\(\int \frac {1}{x^2 (a+b x^8)} \, dx\) [1466]

Optimal result

Integrand size = 13, antiderivative size = 275 \[ \int \frac {1}{x^2 \left (a+b x^8\right )} \, dx=-\frac {1}{a x}+\frac {\sqrt [8]{b} \arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{b} \text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}+\frac {\sqrt [8]{b} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{b} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{9/8}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {331, 307, 303, 1176, 631, 210, 1179, 642, 304, 211, 214} \[ \int \frac {1}{x^2 \left (a+b x^8\right )} \, dx=\frac {\sqrt [8]{b} \arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{b} \text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}+\frac {\sqrt [8]{b} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{b} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{9/8}}-\frac {1}{a x} \]

[In]

Int[1/(x^2*(a + b*x^8)),x]

[Out]

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

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 303

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 304

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

Rule 307

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

Rule 331

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 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 {1}{a x}-\frac {b \int \frac {x^6}{a+b x^8} \, dx}{a} \\ & = -\frac {1}{a x}+\frac {\sqrt {b} \int \frac {x^2}{\sqrt {-a}-\sqrt {b} x^4} \, dx}{2 a}-\frac {\sqrt {b} \int \frac {x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{2 a} \\ & = -\frac {1}{a x}+\frac {\sqrt [4]{b} \int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2} \, dx}{4 a}-\frac {\sqrt [4]{b} \int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{b} x^2} \, dx}{4 a}+\frac {\sqrt [4]{b} \int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{4 a}-\frac {\sqrt [4]{b} \int \frac {\sqrt [4]{-a}+\sqrt [4]{b} x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{4 a} \\ & = -\frac {1}{a x}+\frac {\sqrt [8]{b} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\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}-\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}+\frac {\sqrt [8]{b} \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)^{9/8}}+\frac {\sqrt [8]{b} \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)^{9/8}} \\ & = -\frac {1}{a x}+\frac {\sqrt [8]{b} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}+\frac {\sqrt [8]{b} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{b} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} 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]{b} x}{\sqrt [8]{-a}}\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]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{9/8}} \\ & = -\frac {1}{a x}+\frac {\sqrt [8]{b} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}-\frac {\sqrt [8]{b} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{9/8}}+\frac {\sqrt [8]{b} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{b} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{9/8}}+\frac {\sqrt [8]{b} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{9/8}}-\frac {\sqrt [8]{b} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{9/8}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x^2 \left (a+b x^8\right )} \, dx=-\frac {8 \sqrt [8]{a}+2 \sqrt [8]{b} x \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )+2 \sqrt [8]{b} x \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right ) \cos \left (\frac {\pi }{8}\right )+\sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-\sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right )-2 \sqrt [8]{b} x \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \sin \left (\frac {\pi }{8}\right )+2 \sqrt [8]{b} x \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \sin \left (\frac {\pi }{8}\right )+\sqrt [8]{b} x \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-\sqrt [8]{b} x \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )}{8 a^{9/8} x} \]

[In]

Integrate[1/(x^2*(a + b*x^8)),x]

[Out]

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

Maple [C] (verified)

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

Time = 11.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.13

method	result	size
default	\(-\frac {1}{a x}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}}}{8 a}\)	\(36\)
risch	\(-\frac {1}{a x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{9} \textit {\_Z}^{8}+b \right )}{\sum }\textit {\_R} \ln \left (\left (9 \textit {\_R}^{8} a^{9}+8 b \right ) x +a^{8} \textit {\_R}^{7}\right )\right )}{8}\)	\(50\)

[In]

int(1/x^2/(b*x^8+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.99 \[ \int \frac {1}{x^2 \left (a+b x^8\right )} \, dx=-\frac {-\left (i - 1\right ) \, \sqrt {2} a x \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{8} \left (-\frac {b}{a^{9}}\right )^{\frac {7}{8}} + b x\right ) + \left (i + 1\right ) \, \sqrt {2} a x \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{8} \left (-\frac {b}{a^{9}}\right )^{\frac {7}{8}} + b x\right ) - \left (i + 1\right ) \, \sqrt {2} a x \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{8} \left (-\frac {b}{a^{9}}\right )^{\frac {7}{8}} + b x\right ) + \left (i - 1\right ) \, \sqrt {2} a x \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{8} \left (-\frac {b}{a^{9}}\right )^{\frac {7}{8}} + b x\right ) + 2 \, a x \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (a^{8} \left (-\frac {b}{a^{9}}\right )^{\frac {7}{8}} + b x\right ) - 2 i \, a x \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (i \, a^{8} \left (-\frac {b}{a^{9}}\right )^{\frac {7}{8}} + b x\right ) + 2 i \, a x \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (-i \, a^{8} \left (-\frac {b}{a^{9}}\right )^{\frac {7}{8}} + b x\right ) - 2 \, a x \left (-\frac {b}{a^{9}}\right )^{\frac {1}{8}} \log \left (-a^{8} \left (-\frac {b}{a^{9}}\right )^{\frac {7}{8}} + b x\right ) + 16}{16 \, a x} \]

[In]

integrate(1/x^2/(b*x^8+a),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.11 \[ \int \frac {1}{x^2 \left (a+b x^8\right )} \, dx=\operatorname {RootSum} {\left (16777216 t^{8} a^{9} + b, \left ( t \mapsto t \log {\left (- \frac {2097152 t^{7} a^{8}}{b} + x \right )} \right )\right )} - \frac {1}{a x} \]

[In]

integrate(1/x**2/(b*x**8+a),x)

[Out]

RootSum(16777216*_t**8*a**9 + b, Lambda(_t, _t*log(-2097152*_t**7*a**8/b + x))) - 1/(a*x)

Maxima [F]

\[ \int \frac {1}{x^2 \left (a+b x^8\right )} \, dx=\int { \frac {1}{{\left (b x^{8} + a\right )} x^{2}} \,d x } \]

[In]

integrate(1/x^2/(b*x^8+a),x, algorithm="maxima")

[Out]

-b*integrate(x^6/(b*x^8 + a), x)/a - 1/(a*x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 453 vs. \(2 (190) = 380\).

Time = 0.37 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.65 \[ \int \frac {1}{x^2 \left (a+b x^8\right )} \, dx=-\frac {b \left (\frac {a}{b}\right )^{\frac {7}{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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {b \left (\frac {a}{b}\right )^{\frac {7}{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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {b \left (\frac {a}{b}\right )^{\frac {7}{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^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {b \left (\frac {a}{b}\right )^{\frac {7}{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^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {b \left (\frac {a}{b}\right )^{\frac {7}{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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {b \left (\frac {a}{b}\right )^{\frac {7}{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^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {b \left (\frac {a}{b}\right )^{\frac {7}{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^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {b \left (\frac {a}{b}\right )^{\frac {7}{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^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {1}{a x} \]

[In]

integrate(1/x^2/(b*x^8+a),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.43 \[ \int \frac {1}{x^2 \left (a+b x^8\right )} \, dx=-\frac {1}{a\,x}-\frac {{\left (-b\right )}^{1/8}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/8}\,x}{a^{1/8}}\right )}{4\,a^{9/8}}-\frac {{\left (-b\right )}^{1/8}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/8}\,x\,1{}\mathrm {i}}{a^{1/8}}\right )\,1{}\mathrm {i}}{4\,a^{9/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{a^{9/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{a^{9/8}} \]

[In]

int(1/(x^2*(a + b*x^8)),x)

[Out]

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