \left (-\frac{b}{a^{7}}\right )^{\frac{1}{6}} + x\right ) - 12 \, x}{12 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.331391, size = 22, normalized size = 0.1 \begin{align*} \operatorname{RootSum}{\left (46656 t^{6} a^{7} + b, \left ( t \mapsto t \log{\left (- 6 t a + x \right )} \right )\right )} + \frac{x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.11033, size = 243, normalized size = 1.1 \begin{align*} \frac{x}{a} - \frac{\sqrt{3} \left (a^{5} b\right )^{\frac{1}{6}} \log \left (x^{2} + \sqrt{3} x \left (\frac{b}{a}\right )^{\frac{1}{6}} + \left (\frac{b}{a}\right )^{\frac{1}{3}}\right )}{12 \, a^{2}} + \frac{\sqrt{3} \left (a^{5} b\right )^{\frac{1}{6}} \log \left (x^{2} - \sqrt{3} x \left (\frac{b}{a}\right )^{\frac{1}{6}} + \left (\frac{b}{a}\right )^{\frac{1}{3}}\right )}{12 \, a^{2}} - \frac{\left (a^{5} b\right )^{\frac{1}{6}} \arctan \left (\frac{2 \, x + \sqrt{3} \left (\frac{b}{a}\right )^{\frac{1}{6}}}{\left (\frac{b}{a}\right )^{\frac{1}{6}}}\right )}{6 \, a^{2}} - \frac{\left (a^{5} b\right )^{\frac{1}{6}} \arctan \left (\frac{2 \, x - \sqrt{3} \left (\frac{b}{a}\right )^{\frac{1}{6}}}{\left (\frac{b}{a}\right )^{\frac{1}{6}}}\right )}{6 \, a^{2}} - \frac{\left (a^{5} b\right )^{\frac{1}{6}} \arctan \left (\frac{x}{\left (\frac{b}{a}\right )^{\frac{1}{6}}}\right )}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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