\(\int \frac {1}{a+\frac {b}{x^8}} \, dx\) [2110]

Optimal result

Integrand size = 9, antiderivative size = 272 \[ \int \frac {1}{a+\frac {b}{x^8}} \, dx=\frac {x}{a}+\frac {\sqrt [8]{b} \arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \log \left (\sqrt [4]{b}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2\right )}{8 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \log \left (\sqrt [4]{b}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2\right )}{8 \sqrt {2} (-a)^{9/8}} \]

[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)

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.333, Rules used = {199, 327, 220, 218, 214, 211, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{a+\frac {b}{x^8}} \, dx=\frac {\sqrt [8]{b} \arctan \left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \arctan \left (\frac {\sqrt {2} \sqrt [8]{-a} x}{\sqrt [8]{b}}+1\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \text {arctanh}\left (\frac {\sqrt [8]{-a} x}{\sqrt [8]{b}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{8 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a} x^2+\sqrt [4]{b}\right )}{8 \sqrt {2} (-a)^{9/8}}+\frac {x}{a} \]

[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 199

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 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 211

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

Rule 214

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

Rule 217

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 218

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] &&  !Gt
Q[a/b, 0]

Rule 220

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 327

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 631

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 642

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 1176

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 1179

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*} \text {integral}& = \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} \text {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} \text {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*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.35 \[ \int \frac {1}{a+\frac {b}{x^8}} \, dx=\frac {8 \sqrt [8]{a} x-2 \sqrt [8]{b} \arctan \left (\frac {\sqrt [8]{a} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}-\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )-2 \sqrt [8]{b} \arctan \left (\frac {\sqrt [8]{a} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}+\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )+\sqrt [8]{b} \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{b}+\sqrt [4]{a} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-\sqrt [8]{b} \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{b}+\sqrt [4]{a} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )+2 \sqrt [8]{b} \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{a} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}\right ) \sin \left (\frac {\pi }{8}\right )-2 \sqrt [8]{b} \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{a} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{b}}\right ) \sin \left (\frac {\pi }{8}\right )+\sqrt [8]{b} \log \left (\sqrt [4]{b}+\sqrt [4]{a} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\sqrt [8]{b} \log \left (\sqrt [4]{b}+\sqrt [4]{a} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )}{8 a^{9/8}} \]

[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))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.12

method	result	size
default	\(\frac {x}{a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{8 a^{2}}\)	\(34\)
risch	\(\frac {x}{a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{8}+b \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}\right )}{8 a^{2}}\)	\(34\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.85 \[ \int \frac {1}{a+\frac {b}{x^8}} \, dx=-\frac {\left (i + 1\right ) \, \sqrt {2} a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} + x\right ) - \left (i - 1\right ) \, \sqrt {2} a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} + x\right ) + \left (i - 1\right ) \, \sqrt {2} a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} + x\right ) - \left (i + 1\right ) \, \sqrt {2} a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} + x\right ) + 2 \, a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} + x\right ) + 2 i \, a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (i \, a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} + x\right ) - 2 i \, a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (-i \, a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} + x\right ) - 2 \, a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (-a \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} + x\right ) - 16 \, x}{16 \, a} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.08 \[ \int \frac {1}{a+\frac {b}{x^8}} \, dx=\operatorname {RootSum} {\left (16777216 t^{8} a^{9} + b, \left ( t \mapsto t \log {\left (- 8 t a + x \right )} \right )\right )} + \frac {x}{a} \]

[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

Maxima [F]

\[ \int \frac {1}{a+\frac {b}{x^8}} \, dx=\int { \frac {1}{a + \frac {b}{x^{8}}} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 442 vs. \(2 (187) = 374\).

Time = 0.29 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.62 \[ \int \frac {1}{a+\frac {b}{x^8}} \, dx=\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}} \]

[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))

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.42 \[ \int \frac {1}{a+\frac {b}{x^8}} \, dx=\frac {x}{a}-\frac {{\left (-b\right )}^{1/8}\,\mathrm {atan}\left (\frac {a^{1/8}\,x}{{\left (-b\right )}^{1/8}}\right )}{4\,a^{9/8}}+\frac {{\left (-b\right )}^{1/8}\,\mathrm {atan}\left (\frac {a^{1/8}\,x\,1{}\mathrm {i}}{{\left (-b\right )}^{1/8}}\right )\,1{}\mathrm {i}}{4\,a^{9/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,a^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-b\right )}^{1/8}}\right )\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{a^{9/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,a^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-b\right )}^{1/8}}\right )\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{a^{9/8}} \]

[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)