3.1328 \(\int \frac {1}{x^2 (a+b x^6)} \, dx\)

Optimal. Leaf size=223 \[ -\frac {\sqrt [6]{b} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}}+\frac {\sqrt [6]{b} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{7/6}}+\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac {1}{a x} \]

[Out]

-1/a/x-1/3*b^(1/6)*arctan(b^(1/6)*x/a^(1/6))/a^(7/6)+1/6*b^(1/6)*arctan((-2*b^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6)
)/a^(7/6)-1/6*b^(1/6)*arctan((2*b^(1/6)*x+a^(1/6)*3^(1/2))/a^(1/6))/a^(7/6)-1/12*b^(1/6)*ln(a^(1/3)+b^(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(a^(1/3)+b^(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 = 0.48, antiderivative size = 223, normalized size of antiderivative = 1.00, 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 = {325, 295, 634, 618, 204, 628, 205} \[ -\frac {\sqrt [6]{b} \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}}+\frac {\sqrt [6]{b} \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{7/6}}+\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}-2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [6]{a}+2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac {1}{a x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a + b*x^6)),x]

[Out]

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

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 295

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Module[{r = Numerator[Rt[a/b, n]], s = Denominator[Rt[a/
b, n]], k, u}, Simp[u = Int[(r*Cos[((2*k - 1)*m*Pi)/n] - s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 - 2*r*s*Cos[(
(2*k - 1)*Pi)/n]*x + s^2*x^2), x] + Int[(r*Cos[((2*k - 1)*m*Pi)/n] + s*Cos[((2*k - 1)*(m + 1)*Pi)/n]*x)/(r^2 +
2*r*s*Cos[((2*k - 1)*Pi)/n]*x + s^2*x^2), x]; (2*(-1)^(m/2)*r^(m + 2)*Int[1/(r^2 + s^2*x^2), x])/(a*n*s^m) +
Dist[(2*r^(m + 1))/(a*n*s^m), Sum[u, {k, 1, (n - 2)/4}], x], x]] /; FreeQ[{a, b}, x] && IGtQ[(n - 2)/4, 0] &&
IGtQ[m, 0] && LtQ[m, n - 1] && PosQ[a/b]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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}{x^2 \left (a+b x^6\right )} \, dx &=-\frac {1}{a x}-\frac {b \int \frac {x^4}{a+b x^6} \, dx}{a}\\ &=-\frac {1}{a x}-\frac {\sqrt [3]{b} \int \frac {-\frac {\sqrt [6]{a}}{2}+\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^{7/6}}-\frac {\sqrt [3]{b} \int \frac {-\frac {\sqrt [6]{a}}{2}-\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^{7/6}}-\frac {\sqrt [3]{b} \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} x^2} \, dx}{3 a}\\ &=-\frac {1}{a x}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{7/6}}-\frac {\sqrt [6]{b} \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^{7/6}}+\frac {\sqrt [6]{b} \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^{7/6}}-\frac {\sqrt [3]{b} \int \frac {1}{\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{12 a}-\frac {\sqrt [3]{b} \int \frac {1}{\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2} \, dx}{12 a}\\ &=-\frac {1}{a x}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{7/6}}-\frac {\sqrt [6]{b} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}}+\frac {\sqrt [6]{b} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} 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]{b} x}{\sqrt {3} \sqrt [6]{a}}\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]{b} x}{\sqrt {3} \sqrt [6]{a}}\right )}{6 \sqrt {3} a^{7/6}}\\ &=-\frac {1}{a x}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{3 a^{7/6}}+\frac {\sqrt [6]{b} \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \tan ^{-1}\left (\sqrt {3}+\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )}{6 a^{7/6}}-\frac {\sqrt [6]{b} \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}}+\frac {\sqrt [6]{b} \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{b} x^2\right )}{4 \sqrt {3} a^{7/6}}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 189, normalized size = 0.85 \[ -\frac {\sqrt {3} \sqrt [6]{b} x \log \left (-\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )-\sqrt {3} \sqrt [6]{b} x \log \left (\sqrt {3} \sqrt [6]{a} \sqrt [6]{b} x+\sqrt [3]{a}+\sqrt [3]{b} x^2\right )+4 \sqrt [6]{b} x \tan ^{-1}\left (\frac {\sqrt [6]{b} x}{\sqrt [6]{a}}\right )-2 \sqrt [6]{b} x \tan ^{-1}\left (\sqrt {3}-\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}\right )+2 \sqrt [6]{b} x \tan ^{-1}\left (\frac {2 \sqrt [6]{b} x}{\sqrt [6]{a}}+\sqrt {3}\right )+12 \sqrt [6]{a}}{12 a^{7/6} x} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(a + b*x^6)),x]

[Out]

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

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fricas [B]  time = 0.91, size = 338, normalized size = 1.52 \[ \frac {4 \, \sqrt {3} a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2}{3} \, \sqrt {3} a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + \frac {2}{3} \, \sqrt {3} a \sqrt {-\frac {a^{6} x \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} + a^{5} \left (-\frac {b}{a^{7}}\right )^{\frac {2}{3}} - b x^{2}}{b}} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} - \frac {1}{3} \, \sqrt {3}\right ) + 4 \, \sqrt {3} a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \arctan \left (-\frac {2}{3} \, \sqrt {3} a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + \frac {2}{3} \, \sqrt {3} a \sqrt {\frac {a^{6} x \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} - a^{5} \left (-\frac {b}{a^{7}}\right )^{\frac {2}{3}} + b x^{2}}{b}} \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} + \frac {1}{3} \, \sqrt {3}\right ) - a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (a^{6} x \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} - a^{5} \left (-\frac {b}{a^{7}}\right )^{\frac {2}{3}} + b x^{2}\right ) + a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (-a^{6} x \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} - a^{5} \left (-\frac {b}{a^{7}}\right )^{\frac {2}{3}} + b x^{2}\right ) - 2 \, a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (a^{6} \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} + b x\right ) + 2 \, a x \left (-\frac {b}{a^{7}}\right )^{\frac {1}{6}} \log \left (-a^{6} \left (-\frac {b}{a^{7}}\right )^{\frac {5}{6}} + b x\right ) - 12}{12 \, a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.26, size = 196, normalized size = 0.88 \[ -\frac {b \left (\frac {a}{b}\right )^{\frac {5}{6}} \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \, a^{2}} - \frac {1}{a x} + \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {5}{6}} \log \left (x^{2} + \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, a^{2} b^{4}} - \frac {\sqrt {3} \left (a b^{5}\right )^{\frac {5}{6}} \log \left (x^{2} - \sqrt {3} x \left (\frac {a}{b}\right )^{\frac {1}{6}} + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 \, a^{2} b^{4}} - \frac {\left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x + \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, a^{2} b^{4}} - \frac {\left (a b^{5}\right )^{\frac {5}{6}} \arctan \left (\frac {2 \, x - \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{6}}}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{6 \, a^{2} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.05, size = 169, normalized size = 0.76 \[ -\frac {\arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}\right )}{3 \left (\frac {a}{b}\right )^{\frac {1}{6}} a}-\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{6}} a}-\frac {\arctan \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 \left (\frac {a}{b}\right )^{\frac {1}{6}} a}-\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} b \ln \left (x^{2}-\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a^{2}}+\frac {\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {5}{6}} b \ln \left (x^{2}+\sqrt {3}\, \left (\frac {a}{b}\right )^{\frac {1}{6}} x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{12 a^{2}}-\frac {1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.48, size = 198, normalized size = 0.89 \[ \frac {b {\left (\frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {\sqrt {3} \log \left (b^{\frac {1}{3}} x^{2} - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}} x + a^{\frac {1}{3}}\right )}{a^{\frac {1}{6}} b^{\frac {5}{6}}} - \frac {4 \, \arctan \left (\frac {b^{\frac {1}{3}} x}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x + \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}} - \frac {2 \, \arctan \left (\frac {2 \, b^{\frac {1}{3}} x - \sqrt {3} a^{\frac {1}{6}} b^{\frac {1}{6}}}{\sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}{b^{\frac {2}{3}} \sqrt {a^{\frac {1}{3}} b^{\frac {1}{3}}}}\right )}}{12 \, a} - \frac {1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.18, size = 149, normalized size = 0.67 \[ -\frac {1}{a\,x}-\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/6}\,x\,1{}\mathrm {i}}{a^{1/6}}\right )\,1{}\mathrm {i}}{3\,a^{7/6}}-\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {a^{13/2}\,{\left (-b\right )}^{13/2}\,x\,2{}\mathrm {i}}{a^{20/3}\,{\left (-b\right )}^{19/3}-\sqrt {3}\,a^{20/3}\,{\left (-b\right )}^{19/3}\,1{}\mathrm {i}}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3\,a^{7/6}}+\frac {{\left (-b\right )}^{1/6}\,\mathrm {atan}\left (\frac {a^{13/2}\,{\left (-b\right )}^{13/2}\,x\,2{}\mathrm {i}}{a^{20/3}\,{\left (-b\right )}^{19/3}+\sqrt {3}\,a^{20/3}\,{\left (-b\right )}^{19/3}\,1{}\mathrm {i}}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}}{3\,a^{7/6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a + b*x^6)),x)

[Out]

((-b)^(1/6)*atan((a^(13/2)*(-b)^(13/2)*x*2i)/(a^(20/3)*(-b)^(19/3) + 3^(1/2)*a^(20/3)*(-b)^(19/3)*1i))*((3^(1/
2)*1i)/2 - 1/2)*1i)/(3*a^(7/6)) - ((-b)^(1/6)*atan(((-b)^(1/6)*x*1i)/a^(1/6))*1i)/(3*a^(7/6)) - ((-b)^(1/6)*at
an((a^(13/2)*(-b)^(13/2)*x*2i)/(a^(20/3)*(-b)^(19/3) - 3^(1/2)*a^(20/3)*(-b)^(19/3)*1i))*((3^(1/2)*1i)/2 + 1/2
)*1i)/(3*a^(7/6)) - 1/(a*x)

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sympy [A]  time = 0.46, size = 29, normalized size = 0.13 \[ \operatorname {RootSum} {\left (46656 t^{6} a^{7} + b, \left (t \mapsto t \log {\left (- \frac {7776 t^{5} a^{6}}{b} + x \right )} \right )\right )} - \frac {1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(b*x**6+a),x)

[Out]

RootSum(46656*_t**6*a**7 + b, Lambda(_t, _t*log(-7776*_t**5*a**6/b + x))) - 1/(a*x)

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