3.15.69 \(\int \frac {1}{x^8 (a+b x^8)} \, dx\) [1469]

Optimal. Leaf size=277 \[ -\frac {1}{7 a x^7}-\frac {b^{7/8} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}+\frac {b^{7/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{15/8}}-\frac {b^{7/8} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{15/8}}-\frac {b^{7/8} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}+\frac {b^{7/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{15/8}}-\frac {b^{7/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{15/8}} \]

[Out]

-1/7/a/x^7-1/4*b^(7/8)*arctan(b^(1/8)*x/(-a)^(1/8))/(-a)^(15/8)-1/4*b^(7/8)*arctanh(b^(1/8)*x/(-a)^(1/8))/(-a)
^(15/8)-1/8*b^(7/8)*arctan(-1+b^(1/8)*x*2^(1/2)/(-a)^(1/8))/(-a)^(15/8)*2^(1/2)-1/8*b^(7/8)*arctan(1+b^(1/8)*x
*2^(1/2)/(-a)^(1/8))/(-a)^(15/8)*2^(1/2)+1/16*b^(7/8)*ln((-a)^(1/4)+b^(1/4)*x^2-(-a)^(1/8)*b^(1/8)*x*2^(1/2))/
(-a)^(15/8)*2^(1/2)-1/16*b^(7/8)*ln((-a)^(1/4)+b^(1/4)*x^2+(-a)^(1/8)*b^(1/8)*x*2^(1/2))/(-a)^(15/8)*2^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.16, antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.846, Rules used = {331, 220, 218, 214, 211, 217, 1179, 642, 1176, 631, 210} \begin {gather*} -\frac {b^{7/8} \text {ArcTan}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}+\frac {b^{7/8} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{15/8}}-\frac {b^{7/8} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}+1\right )}{4 \sqrt {2} (-a)^{15/8}}+\frac {b^{7/8} \log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{15/8}}-\frac {b^{7/8} \log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{-a}+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{15/8}}-\frac {b^{7/8} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}-\frac {1}{7 a x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^8*(a + b*x^8)),x]

[Out]

-1/7*1/(a*x^7) - (b^(7/8)*ArcTan[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(15/8)) + (b^(7/8)*ArcTan[1 - (Sqrt[2]*b^(1/
8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(-a)^(15/8)) - (b^(7/8)*ArcTan[1 + (Sqrt[2]*b^(1/8)*x)/(-a)^(1/8)])/(4*Sqrt[2]*(
-a)^(15/8)) - (b^(7/8)*ArcTanh[(b^(1/8)*x)/(-a)^(1/8)])/(4*(-a)^(15/8)) + (b^(7/8)*Log[(-a)^(1/4) - Sqrt[2]*(-
a)^(1/8)*b^(1/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(15/8)) - (b^(7/8)*Log[(-a)^(1/4) + Sqrt[2]*(-a)^(1/8)*b^(1
/8)*x + b^(1/4)*x^2])/(8*Sqrt[2]*(-a)^(15/8))

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 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*} \int \frac {1}{x^8 \left (a+b x^8\right )} \, dx &=-\frac {1}{7 a x^7}-\frac {b \int \frac {1}{a+b x^8} \, dx}{a}\\ &=-\frac {1}{7 a x^7}-\frac {b \int \frac {1}{\sqrt {-a}-\sqrt {b} x^4} \, dx}{2 (-a)^{3/2}}-\frac {b \int \frac {1}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{2 (-a)^{3/2}}\\ &=-\frac {1}{7 a x^7}-\frac {b \int \frac {1}{\sqrt [4]{-a}-\sqrt [4]{b} x^2} \, dx}{4 (-a)^{7/4}}-\frac {b \int \frac {1}{\sqrt [4]{-a}+\sqrt [4]{b} x^2} \, dx}{4 (-a)^{7/4}}-\frac {b \int \frac {\sqrt [4]{-a}-\sqrt [4]{b} x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{4 (-a)^{7/4}}-\frac {b \int \frac {\sqrt [4]{-a}+\sqrt [4]{b} x^2}{\sqrt {-a}+\sqrt {b} x^4} \, dx}{4 (-a)^{7/4}}\\ &=-\frac {1}{7 a x^7}-\frac {b^{7/8} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}-\frac {b^{7/8} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}-\frac {b^{3/4} \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)^{7/4}}-\frac {b^{3/4} \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)^{7/4}}+\frac {b^{7/8} \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)^{15/8}}+\frac {b^{7/8} \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)^{15/8}}\\ &=-\frac {1}{7 a x^7}-\frac {b^{7/8} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}-\frac {b^{7/8} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}+\frac {b^{7/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{15/8}}-\frac {b^{7/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{15/8}}-\frac {b^{7/8} \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)^{15/8}}+\frac {b^{7/8} \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)^{15/8}}\\ &=-\frac {1}{7 a x^7}-\frac {b^{7/8} \tan ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}+\frac {b^{7/8} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{15/8}}-\frac {b^{7/8} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 \sqrt {2} (-a)^{15/8}}-\frac {b^{7/8} \tanh ^{-1}\left (\frac {\sqrt [8]{b} x}{\sqrt [8]{-a}}\right )}{4 (-a)^{15/8}}+\frac {b^{7/8} \log \left (\sqrt [4]{-a}-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{15/8}}-\frac {b^{7/8} \log \left (\sqrt [4]{-a}+\sqrt {2} \sqrt [8]{-a} \sqrt [8]{b} x+\sqrt [4]{b} x^2\right )}{8 \sqrt {2} (-a)^{15/8}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.11, size = 395, normalized size = 1.43 \begin {gather*} -\frac {8 a^{7/8}+14 b^{7/8} x^7 \tan ^{-1}\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 )+14 b^{7/8} x^7 \tan ^{-1}\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 )-7 b^{7/8} x^7 \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 )+7 b^{7/8} x^7 \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 )-14 b^{7/8} x^7 \tan ^{-1}\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 )+14 b^{7/8} x^7 \tan ^{-1}\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 )-7 b^{7/8} x^7 \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 )+7 b^{7/8} x^7 \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 )}{56 a^{15/8} x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^8*(a + b*x^8)),x]

[Out]

-1/56*(8*a^(7/8) + 14*b^(7/8)*x^7*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]]*Cos[Pi/8] + 14*b^(7/8)*x^7
*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]]*Cos[Pi/8] - 7*b^(7/8)*x^7*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x
^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] + 7*b^(7/8)*x^7*Cos[Pi/8]*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*
Cos[Pi/8]] - 14*b^(7/8)*x^7*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] + 14*b^(7/8)*x^7*ArcTa
n[Cot[Pi/8] + (b^(1/8)*x*Csc[Pi/8])/a^(1/8)]*Sin[Pi/8] - 7*b^(7/8)*x^7*Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b
^(1/8)*x*Sin[Pi/8]]*Sin[Pi/8] + 7*b^(7/8)*x^7*Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]]*Sin[P
i/8])/(a^(15/8)*x^7)

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.19, size = 36, normalized size = 0.13

method	result	size
default	\(-\frac {1}{7 a \,x^{7}}-\frac {\munderset {\textit {\_R} =\RootOf \left (b \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{7}}}{8 a}\)	\(36\)
risch	\(-\frac {1}{7 a \,x^{7}}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (a^{15} \textit {\_Z}^{8}+b^{7}\right )}{\sum }\textit {\_R} \ln \left (\left (-9 \textit {\_R}^{8} a^{15}-8 b^{7}\right ) x -a^{2} b^{6} \textit {\_R} \right )\right )}{8}\)	\(56\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*integrate(1/(b*x^8 + a), x)/a - 1/7/(a*x^7)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 503 vs. \(2 (190) = 380\).
time = 0.39, size = 503, normalized size = 1.82 \begin {gather*} -\frac {28 \, \sqrt {2} a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a^{13} b x \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {7}{8}} - \sqrt {2} \sqrt {\sqrt {2} a^{2} b x \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} + a^{4} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{4}} + b^{2} x^{2}} a^{13} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {7}{8}} - b^{7}}{b^{7}}\right ) + 28 \, \sqrt {2} a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \arctan \left (-\frac {\sqrt {2} a^{13} b x \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {7}{8}} - \sqrt {2} \sqrt {-\sqrt {2} a^{2} b x \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} + a^{4} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{4}} + b^{2} x^{2}} a^{13} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {7}{8}} + b^{7}}{b^{7}}\right ) + 7 \, \sqrt {2} a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \log \left (\sqrt {2} a^{2} b x \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} + a^{4} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{4}} + b^{2} x^{2}\right ) - 7 \, \sqrt {2} a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \log \left (-\sqrt {2} a^{2} b x \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} + a^{4} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{4}} + b^{2} x^{2}\right ) + 56 \, a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \arctan \left (-\frac {a^{13} b x \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {7}{8}} - \sqrt {a^{4} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{4}} + b^{2} x^{2}} a^{13} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {7}{8}}}{b^{7}}\right ) + 14 \, a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \log \left (a^{2} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} + b x\right ) - 14 \, a x^{7} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} \log \left (-a^{2} \left (-\frac {b^{7}}{a^{15}}\right )^{\frac {1}{8}} + b x\right ) + 16}{112 \, a x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/112*(28*sqrt(2)*a*x^7*(-b^7/a^15)^(1/8)*arctan(-(sqrt(2)*a^13*b*x*(-b^7/a^15)^(7/8) - sqrt(2)*sqrt(sqrt(2)*
a^2*b*x*(-b^7/a^15)^(1/8) + a^4*(-b^7/a^15)^(1/4) + b^2*x^2)*a^13*(-b^7/a^15)^(7/8) - b^7)/b^7) + 28*sqrt(2)*a
*x^7*(-b^7/a^15)^(1/8)*arctan(-(sqrt(2)*a^13*b*x*(-b^7/a^15)^(7/8) - sqrt(2)*sqrt(-sqrt(2)*a^2*b*x*(-b^7/a^15)
^(1/8) + a^4*(-b^7/a^15)^(1/4) + b^2*x^2)*a^13*(-b^7/a^15)^(7/8) + b^7)/b^7) + 7*sqrt(2)*a*x^7*(-b^7/a^15)^(1/
8)*log(sqrt(2)*a^2*b*x*(-b^7/a^15)^(1/8) + a^4*(-b^7/a^15)^(1/4) + b^2*x^2) - 7*sqrt(2)*a*x^7*(-b^7/a^15)^(1/8
)*log(-sqrt(2)*a^2*b*x*(-b^7/a^15)^(1/8) + a^4*(-b^7/a^15)^(1/4) + b^2*x^2) + 56*a*x^7*(-b^7/a^15)^(1/8)*arcta
n(-(a^13*b*x*(-b^7/a^15)^(7/8) - sqrt(a^4*(-b^7/a^15)^(1/4) + b^2*x^2)*a^13*(-b^7/a^15)^(7/8))/b^7) + 14*a*x^7
*(-b^7/a^15)^(1/8)*log(a^2*(-b^7/a^15)^(1/8) + b*x) - 14*a*x^7*(-b^7/a^15)^(1/8)*log(-a^2*(-b^7/a^15)^(1/8) +
b*x) + 16)/(a*x^7)

________________________________________________________________________________________

Sympy [A]
time = 0.15, size = 32, normalized size = 0.12 \begin {gather*} \operatorname {RootSum} {\left (16777216 t^{8} a^{15} + b^{7}, \left ( t \mapsto t \log {\left (- \frac {8 t a^{2}}{b} + x \right )} \right )\right )} - \frac {1}{7 a x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

RootSum(16777216*_t**8*a**15 + b**7, Lambda(_t, _t*log(-8*_t*a**2/b + x))) - 1/(7*a*x**7)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 453 vs. \(2 (190) = 380\).
time = 1.32, size = 453, normalized size = 1.64 \begin {gather*} -\frac {b \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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {b \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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {b \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^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {b \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^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {b \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^{2} \sqrt {-2 \, \sqrt {2} + 4}} + \frac {b \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^{2} \sqrt {-2 \, \sqrt {2} + 4}} - \frac {b \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^{2} \sqrt {2 \, \sqrt {2} + 4}} + \frac {b \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^{2} \sqrt {2 \, \sqrt {2} + 4}} - \frac {1}{7 \, a x^{7}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 1.14, size = 118, normalized size = 0.43 \begin {gather*} \frac {{\left (-b\right )}^{7/8}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/8}\,x}{a^{1/8}}\right )}{4\,a^{15/8}}-\frac {1}{7\,a\,x^7}-\frac {{\left (-b\right )}^{7/8}\,\mathrm {atan}\left (\frac {{\left (-b\right )}^{1/8}\,x\,1{}\mathrm {i}}{a^{1/8}}\right )\,1{}\mathrm {i}}{4\,a^{15/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{7/8}\,\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^{15/8}}+\frac {\sqrt {2}\,{\left (-b\right )}^{7/8}\,\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^{15/8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^8*(a + b*x^8)),x)

[Out]

((-b)^(7/8)*atan(((-b)^(1/8)*x)/a^(1/8)))/(4*a^(15/8)) - 1/(7*a*x^7) - ((-b)^(7/8)*atan(((-b)^(1/8)*x*1i)/a^(1
/8))*1i)/(4*a^(15/8)) + (2^(1/2)*(-b)^(7/8)*atan((2^(1/2)*(-b)^(1/8)*x*(1/2 - 1i/2))/a^(1/8))*(1/8 + 1i/8))/a^
(15/8) + (2^(1/2)*(-b)^(7/8)*atan((2^(1/2)*(-b)^(1/8)*x*(1/2 + 1i/2))/a^(1/8))*(1/8 - 1i/8))/a^(15/8)

________________________________________________________________________________________