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-1/3*b^(1/6)*arctan(a^(1/6)*x/b^(1/6))/a^(7/6)+1/6*b^(1/6)*arctan((-2*a^(1/6)*x+b^(1/6)*3^(1/2))/b^(1/6))/a
^(7/6)-1/6*b^(1/6)*arctan((2*a^(1/6)*x+b^(1/6)*3^(1/2))/b^(1/6))/a^(7/6)+1/12*b^(1/6)*ln(b^(1/3)+a^(1/3)*x^2-a
^(1/6)*b^(1/6)*x*3^(1/2))/a^(7/6)*3^(1/2)-1/12*b^(1/6)*ln(b^(1/3)+a^(1/3)*x^2+a^(1/6)*b^(1/6)*x*3^(1/2))/a^(7/
6)*3^(1/2)

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Rubi [A]  time = 0.44, antiderivative size = 220, normalized size of antiderivative = 1.00, 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 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 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]

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 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 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 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 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]

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*}

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Mathematica [A]  time = 0.08, 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))

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fricas [B]  time = 1.30, size = 314, normalized size = 1.43 \[ -\frac {4 \, \sqrt {3} a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} a^{6} x \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} \sqrt {a^{2} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{3}} + a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x^{2}} a^{6} \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} - \sqrt {3} b}{3 \, b}\right ) + 4 \, \sqrt {3} a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2 \, \sqrt {3} a^{6} x \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} - 2 \, \sqrt {3} \sqrt {a^{2} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{3}} - a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x^{2}} a^{6} \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} + \sqrt {3} b}{3 \, b}\right ) + a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (a^{2} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{3}} + a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x^{2}\right ) - a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (a^{2} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{3}} - a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x^{2}\right ) + 2 \, a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x\right ) - 2 \, a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (-a \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + x\right ) - 12 \, x}{12 \, a} \]

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

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giac [A]  time = 0.17, size = 180, normalized size = 0.82 \[ \frac {x}{a} - \frac {\sqrt {3} \left (a^{5} b\right )^{\frac {1}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{6}} + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 \, a^{2}} + \frac {\sqrt {3} \left (a^{5} b\right )^{\frac {1}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {b}{a}\right )^{\frac {1}{6}} + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 \, a^{2}} - \frac {\left (a^{5} b\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {b}{a}\right )^{\frac {1}{6}}}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{6 \, a^{2}} - \frac {\left (a^{5} b\right )^{\frac {1}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {b}{a}\right )^{\frac {1}{6}}}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{6 \, a^{2}} - \frac {\left (a^{5} b\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{3 \, a^{2}} \]

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

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maple [A]  time = 0.10, size = 164, normalized size = 0.75 \[ \frac {x}{a}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{6}} \arctan \left (\frac {x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}\right )}{3 a}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 a}-\frac {\left (\frac {b}{a}\right )^{\frac {1}{6}} \arctan \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 a}+\frac {\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} \ln \left (x^{2}-\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 a}-\frac {\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} \ln \left (x^{2}+\sqrt {3}\, \left (\frac {b}{a}\right )^{\frac {1}{6}} x +\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{12 a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.77, size = 194, normalized size = 0.88 \[ -\frac {\frac {\sqrt {3} b^{\frac {1}{6}} \log \left (a^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + b^{\frac {1}{3}}\right )}{a^{\frac {1}{6}}} - \frac {\sqrt {3} b^{\frac {1}{6}} \log \left (a^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + b^{\frac {1}{3}}\right )}{a^{\frac {1}{6}}} + \frac {4 \, b^{\frac {1}{3}} \arctan \left (\frac {a^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, b^{\frac {1}{3}} \arctan \left (\frac {2 \, a^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} + \frac {2 \, b^{\frac {1}{3}} \arctan \left (\frac {2 \, a^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}}{12 \, a} + \frac {x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.23, size = 227, normalized size = 1.03 \[ \frac {x}{a}+\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {a^{1/6}\,x\,1{}\mathrm {i}}{{\left (-b\right )}^{1/6}}\right )\,1{}\mathrm {i}}{3\,a^{7/6}}+\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{25/6}\,x\,1{}\mathrm {i}}{a^{1/6}\,\left (\frac {{\left (-b\right )}^{13/3}}{a^{1/3}}+\frac {\sqrt {3}\,{\left (-b\right )}^{13/3}\,1{}\mathrm {i}}{a^{1/3}}\right )}+\frac {\sqrt {3}\,{\left (-b\right )}^{25/6}\,x}{a^{1/6}\,\left (\frac {{\left (-b\right )}^{13/3}}{a^{1/3}}+\frac {\sqrt {3}\,{\left (-b\right )}^{13/3}\,1{}\mathrm {i}}{a^{1/3}}\right )}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3\,a^{7/6}}-\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{25/6}\,x\,1{}\mathrm {i}}{a^{1/6}\,\left (\frac {{\left (-b\right )}^{13/3}}{a^{1/3}}-\frac {\sqrt {3}\,{\left (-b\right )}^{13/3}\,1{}\mathrm {i}}{a^{1/3}}\right )}-\frac {\sqrt {3}\,{\left (-b\right )}^{25/6}\,x}{a^{1/6}\,\left (\frac {{\left (-b\right )}^{13/3}}{a^{1/3}}-\frac {\sqrt {3}\,{\left (-b\right )}^{13/3}\,1{}\mathrm {i}}{a^{1/3}}\right )}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3\,a^{7/6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/x^6),x)

[Out]

x/a + ((-b)^(1/6)*atan((a^(1/6)*x*1i)/(-b)^(1/6))*1i)/(3*a^(7/6)) + ((-b)^(1/6)*atan(((-b)^(25/6)*x*1i)/(a^(1/
6)*((-b)^(13/3)/a^(1/3) + (3^(1/2)*(-b)^(13/3)*1i)/a^(1/3))) + (3^(1/2)*(-b)^(25/6)*x)/(a^(1/6)*((-b)^(13/3)/a
^(1/3) + (3^(1/2)*(-b)^(13/3)*1i)/a^(1/3))))*((3^(1/2)*1i)/2 - 1/2)*1i)/(3*a^(7/6)) - ((-b)^(1/6)*atan(((-b)^(
25/6)*x*1i)/(a^(1/6)*((-b)^(13/3)/a^(1/3) - (3^(1/2)*(-b)^(13/3)*1i)/a^(1/3))) - (3^(1/2)*(-b)^(25/6)*x)/(a^(1
/6)*((-b)^(13/3)/a^(1/3) - (3^(1/2)*(-b)^(13/3)*1i)/a^(1/3))))*((3^(1/2)*1i)/2 + 1/2)*1i)/(3*a^(7/6))

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sympy [A]  time = 0.21, size = 22, normalized size = 0.10 \[ \operatorname {RootSum} {\left (46656 t^{6} a^{7} + b, \left (t \mapsto t \log {\left (- 6 t a + x \right )} \right )\right )} + \frac {x}{a} \]

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

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