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)) - (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))

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Rubi [A]  time = 0.479936, antiderivative size = 223, normalized size of antiderivative = 1., 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 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 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 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}{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*}

Mathematica [A]  time = 0.0430273, 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]

-(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])/(12*a^(7/6)*x)

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Maple [A]  time = 0.03, size = 169, normalized size = 0.8 \begin{align*}{\frac{b\sqrt{3}}{12\,{a}^{2}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{1}{6\,a}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}+\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-{\frac{1}{3\,a}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-{\frac{b\sqrt{3}}{12\,{a}^{2}} \left ({\frac{a}{b}} \right ) ^{{\frac{5}{6}}}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{b}}}x+\sqrt [3]{{\frac{a}{b}}} \right ) }-{\frac{1}{6\,a}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-\sqrt{3} \right ){\frac{1}{\sqrt [6]{{\frac{a}{b}}}}}}-{\frac{1}{ax}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*b/a^2*3^(1/2)*(1/b*a)^(5/6)*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*b/a^2*3^(1/2)*(1/b*a)^(5/6)*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/a/x

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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/x^2/(b*x^6+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.81757, size = 851, normalized size = 3.82 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 0.364808, size = 29, normalized size = 0.13 \begin{align*} \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} \end{align*}

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError