3.1457 \(\int \frac{1}{x^3 (a+b x^8)} \, dx\)

Optimal. Leaf size=203 \[ -\frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4}}-\frac{1}{2 a x^2} \]

[Out]

-1/(2*a*x^2) + (b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*a^(5/4)) - (b^(1/4)*ArcTan[1 + (
Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*a^(5/4)) - (b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt
[b]*x^4])/(8*Sqrt[2]*a^(5/4)) + (b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(8*Sqrt[2]*
a^(5/4))

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Rubi [A]  time = 0.174804, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {275, 325, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\sqrt [4]{b} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} a^{5/4}}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a + b*x^8)),x]

[Out]

-1/(2*a*x^2) + (b^(1/4)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*a^(5/4)) - (b^(1/4)*ArcTan[1 + (
Sqrt[2]*b^(1/4)*x^2)/a^(1/4)])/(4*Sqrt[2]*a^(5/4)) - (b^(1/4)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt
[b]*x^4])/(8*Sqrt[2]*a^(5/4)) + (b^(1/4)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x^2 + Sqrt[b]*x^4])/(8*Sqrt[2]*
a^(5/4))

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
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 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

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]

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (a+b x^8\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 \left (a+b x^4\right )} \, dx,x,x^2\right )\\ &=-\frac{1}{2 a x^2}-\frac{b \operatorname{Subst}\left (\int \frac{x^2}{a+b x^4} \, dx,x,x^2\right )}{2 a}\\ &=-\frac{1}{2 a x^2}+\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{\sqrt{a}-\sqrt{b} x^2}{a+b x^4} \, dx,x,x^2\right )}{4 a}-\frac{\sqrt{b} \operatorname{Subst}\left (\int \frac{\sqrt{a}+\sqrt{b} x^2}{a+b x^4} \, dx,x,x^2\right )}{4 a}\\ &=-\frac{1}{2 a x^2}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^2\right )}{8 a}-\frac{\operatorname{Subst}\left (\int \frac{1}{\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}+x^2} \, dx,x,x^2\right )}{8 a}-\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}+2 x}{-\frac{\sqrt{a}}{\sqrt{b}}-\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^2\right )}{8 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{\frac{\sqrt{2} \sqrt [4]{a}}{\sqrt [4]{b}}-2 x}{-\frac{\sqrt{a}}{\sqrt{b}}+\frac{\sqrt{2} \sqrt [4]{a} x}{\sqrt [4]{b}}-x^2} \, dx,x,x^2\right )}{8 \sqrt{2} a^{5/4}}\\ &=-\frac{1}{2 a x^2}-\frac{\sqrt [4]{b} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4}}\\ &=-\frac{1}{2 a x^2}+\frac{\sqrt [4]{b} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} a^{5/4}}-\frac{\sqrt [4]{b} \log \left (\sqrt{a}-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{5/4}}+\frac{\sqrt [4]{b} \log \left (\sqrt{a}+\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{b} x^4\right )}{8 \sqrt{2} a^{5/4}}\\ \end{align*}

Mathematica [A]  time = 0.0750748, size = 385, normalized size = 1.9 \[ \frac{\sqrt{2} \sqrt [4]{b} x^2 \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\sqrt{2} \sqrt [4]{b} x^2 \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sqrt{2} \sqrt [4]{b} x^2 \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\sqrt{2} \sqrt [4]{b} x^2 \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-2 \sqrt{2} \sqrt [4]{b} x^2 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \sqrt{2} \sqrt [4]{b} x^2 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+2 \sqrt{2} \sqrt [4]{b} x^2 \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \sqrt{2} \sqrt [4]{b} x^2 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )-8 \sqrt [4]{a}}{16 a^{5/4} x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a + b*x^8)),x]

[Out]

(-8*a^(1/4) + 2*Sqrt[2]*b^(1/4)*x^2*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)] + 2*Sqrt[2]*b^(1/4)*x^2*
ArcTan[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)] - 2*Sqrt[2]*b^(1/4)*x^2*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8)
- Tan[Pi/8]] + 2*Sqrt[2]*b^(1/4)*x^2*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]] - Sqrt[2]*b^(1/4)*x^2*
Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] - Sqrt[2]*b^(1/4)*x^2*Log[a^(1/4) + b^(1/4)*x^2 + 2
*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + Sqrt[2]*b^(1/4)*x^2*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]
+ Sqrt[2]*b^(1/4)*x^2*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]])/(16*a^(5/4)*x^2)

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Maple [A]  time = 0.004, size = 144, normalized size = 0.7 \begin{align*} -{\frac{\sqrt{2}}{16\,a}\ln \left ({ \left ({x}^{4}-\sqrt [4]{{\frac{a}{b}}}{x}^{2}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{4}+\sqrt [4]{{\frac{a}{b}}}{x}^{2}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}}{8\,a}\arctan \left ({{x}^{2}\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{\sqrt{2}}{8\,a}\arctan \left ({{x}^{2}\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-{\frac{1}{2\,a{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b*x^8+a),x)

[Out]

-1/16/a/(1/b*a)^(1/4)*2^(1/2)*ln((x^4-(1/b*a)^(1/4)*x^2*2^(1/2)+(1/b*a)^(1/2))/(x^4+(1/b*a)^(1/4)*x^2*2^(1/2)+
(1/b*a)^(1/2)))-1/8/a/(1/b*a)^(1/4)*2^(1/2)*arctan(2^(1/2)/(1/b*a)^(1/4)*x^2+1)-1/8/a/(1/b*a)^(1/4)*2^(1/2)*ar
ctan(2^(1/2)/(1/b*a)^(1/4)*x^2-1)-1/2/a/x^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/x^3/(b*x^8+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.2873, size = 320, normalized size = 1.58 \begin{align*} \frac{4 \, a x^{2} \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} \arctan \left (-a x^{2} \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} + a \sqrt{\frac{b x^{4} - a^{3} \sqrt{-\frac{b}{a^{5}}}}{b}} \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}}\right ) - a x^{2} \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} \log \left (a^{4} \left (-\frac{b}{a^{5}}\right )^{\frac{3}{4}} + b x^{2}\right ) + a x^{2} \left (-\frac{b}{a^{5}}\right )^{\frac{1}{4}} \log \left (-a^{4} \left (-\frac{b}{a^{5}}\right )^{\frac{3}{4}} + b x^{2}\right ) - 4}{8 \, a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^8+a),x, algorithm="fricas")

[Out]

1/8*(4*a*x^2*(-b/a^5)^(1/4)*arctan(-a*x^2*(-b/a^5)^(1/4) + a*sqrt((b*x^4 - a^3*sqrt(-b/a^5))/b)*(-b/a^5)^(1/4)
) - a*x^2*(-b/a^5)^(1/4)*log(a^4*(-b/a^5)^(3/4) + b*x^2) + a*x^2*(-b/a^5)^(1/4)*log(-a^4*(-b/a^5)^(3/4) + b*x^
2) - 4)/(a*x^2)

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Sympy [A]  time = 0.482109, size = 34, normalized size = 0.17 \begin{align*} \operatorname{RootSum}{\left (4096 t^{4} a^{5} + b, \left ( t \mapsto t \log{\left (- \frac{512 t^{3} a^{4}}{b} + x^{2} \right )} \right )\right )} - \frac{1}{2 a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(b*x**8+a),x)

[Out]

RootSum(4096*_t**4*a**5 + b, Lambda(_t, _t*log(-512*_t**3*a**4/b + x**2))) - 1/(2*a*x**2)

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Giac [A]  time = 1.18092, size = 263, normalized size = 1.3 \begin{align*} -\frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} x^{4} \arctan \left (\frac{\sqrt{2}{\left (2 \, x^{2} + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} x^{4} \arctan \left (\frac{\sqrt{2}{\left (2 \, x^{2} - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a^{2}} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} x^{4} \log \left (x^{4} + \sqrt{2} x^{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2}} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} x^{4} \log \left (x^{4} - \sqrt{2} x^{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{16 \, a^{2}} - \frac{1}{2 \, a x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b*x^8+a),x, algorithm="giac")

[Out]

-1/8*sqrt(2)*(a*b^3)^(1/4)*x^4*arctan(1/2*sqrt(2)*(2*x^2 + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/a^2 - 1/8*sqrt(2)
*(a*b^3)^(1/4)*x^4*arctan(1/2*sqrt(2)*(2*x^2 - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/a^2 - 1/16*sqrt(2)*(a*b^3)^(1
/4)*x^4*log(x^4 + sqrt(2)*x^2*(a/b)^(1/4) + sqrt(a/b))/a^2 + 1/16*sqrt(2)*(a*b^3)^(1/4)*x^4*log(x^4 - sqrt(2)*
x^2*(a/b)^(1/4) + sqrt(a/b))/a^2 - 1/2/(a*x^2)