\(\int \frac {1}{x^4 (a+b x^8)} \, dx\) [1467]

Optimal result

Integrand size = 13, antiderivative size = 277 \[ \int \frac {1}{x^4 \left (a+b x^8\right )} \, dx=-\frac {1}{3 a x^3}-\frac {b^{3/8} \arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac {b^{3/8} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{11/8}}+\frac {b^{3/8} \arctan \left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{11/8}}-\frac {b^{3/8} \text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac {b^{3/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{11/8}}+\frac {b^{3/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{11/8}} \]

[Out]

-1/3/a/x^3-1/4*b^(3/8)*arctan(b^(1/8)*x/(-a)^(1/8))/(-a)^(11/8)-1/4*b^(3/8)*arctanh(b^(1/8)*x/(-a)^(1/8))/(-a)
^(11/8)+1/8*b^(3/8)*arctan(-1+b^(1/8)*x*2^(1/2)/(-a)^(1/8))/(-a)^(11/8)*2^(1/2)+1/8*b^(3/8)*arctan(1+b^(1/8)*x
*2^(1/2)/(-a)^(1/8))/(-a)^(11/8)*2^(1/2)-1/16*b^(3/8)*ln((-a)^(1/4)+b^(1/4)*x^2-(-a)^(1/8)*b^(1/8)*x*2^(1/2))/
(-a)^(11/8)*2^(1/2)+1/16*b^(3/8)*ln((-a)^(1/4)+b^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*2^(1/2))/(-a)^(11/8)*2^(1/2)

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 277, 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, 217, 1179, 642, 1176, 631, 210, 218, 214, 211} \[ \int \frac {1}{x^4 \left (a+b x^8\right )} \, dx=-\frac {b^{3/8} \arctan \left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac {b^{3/8} \arctan \left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{11/8}}+\frac {b^{3/8} \arctan \left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt {2} (-a)^{11/8}}-\frac {b^{3/8} \text {arctanh}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac {b^{3/8} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{11/8}}+\frac {b^{3/8} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{11/8}}-\frac {1}{3 a x^3} \]

[In]

Int[1/(x^4*(a + b*x^8)),x]

[Out]

-1/3*1/(a*x^3) - (b^(3/8)*ArcTan[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(11/8)) - (b^(3/8)*ArcTan[1 - (Sqrt[2]*b^(1/
8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(11/8)) + (b^(3/8)*ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(
-a)^(11/8)) - (b^(3/8)*ArcTanh[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(11/8)) - (b^(3/8)*Log[(-a)^(1/4) - Sqrt[2]*(-
a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(11/8)) + (b^(3/8)*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1
/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(11/8))

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 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}{3 a x^3}-\frac {b \int \frac {x^4}{a+b x^8} \, dx}{a} \\ & = -\frac {1}{3 a x^3}+\frac {\sqrt {b} \int \frac {1}{\sqrt {-a}-\sqrt {b} x^4} \, dx}{2 a}-\frac {\sqrt {b} \int \frac {1}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{2 a} \\ & = -\frac {1}{3 a x^3}-\frac {\sqrt {b} \int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2} \, dx}{4 (-a)^{5/4}}-\frac {\sqrt {b} \int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{b} x^2} \, dx}{4 (-a)^{5/4}}+\frac {\sqrt {b} \int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{4 (-a)^{5/4}}+\frac {\sqrt {b} \int \frac {\sqrt [4]{-a}+\sqrt [4]{b} x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{4 (-a)^{5/4}} \\ & = -\frac {1}{3 a x^3}-\frac {b^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac {b^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}+\frac {\sqrt [4]{b} \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)^{5/4}}+\frac {\sqrt [4]{b} \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)^{5/4}}-\frac {b^{3/8} \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)^{11/8}}-\frac {b^{3/8} \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)^{11/8}} \\ & = -\frac {1}{3 a x^3}-\frac {b^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac {b^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac {b^{3/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{11/8}}+\frac {b^{3/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{11/8}}+\frac {b^{3/8} \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)^{11/8}}-\frac {b^{3/8} \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)^{11/8}} \\ & = -\frac {1}{3 a x^3}-\frac {b^{3/8} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac {b^{3/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{11/8}}+\frac {b^{3/8} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{11/8}}-\frac {b^{3/8} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{11/8}}-\frac {b^{3/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{11/8}}+\frac {b^{3/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{11/8}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.43 \[ \int \frac {1}{x^4 \left (a+b x^8\right )} \, dx=\frac {-8 a^{3/8}+6 b^{3/8} x^3 \arctan \left (\cot \left (\frac {\pi }{8}\right )-\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \cos \left (\frac {\pi }{8}\right )-6 b^{3/8} x^3 \arctan \left (\cot \left (\frac {\pi }{8}\right )+\frac {\sqrt [8]{b} x \csc \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}\right ) \cos \left (\frac {\pi }{8}\right )+3 b^{3/8} x^3 \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right )-3 b^{3/8} x^3 \cos \left (\frac {\pi }{8}\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac {\pi }{8}\right )\right )+6 b^{3/8} x^3 \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+6 b^{3/8} x^3 \arctan \left (\frac {\sqrt [8]{b} x \sec \left (\frac {\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )-3 b^{3/8} x^3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )+3 b^{3/8} x^3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x^2+2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac {\pi }{8}\right )\right ) \sin \left (\frac {\pi }{8}\right )}{24 a^{11/8} x^3} \]

[In]

Integrate[1/(x^4*(a + b*x^8)),x]

[Out]

(-8*a^(3/8) + 6*b^(3/8)*x^3*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Cos[Pi/8] - 6*b^(3/8)*x^3*ArcTan
[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Cos[Pi/8] + 3*b^(3/8)*x^3*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 - 2*
a^(1/8)*b^(1/8)*x*Sin[Pi/8]] - 3*b^(3/8)*x^3*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/
8]] + 6*b^(3/8)*x^3*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Sin[Pi/8] + 6*b^(3/8)*x^3*ArcTan[(b^(1/8
)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Sin[Pi/8] - 3*b^(3/8)*x^3*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*
Cos[Pi/8]]*Sin[Pi/8] + 3*b^(3/8)*x^3*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]]*Sin[Pi/8])/(24
*a^(11/8)*x^3)

Maple [C] (verified)

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

Time = 5.33 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.13

method	result	size
default	\(-\frac {1}{3 a \,x^{3}}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{8 a}\)	\(36\)
risch	\(-\frac {1}{3 a \,x^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{11} \textit {\_Z}^{8}+b^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-9 \textit {\_R}^{8} a^{11}-8 b^{3}\right ) x +a^{7} b \,\textit {\_R}^{5}\right )\right )}{8}\)	\(55\)

[In]

int(1/x^4/(b*x^8+a),x,method=_RETURNVERBOSE)

[Out]

-1/3/a/x^3-1/8/a*sum(1/_R^3*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 = 337, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x^4 \left (a+b x^8\right )} \, dx=-\frac {\left (3 i + 3\right ) \, \sqrt {2} a x^{3} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{7} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {5}{8}} + b^{2} x\right ) - \left (3 i - 3\right ) \, \sqrt {2} a x^{3} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{7} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {5}{8}} + b^{2} x\right ) + \left (3 i - 3\right ) \, \sqrt {2} a x^{3} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{7} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {5}{8}} + b^{2} x\right ) - \left (3 i + 3\right ) \, \sqrt {2} a x^{3} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{7} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {5}{8}} + b^{2} x\right ) - 6 \, a x^{3} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (a^{7} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {5}{8}} + b^{2} x\right ) - 6 i \, a x^{3} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (i \, a^{7} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {5}{8}} + b^{2} x\right ) + 6 i \, a x^{3} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (-i \, a^{7} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {5}{8}} + b^{2} x\right ) + 6 \, a x^{3} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {1}{8}} \log \left (-a^{7} \left (-\frac {b^{3}}{a^{11}}\right )^{\frac {5}{8}} + b^{2} x\right ) + 16}{48 \, a x^{3}} \]

[In]

integrate(1/x^4/(b*x^8+a),x, algorithm="fricas")

[Out]

-1/48*((3*I + 3)*sqrt(2)*a*x^3*(-b^3/a^11)^(1/8)*log((1/2*I + 1/2)*sqrt(2)*a^7*(-b^3/a^11)^(5/8) + b^2*x) - (3
*I - 3)*sqrt(2)*a*x^3*(-b^3/a^11)^(1/8)*log(-(1/2*I - 1/2)*sqrt(2)*a^7*(-b^3/a^11)^(5/8) + b^2*x) + (3*I - 3)*
sqrt(2)*a*x^3*(-b^3/a^11)^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a^7*(-b^3/a^11)^(5/8) + b^2*x) - (3*I + 3)*sqrt(2)*a
*x^3*(-b^3/a^11)^(1/8)*log(-(1/2*I + 1/2)*sqrt(2)*a^7*(-b^3/a^11)^(5/8) + b^2*x) - 6*a*x^3*(-b^3/a^11)^(1/8)*l
og(a^7*(-b^3/a^11)^(5/8) + b^2*x) - 6*I*a*x^3*(-b^3/a^11)^(1/8)*log(I*a^7*(-b^3/a^11)^(5/8) + b^2*x) + 6*I*a*x
^3*(-b^3/a^11)^(1/8)*log(-I*a^7*(-b^3/a^11)^(5/8) + b^2*x) + 6*a*x^3*(-b^3/a^11)^(1/8)*log(-a^7*(-b^3/a^11)^(5
/8) + b^2*x) + 16)/(a*x^3)

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.13 \[ \int \frac {1}{x^4 \left (a+b x^8\right )} \, dx=\operatorname {RootSum} {\left (16777216 t^{8} a^{11} + b^{3}, \left ( t \mapsto t \log {\left (\frac {32768 t^{5} a^{7}}{b^{2}} + x \right )} \right )\right )} - \frac {1}{3 a x^{3}} \]

[In]

integrate(1/x**4/(b*x**8+a),x)

[Out]

RootSum(16777216*_t**8*a**11 + b**3, Lambda(_t, _t*log(32768*_t**5*a**7/b**2 + x))) - 1/(3*a*x**3)

Maxima [F]

\[ \int \frac {1}{x^4 \left (a+b x^8\right )} \, dx=\int { \frac {1}{{\left (b x^{8} + a\right )} x^{4}} \,d x } \]

[In]

integrate(1/x^4/(b*x^8+a),x, algorithm="maxima")

[Out]

-b*integrate(x^4/(b*x^8 + a), x)/a - 1/3/(a*x^3)

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 = 1.64 \[ \int \frac {1}{x^4 \left (a+b x^8\right )} \, dx=\frac {b \left (\frac {a}{b}\right )^{\frac {5}{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 {5}{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 {5}{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 {5}{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 {5}{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 {5}{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 {5}{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 {5}{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}{3 \, a x^{3}} \]

[In]

integrate(1/x^4/(b*x^8+a),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.38 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.43 \[ \int \frac {1}{x^4 \left (a+b x^8\right )} \, dx=\frac {{\left (-b\right )}^{3/8}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/8}\,x}{a^{1/8}}\right )}{4\,a^{11/8}}-\frac {1}{3\,a\,x^3}-\frac {{\left (-b\right )}^{3/8}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/8}\,x\,1{}\mathrm {i}}{a^{1/8}}\right )\,1{}\mathrm {i}}{4\,a^{11/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{3/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,x\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {1}{8}-\frac {1}{8}{}\mathrm {i}\right )}{a^{11/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{3/8}\,\mathrm {atan}\left (\frac {\sqrt {2}\,{\left (-b\right )}^{1/8}\,x\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{a^{1/8}}\right )\,\left (-\frac {1}{8}+\frac {1}{8}{}\mathrm {i}\right )}{a^{11/8}} \]

[In]

int(1/(x^4*(a + b*x^8)),x)

[Out]

((-b)^(3/8)*atan(((-b)^(1/8)*x)/a^(1/8)))/(4*a^(11/8)) - 1/(3*a*x^3) - ((-b)^(3/8)*atan(((-b)^(1/8)*x*1i)/a^(1
/8))*1i)/(4*a^(11/8)) - (2^(1/2)*(-b)^(3/8)*atan((2^(1/2)*(-b)^(1/8)*x*(1/2 - 1i/2))/a^(1/8))*(1/8 + 1i/8))/a^
(11/8) - (2^(1/2)*(-b)^(3/8)*atan((2^(1/2)*(-b)^(1/8)*x*(1/2 + 1i/2))/a^(1/8))*(1/8 - 1i/8))/a^(11/8)