3.2110 \(\int \frac {1}{a+\frac {b}{x^8}} \, dx\)

Optimal. Leaf size=272 \[ -\frac {\sqrt [8]{b} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{8 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{8 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \tan ^{-1}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \tanh ^{-1}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}+\frac {x}{a} \]

[Out]

x/a+1/4*b^(1/8)*arctan((-a)^(1/8)*x/b^(1/8))/(-a)^(9/8)+1/4*b^(1/8)*arctanh((-a)^(1/8)*x/b^(1/8))/(-a)^(9/8)+1
/8*b^(1/8)*arctan(-1+(-a)^(1/8)*x*2^(1/2)/b^(1/8))/(-a)^(9/8)*2^(1/2)+1/8*b^(1/8)*arctan(1+(-a)^(1/8)*x*2^(1/2
)/b^(1/8))/(-a)^(9/8)*2^(1/2)-1/16*b^(1/8)*ln(b^(1/4)+(-a)^(1/4)*x^2-(-a)^(1/8)*b^(1/8)*x*2^(1/2))/(-a)^(9/8)*
2^(1/2)+1/16*b^(1/8)*ln(b^(1/4)+(-a)^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*2^(1/2))/(-a)^(9/8)*2^(1/2)

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Rubi [A]  time = 0.27, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {193, 321, 214, 212, 208, 205, 211, 1165, 628, 1162, 617, 204} \[ -\frac {\sqrt [8]{b} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{8 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{8 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \tan ^{-1}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \tanh ^{-1}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}+\frac {x}{a} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x^8)^(-1),x]

[Out]

x/a + (b^(1/8)*ArcTan[((-a)^(1/8)*x)/b^(1/8)])/(4*(-a)^(9/8)) - (b^(1/8)*ArcTan[1 - (Sqrt[2]*(-a)^(1/8)*x)/b^(
1/8)])/(4*Sqrt[2]*(-a)^(9/8)) + (b^(1/8)*ArcTan[1 + (Sqrt[2]*(-a)^(1/8)*x)/b^(1/8)])/(4*Sqrt[2]*(-a)^(9/8)) +
(b^(1/8)*ArcTanh[((-a)^(1/8)*x)/b^(1/8)])/(4*(-a)^(9/8)) - (b^(1/8)*Log[b^(1/4) - Sqrt[2]*(-a)^(1/8)*b^(1/8)*x
+ (-a)^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(9/8)) + (b^(1/8)*Log[b^(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1/8)*x + (-a)^(1/4)*
x^2])/(8*Sqrt[2]*(-a)^(9/8))

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 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 211

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

Rule 212

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

Rule 214

Int[((a_) + (b_.)*(x_)^(n_))^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[r/(2*a), Int[1/(r - s*x^(n/2)), x], x] + Dist[r/(2*a), Int[1/(r + s*x^(n/2)), x], x]] /; FreeQ[{a,
b}, x] && IGtQ[n/4, 1] &&  !GtQ[a/b, 0]

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 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 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 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 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]

Rubi steps

\begin {align*} \int \frac {1}{a+\frac {b}{x^8}} \, dx &=\int \frac {x^8}{b+a x^8} \, dx\\ &=\frac {x}{a}-\frac {b \int \frac {1}{b+a x^8} \, dx}{a}\\ &=\frac {x}{a}-\frac {\sqrt {b} \int \frac {1}{\sqrt {b}-\sqrt {-a} x^4} \, dx}{2 a}-\frac {\sqrt {b} \int \frac {1}{\sqrt {b}+\sqrt {-a} x^4} \, dx}{2 a}\\ &=\frac {x}{a}-\frac {\sqrt [4]{b} \int \frac {1}{\sqrt [4]{b}-\sqrt [4]{-a} x^2} \, dx}{4 a}-\frac {\sqrt [4]{b} \int \frac {1}{\sqrt [4]{b}+\sqrt [4]{-a} x^2} \, dx}{4 a}-\frac {\sqrt [4]{b} \int \frac {\sqrt [4]{b}-\sqrt [4]{-a} x^2}{\sqrt {b}+\sqrt {-a} x^4} \, dx}{4 a}-\frac {\sqrt [4]{b} \int \frac {\sqrt [4]{b}+\sqrt [4]{-a} x^2}{\sqrt {b}+\sqrt {-a} x^4} \, dx}{4 a}\\ &=\frac {x}{a}+\frac {\sqrt [8]{b} \tan ^{-1}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}+\frac {\sqrt [8]{b} \tanh ^{-1}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \int \frac {\frac {\sqrt {2} \sqrt [8]{b}}{\sqrt [8]{-a}}+2 x}{-\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}-x^2} \, dx}{8 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{b} \int \frac {\frac {\sqrt {2} \sqrt [8]{b}}{\sqrt [8]{-a}}-2 x}{-\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}-x^2} \, dx}{8 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [4]{b} \int \frac {1}{\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+x^2} \, dx}{8 (-a)^{5/4}}+\frac {\sqrt [4]{b} \int \frac {1}{\frac {\sqrt [4]{b}}{\sqrt [4]{-a}}+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+x^2} \, dx}{8 (-a)^{5/4}}\\ &=\frac {x}{a}+\frac {\sqrt [8]{b} \tan ^{-1}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}+\frac {\sqrt [8]{b} \tanh ^{-1}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \log \left (\sqrt [4]{b}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2\right )}{8 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \log \left (\sqrt [4]{b}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2\right )}{8 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{b} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt {2} (-a)^{9/8}}\\ &=\frac {x}{a}+\frac {\sqrt [8]{b} \tan ^{-1}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \tanh ^{-1}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \log \left (\sqrt [4]{b}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2\right )}{8 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \log \left (\sqrt [4]{b}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2\right )}{8 \sqrt {2} (-a)^{9/8}}\\ \end {align*}

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Mathematica [A]  time = 0.34, size = 367, normalized size = 1.35 \[ \frac {\sqrt [8]{b} \sin \left (\frac {\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )+\sqrt [4]{a} x^2+\sqrt [4]{b}\right )-\sqrt [8]{b} \sin \left (\frac {\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )+\sqrt [4]{a} x^2+\sqrt [4]{b}\right )+\sqrt [8]{b} \cos \left (\frac {\pi }{8}\right ) \log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )+\sqrt [4]{a} x^2+\sqrt [4]{b}\right )-\sqrt [8]{b} \cos \left (\frac {\pi }{8}\right ) \log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )+\sqrt [4]{a} x^2+\sqrt [4]{b}\right )-2 \sqrt [8]{b} \cos \left (\frac {\pi }{8}\right ) \tan ^{-1}\left (\frac {\sqrt [8]{a} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}-\tan \left (\frac {\pi }{8}\right )\right )-2 \sqrt [8]{b} \cos \left (\frac {\pi }{8}\right ) \tan ^{-1}\left (\frac {\sqrt [8]{a} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}+\tan \left (\frac {\pi }{8}\right )\right )+2 \sqrt [8]{b} \sin \left (\frac {\pi }{8}\right ) \tan ^{-1}\left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{a} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}\right )-2 \sqrt [8]{b} \sin \left (\frac {\pi }{8}\right ) \tan ^{-1}\left (\frac {\sqrt [8]{a} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}+\cot \left (\frac {\pi }{8}\right )\right )+8 \sqrt [8]{a} x}{8 a^{9/8}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x^8)^(-1),x]

[Out]

(8*a^(1/8)*x - 2*b^(1/8)*ArcTan[(a^(1/8)*x*Sec[Pi/8])/b^(1/8) - Tan[Pi/8]]*Cos[Pi/8] - 2*b^(1/8)*ArcTan[(a^(1/
8)*x*Sec[Pi/8])/b^(1/8) + Tan[Pi/8]]*Cos[Pi/8] + b^(1/8)*Cos[Pi/8]*Log[b^(1/4) + a^(1/4)*x^2 - 2*a^(1/8)*b^(1/
8)*x*Cos[Pi/8]] - b^(1/8)*Cos[Pi/8]*Log[b^(1/4) + a^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + 2*b^(1/8)*Arc
Tan[Cot[Pi/8] - (a^(1/8)*x*Csc[Pi/8])/b^(1/8)]*Sin[Pi/8] - 2*b^(1/8)*ArcTan[Cot[Pi/8] + (a^(1/8)*x*Csc[Pi/8])/
b^(1/8)]*Sin[Pi/8] + b^(1/8)*Log[b^(1/4) + a^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8] - b^(1/8)*Lo
g[b^(1/4) + a^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8])/(8*a^(9/8))

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fricas [B]  time = 1.01, size = 385, normalized size = 1.42 \[ -\frac {4 \, \sqrt {2} a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a^{8} x \left (-\frac {b}{a^{9}}\right )^{\frac {7}{8}} - \sqrt {2} \sqrt {\sqrt {2} a x \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} + a^{2} \left (-\frac {b}{a^{9}}\right )^{\frac {1}{4}} + x^{2}} a^{8} \left (-\frac {b}{a^{9}}\right )^{\frac {7}{8}} - b}{b}\right ) + 4 \, \sqrt {2} a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a^{8} x \left (-\frac {b}{a^{9}}\right )^{\frac {7}{8}} - \sqrt {2} \sqrt {-\sqrt {2} a x \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} + a^{2} \left (-\frac {b}{a^{9}}\right )^{\frac {1}{4}} + x^{2}} a^{8} \left (-\frac {b}{a^{9}}\right )^{\frac {7}{8}} + b}{b}\right ) + \sqrt {2} a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (\sqrt {2} a x \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} + a^{2} \left (-\frac {b}{a^{9}}\right )^{\frac {1}{4}} + x^{2}\right ) - \sqrt {2} a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (-\sqrt {2} a x \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} + a^{2} \left (-\frac {b}{a^{9}}\right )^{\frac {1}{4}} + x^{2}\right ) + 8 \, a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \arctan \left (-\frac {a^{8} x \left (-\frac {b}{a^{9}}\right )^{\frac {7}{8}} - \sqrt {a^{2} \left (-\frac {b}{a^{9}}\right )^{\frac {1}{4}} + x^{2}} a^{8} \left (-\frac {b}{a^{9}}\right )^{\frac {7}{8}}}{b}\right ) + 2 \, a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} + x\right ) - 2 \, a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (-a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} + x\right ) - 16 \, x}{16 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(4*sqrt(2)*a*(-b/a^9)^(1/8)*arctan(-(sqrt(2)*a^8*x*(-b/a^9)^(7/8) - sqrt(2)*sqrt(sqrt(2)*a*x*(-b/a^9)^(1
/8) + a^2*(-b/a^9)^(1/4) + x^2)*a^8*(-b/a^9)^(7/8) - b)/b) + 4*sqrt(2)*a*(-b/a^9)^(1/8)*arctan(-(sqrt(2)*a^8*x
*(-b/a^9)^(7/8) - sqrt(2)*sqrt(-sqrt(2)*a*x*(-b/a^9)^(1/8) + a^2*(-b/a^9)^(1/4) + x^2)*a^8*(-b/a^9)^(7/8) + b)
/b) + sqrt(2)*a*(-b/a^9)^(1/8)*log(sqrt(2)*a*x*(-b/a^9)^(1/8) + a^2*(-b/a^9)^(1/4) + x^2) - sqrt(2)*a*(-b/a^9)
^(1/8)*log(-sqrt(2)*a*x*(-b/a^9)^(1/8) + a^2*(-b/a^9)^(1/4) + x^2) + 8*a*(-b/a^9)^(1/8)*arctan(-(a^8*x*(-b/a^9
)^(7/8) - sqrt(a^2*(-b/a^9)^(1/4) + x^2)*a^8*(-b/a^9)^(7/8))/b) + 2*a*(-b/a^9)^(1/8)*log(a*(-b/a^9)^(1/8) + x)
- 2*a*(-b/a^9)^(1/8)*log(-a*(-b/a^9)^(1/8) + x) - 16*x)/a

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giac [B]  time = 0.21, size = 442, normalized size = 1.62 \[ \frac {x}{a} - \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {2 \, \sqrt {2} + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.02, size = 34, normalized size = 0.12 \[ \frac {x}{a}-\frac {b \ln \left (-\RootOf \left (a \,\textit {\_Z}^{8}+b \right )+x \right )}{8 a^{2} \RootOf \left (a \,\textit {\_Z}^{8}+b \right )^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*x-1/8/a^2*b*sum(1/_R^7*ln(-_R+x),_R=RootOf(_Z^8*a+b))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\frac {1}{8} \, b {\left (\frac {2 \, \left (\frac {b}{a}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{b \sqrt {-2 \, \sqrt {2} + 4}} + \frac {2 \, \left (\frac {b}{a}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{b \sqrt {-2 \, \sqrt {2} + 4}} + \frac {2 \, \left (\frac {b}{a}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{b \sqrt {2 \, \sqrt {2} + 4}} + \frac {2 \, \left (\frac {b}{a}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}}}\right )}{b \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{b \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{b \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{b \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {b}{a}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} \left (\frac {b}{a}\right )^{\frac {1}{8}} + \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )}{b \sqrt {2 \, \sqrt {2} + 4}}\right )}}{a} + \frac {x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b/x^8),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.21, size = 115, normalized size = 0.42 \[ \frac {x}{a}-\frac {{\left (-b\right )}^{1/8}\,\mathrm {atan}\left (\frac {a^{1/8}\,x}{{\left (-b\right )}^{1/8}}\right )}{4\,a^{9/8}}+\frac {{\left (-b\right )}^{1/8}\,\mathrm {atan}\left (\frac {a^{1/8}\,x\,1{}\mathrm {i}}{{\left (-b\right )}^{1/8}}\right )\,1{}\mathrm {i}}{4\,a^{9/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,a^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-b\right )}^{1/8}}\right )\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{a^{9/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,a^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-b\right )}^{1/8}}\right )\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{a^{9/8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a + b/x^8),x)

[Out]

x/a - ((-b)^(1/8)*atan((a^(1/8)*x)/(-b)^(1/8)))/(4*a^(9/8)) + ((-b)^(1/8)*atan((a^(1/8)*x*1i)/(-b)^(1/8))*1i)/
(4*a^(9/8)) - (2^(1/2)*(-b)^(1/8)*atan((2^(1/2)*a^(1/8)*x*(1/2 - 1i/2))/(-b)^(1/8))*(1/8 + 1i/8))/a^(9/8) - (2
^(1/2)*(-b)^(1/8)*atan((2^(1/2)*a^(1/8)*x*(1/2 + 1i/2))/(-b)^(1/8))*(1/8 - 1i/8))/a^(9/8)

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sympy [A]  time = 0.22, size = 22, normalized size = 0.08 \[ \operatorname {RootSum} {\left (16777216 t^{8} a^{9} + b, \left (t \mapsto t \log {\left (- 8 t a + x \right )} \right )\right )} + \frac {x}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(16777216*_t**8*a**9 + b, Lambda(_t, _t*log(-8*_t*a + x))) + x/a

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