\(\int \frac {1}{a-b x^8} \, dx\) [1470]

Optimal result

Integrand size = 10, antiderivative size = 239 \[ \int \frac {1}{a-b x^8} \, dx=\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt {2} a^{7/8} \sqrt [8]{b}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt {2} a^{7/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac {\log \left (\sqrt [4]{a}-\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} a^{7/8} \sqrt [8]{b}}+\frac {\log \left (\sqrt [4]{a}+\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} a^{7/8} \sqrt [8]{b}} \]

[Out]

1/4*arctan(b^(1/8)*x/a^(1/8))/a^(7/8)/b^(1/8)+1/4*arctanh(b^(1/8)*x/a^(1/8))/a^(7/8)/b^(1/8)+1/8*arctan(-1+b^(
1/8)*x*2^(1/2)/a^(1/8))/a^(7/8)/b^(1/8)*2^(1/2)+1/8*arctan(1+b^(1/8)*x*2^(1/2)/a^(1/8))/a^(7/8)/b^(1/8)*2^(1/2
)-1/16*ln(a^(1/4)+b^(1/4)*x^2-a^(1/8)*b^(1/8)*x*2^(1/2))/a^(7/8)/b^(1/8)*2^(1/2)+1/16*ln(a^(1/4)+b^(1/4)*x^2+a
^(1/8)*b^(1/8)*x*2^(1/2))/a^(7/8)/b^(1/8)*2^(1/2)

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {220, 218, 214, 211, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{a-b x^8} \, dx=\frac {\arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt {2} a^{7/8} \sqrt [8]{b}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}+1\right )}{4 \sqrt {2} a^{7/8} \sqrt [8]{b}}+\frac {\text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} a^{7/8} \sqrt [8]{b}}+\frac {\log \left (\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} a^{7/8} \sqrt [8]{b}} \]

[In]

Int[(a - b*x^8)^(-1),x]

[Out]

ArcTan[(b^(1/8)*x)/a^(1/8)]/(4*a^(7/8)*b^(1/8)) - ArcTan[1 - (Sqrt[2]*b^(1/8)*x)/a^(1/8)]/(4*Sqrt[2]*a^(7/8)*b
^(1/8)) + ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/a^(1/8)]/(4*Sqrt[2]*a^(7/8)*b^(1/8)) + ArcTanh[(b^(1/8)*x)/a^(1/8)]/(
4*a^(7/8)*b^(1/8)) - Log[a^(1/4) - Sqrt[2]*a^(1/8)*b^(1/8)*x + b^(1/4)*x^2]/(8*Sqrt[2]*a^(7/8)*b^(1/8)) + Log[
a^(1/4) + Sqrt[2]*a^(1/8)*b^(1/8)*x + b^(1/4)*x^2]/(8*Sqrt[2]*a^(7/8)*b^(1/8))

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 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}& = \frac {\int \frac {1}{\sqrt {a}-\sqrt {b} x^4} \, dx}{2 \sqrt {a}}+\frac {\int \frac {1}{\sqrt {a}+\sqrt {b} x^4} \, dx}{2 \sqrt {a}} \\ & = \frac {\int \frac {1}{\sqrt [4]{a}-\sqrt [4]{b} x^2} \, dx}{4 a^{3/4}}+\frac {\int \frac {1}{\sqrt [4]{a}+\sqrt [4]{b} x^2} \, dx}{4 a^{3/4}}+\frac {\int \frac {\sqrt [4]{a}-\sqrt [4]{b} x^2}{\sqrt {a}+\sqrt {b} x^4} \, dx}{4 a^{3/4}}+\frac {\int \frac {\sqrt [4]{a}+\sqrt [4]{b} x^2}{\sqrt {a}+\sqrt {b} x^4} \, dx}{4 a^{3/4}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}+\frac {\int \frac {1}{\frac {\sqrt [4]{a}}{\sqrt [4]{b}}-\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 a^{3/4} \sqrt [4]{b}}+\frac {\int \frac {1}{\frac {\sqrt [4]{a}}{\sqrt [4]{b}}+\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{b}}+x^2} \, dx}{8 a^{3/4} \sqrt [4]{b}}-\frac {\int \frac {\frac {\sqrt {2} \sqrt [8]{a}}{\sqrt [8]{b}}+2 x}{-\frac {\sqrt [4]{a}}{\sqrt [4]{b}}-\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{b}}-x^2} \, dx}{8 \sqrt {2} a^{7/8} \sqrt [8]{b}}-\frac {\int \frac {\frac {\sqrt {2} \sqrt [8]{a}}{\sqrt [8]{b}}-2 x}{-\frac {\sqrt [4]{a}}{\sqrt [4]{b}}+\frac {\sqrt {2} \sqrt [8]{a} x}{\sqrt [8]{b}}-x^2} \, dx}{8 \sqrt {2} a^{7/8} \sqrt [8]{b}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac {\log \left (\sqrt [4]{a}-\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} a^{7/8} \sqrt [8]{b}}+\frac {\log \left (\sqrt [4]{a}+\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} a^{7/8} \sqrt [8]{b}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt {2} a^{7/8} \sqrt [8]{b}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt {2} a^{7/8} \sqrt [8]{b}} \\ & = \frac {\tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt {2} a^{7/8} \sqrt [8]{b}}+\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 \sqrt {2} a^{7/8} \sqrt [8]{b}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )}{4 a^{7/8} \sqrt [8]{b}}-\frac {\log \left (\sqrt [4]{a}-\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} a^{7/8} \sqrt [8]{b}}+\frac {\log \left (\sqrt [4]{a}+\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} a^{7/8} \sqrt [8]{b}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.83 \[ \int \frac {1}{a-b x^8} \, dx=\frac {4 \arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{a}}\right )-2 \sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )+2 \sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{a}}\right )-2 \log \left (\sqrt [8]{a}-\sqrt [8]{b} x\right )+2 \log \left (\sqrt [8]{a}+\sqrt [8]{b} x\right )-\sqrt {2} \log \left (\sqrt [4]{a}-\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )+\sqrt {2} \log \left (\sqrt [4]{a}+\sqrt {2} \sqrt [8]{a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{16 a^{7/8} \sqrt [8]{b}} \]

[In]

Integrate[(a - b*x^8)^(-1),x]

[Out]

(4*ArcTan[(b^(1/8)*x)/a^(1/8)] - 2*Sqrt[2]*ArcTan[1 - (Sqrt[2]*b^(1/8)*x)/a^(1/8)] + 2*Sqrt[2]*ArcTan[1 + (Sqr
t[2]*b^(1/8)*x)/a^(1/8)] - 2*Log[a^(1/8) - b^(1/8)*x] + 2*Log[a^(1/8) + b^(1/8)*x] - Sqrt[2]*Log[a^(1/4) - Sqr
t[2]*a^(1/8)*b^(1/8)*x + b^(1/4)*x^2] + Sqrt[2]*Log[a^(1/4) + Sqrt[2]*a^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(16*a^
(7/8)*b^(1/8))

Maple [C] (verified)

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

Time = 5.47 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.12

method	result	size
default	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}-a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 b}\)	\(29\)
risch	\(-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}-a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 b}\)	\(29\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.29 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.97 \[ \int \frac {1}{a-b x^8} \, dx=\left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) + \left (\frac {1}{16} i - \frac {1}{16}\right ) \, \sqrt {2} \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \left (\frac {1}{16} i + \frac {1}{16}\right ) \, \sqrt {2} \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) + \frac {1}{8} \, \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) + \frac {1}{8} i \, \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (i \, a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \frac {1}{8} i \, \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-i \, a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) - \frac {1}{8} \, \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} \log \left (-a \left (\frac {1}{a^{7} b}\right )^{\frac {1}{8}} + x\right ) \]

[In]

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

[Out]

(1/16*I + 1/16)*sqrt(2)*(1/(a^7*b))^(1/8)*log((1/2*I + 1/2)*sqrt(2)*a*(1/(a^7*b))^(1/8) + x) - (1/16*I - 1/16)
*sqrt(2)*(1/(a^7*b))^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*a*(1/(a^7*b))^(1/8) + x) + (1/16*I - 1/16)*sqrt(2)*(1/(a
^7*b))^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a*(1/(a^7*b))^(1/8) + x) - (1/16*I + 1/16)*sqrt(2)*(1/(a^7*b))^(1/8)*lo
g(-(1/2*I + 1/2)*sqrt(2)*a*(1/(a^7*b))^(1/8) + x) + 1/8*(1/(a^7*b))^(1/8)*log(a*(1/(a^7*b))^(1/8) + x) + 1/8*I
*(1/(a^7*b))^(1/8)*log(I*a*(1/(a^7*b))^(1/8) + x) - 1/8*I*(1/(a^7*b))^(1/8)*log(-I*a*(1/(a^7*b))^(1/8) + x) -
1/8*(1/(a^7*b))^(1/8)*log(-a*(1/(a^7*b))^(1/8) + x)

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.09 \[ \int \frac {1}{a-b x^8} \, dx=- \operatorname {RootSum} {\left (16777216 t^{8} a^{7} b - 1, \left ( t \mapsto t \log {\left (- 8 t a + x \right )} \right )\right )} \]

[In]

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

[Out]

-RootSum(16777216*_t**8*a**7*b - 1, Lambda(_t, _t*log(-8*_t*a + x)))

Maxima [F]

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

[In]

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

[Out]

-integrate(1/(b*x^8 - a), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (154) = 308\).

Time = 0.29 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.93 \[ \int \frac {1}{a-b x^8} \, dx=\frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {-\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {-\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x + \sqrt {\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \arctan \left (\frac {2 \, x - \sqrt {\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}{\sqrt {-\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}} + \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {-2 \, \sqrt {2} + 4}} - \frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}} + \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} + x \sqrt {-\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}} + \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (-\frac {a}{b}\right )^{\frac {1}{8}} \log \left (x^{2} - x \sqrt {-\sqrt {2} + 2} \left (-\frac {a}{b}\right )^{\frac {1}{8}} + \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}{8 \, a \sqrt {2 \, \sqrt {2} + 4}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.39 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.39 \[ \int \frac {1}{a-b x^8} \, dx=\frac {\mathrm {atan}\left (\frac {b^{1/8}\,x}{a^{1/8}}\right )}{4\,a^{7/8}\,b^{1/8}}-\frac {\mathrm {atan}\left (\frac {b^{1/8}\,x\,1{}\mathrm {i}}{a^{1/8}}\right )\,1{}\mathrm {i}}{4\,a^{7/8}\,b^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{a^{7/8}\,b^{1/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,b^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{a^{7/8}\,b^{1/8}} \]

[In]

int(1/(a - b*x^8),x)

[Out]

atan((b^(1/8)*x)/a^(1/8))/(4*a^(7/8)*b^(1/8)) - (atan((b^(1/8)*x*1i)/a^(1/8))*1i)/(4*a^(7/8)*b^(1/8)) + (2^(1/
2)*atan((2^(1/2)*b^(1/8)*x*(1/2 - 1i/2))/a^(1/8))*(1/8 + 1i/8))/(a^(7/8)*b^(1/8)) + (2^(1/2)*atan((2^(1/2)*b^(
1/8)*x*(1/2 + 1i/2))/a^(1/8))*(1/8 - 1i/8))/(a^(7/8)*b^(1/8))