\(\int \frac {(a+b \coth ^{-1}(c x)) (d+e \log (f+g x^2))}{x^3} \, dx\) [282]

Optimal result

Integrand size = 24, antiderivative size = 712 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\frac {b c e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}+\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )}{f}+b c^2 e \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b e g \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )}{2 f}-\frac {b e g \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )}{2 f}-\frac {1}{2} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )-\frac {b e g \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 f}+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )+\frac {b e g \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{4 f}+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )+\frac {b e g \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{4 f} \]

[Out]

a*e*g*ln(x)/f+b*e*g*arccoth(c*x)*ln(2/(c*x+1))/f+b*c^2*e*arctanh(c*x)*ln(2/(c*x+1))-1/2*a*e*g*ln(g*x^2+f)/f-1/
2*b*c*(d+e*ln(g*x^2+f))/x-1/2*(a+b*arccoth(c*x))*(d+e*ln(g*x^2+f))/x^2+1/2*b*c^2*arctanh(c*x)*(d+e*ln(g*x^2+f)
)-1/2*b*e*g*arccoth(c*x)*ln(2*c*((-f)^(1/2)-x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2)-g^(1/2)))/f-1/2*b*c^2*e*arctanh(c
*x)*ln(2*c*((-f)^(1/2)-x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2)-g^(1/2)))-1/2*b*e*g*arccoth(c*x)*ln(2*c*((-f)^(1/2)+x*
g^(1/2))/(c*x+1)/(c*(-f)^(1/2)+g^(1/2)))/f-1/2*b*c^2*e*arctanh(c*x)*ln(2*c*((-f)^(1/2)+x*g^(1/2))/(c*x+1)/(c*(
-f)^(1/2)+g^(1/2)))+1/2*b*e*g*polylog(2,-1/c/x)/f-1/2*b*e*g*polylog(2,1/c/x)/f-1/2*b*c^2*e*polylog(2,1-2/(c*x+
1))-1/2*b*e*g*polylog(2,1-2/(c*x+1))/f+1/4*b*c^2*e*polylog(2,1-2*c*((-f)^(1/2)-x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2
)-g^(1/2)))+1/4*b*e*g*polylog(2,1-2*c*((-f)^(1/2)-x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2)-g^(1/2)))/f+1/4*b*c^2*e*pol
ylog(2,1-2*c*((-f)^(1/2)+x*g^(1/2))/(c*x+1)/(c*(-f)^(1/2)+g^(1/2)))+1/4*b*e*g*polylog(2,1-2*c*((-f)^(1/2)+x*g^
(1/2))/(c*x+1)/(c*(-f)^(1/2)+g^(1/2)))/f+b*c*e*arctan(x*g^(1/2)/f^(1/2))*g^(1/2)/f^(1/2)

Rubi [A] (verified)

Time = 0.84 (sec) , antiderivative size = 712, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {6038, 331, 212, 6233, 6857, 815, 649, 211, 266, 6140, 6032, 6058, 2449, 2352, 2497, 6139, 6057} \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac {a e g \log \left (f+g x^2\right )}{2 f}+\frac {a e g \log (x)}{f}+\frac {b c e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )+b c^2 e \text {arctanh}(c x) \log \left (\frac {2}{c x+1}\right )+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (c x+1)}\right )+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {g} x+\sqrt {-f}\right )}{\left (\sqrt {-f} c+\sqrt {g}\right ) (c x+1)}\right )-\frac {1}{2} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac {b e g \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )}{2 f}-\frac {b e g \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )}{2 f}-\frac {b e g \operatorname {PolyLog}\left (2,1-\frac {2}{c x+1}\right )}{2 f}+\frac {b e g \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (c x+1)}\right )}{4 f}+\frac {b e g \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {g} x+\sqrt {-f}\right )}{\left (\sqrt {-f} c+\sqrt {g}\right ) (c x+1)}\right )}{4 f}+\frac {b e g \log \left (\frac {2}{c x+1}\right ) \coth ^{-1}(c x)}{f}-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}-\sqrt {g}\right )}\right )}{2 f}-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{(c x+1) \left (c \sqrt {-f}+\sqrt {g}\right )}\right )}{2 f} \]

[In]

Int[((a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]))/x^3,x]

[Out]

(b*c*e*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[f] + (a*e*g*Log[x])/f + (b*e*g*ArcCoth[c*x]*Log[2/(1 + c*x)])
/f + b*c^2*e*ArcTanh[c*x]*Log[2/(1 + c*x)] - (b*e*g*ArcCoth[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f]
- Sqrt[g])*(1 + c*x))])/(2*f) - (b*c^2*e*ArcTanh[c*x]*Log[(2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g]
)*(1 + c*x))])/2 - (b*e*g*ArcCoth[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(
2*f) - (b*c^2*e*ArcTanh[c*x]*Log[(2*c*(Sqrt[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/2 - (a*e*g*
Log[f + g*x^2])/(2*f) - (b*c*(d + e*Log[f + g*x^2]))/(2*x) - ((a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]))/(2*
x^2) + (b*c^2*ArcTanh[c*x]*(d + e*Log[f + g*x^2]))/2 + (b*e*g*PolyLog[2, -(1/(c*x))])/(2*f) - (b*e*g*PolyLog[2
, 1/(c*x)])/(2*f) - (b*c^2*e*PolyLog[2, 1 - 2/(1 + c*x)])/2 - (b*e*g*PolyLog[2, 1 - 2/(1 + c*x)])/(2*f) + (b*c
^2*e*PolyLog[2, 1 - (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/4 + (b*e*g*PolyLog[2, 1
- (2*c*(Sqrt[-f] - Sqrt[g]*x))/((c*Sqrt[-f] - Sqrt[g])*(1 + c*x))])/(4*f) + (b*c^2*e*PolyLog[2, 1 - (2*c*(Sqrt
[-f] + Sqrt[g]*x))/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/4 + (b*e*g*PolyLog[2, 1 - (2*c*(Sqrt[-f] + Sqrt[g]*x))
/((c*Sqrt[-f] + Sqrt[g])*(1 + c*x))])/(4*f)

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 212

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

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

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 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 815

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m*((f + g*x)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

Rule 2449

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Dist[-e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*g, 0]

Rule 2497

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/D[u, x])]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rule 6032

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Simp[(b/2)*PolyLog[2, -(c*x)^(
-1)], x] - Simp[(b/2)*PolyLog[2, 1/(c*x)], x]) /; FreeQ[{a, b, c}, x]

Rule 6038

Int[((a_.) + ArcCoth[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCoth[c*
x^n])^p/(m + 1)), x] - Dist[b*c*n*(p/(m + 1)), Int[x^(m + n)*((a + b*ArcCoth[c*x^n])^(p - 1)/(1 - c^2*x^(2*n))
), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] && IntegerQ[m])) && NeQ[m, -1
]

Rule 6057

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcTanh[c*x]))*(Log[2/
(1 + c*x)]/e), x] + (Dist[b*(c/e), Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Dist[b*(c/e), Int[Log[2*c*((d
+ e*x)/((c*d + e)*(1 + c*x)))]/(1 - c^2*x^2), x], x] + Simp[(a + b*ArcTanh[c*x])*(Log[2*c*((d + e*x)/((c*d + e
)*(1 + c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]

Rule 6058

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-(a + b*ArcCoth[c*x]))*(Log[2/
(1 + c*x)]/e), x] + (Dist[b*(c/e), Int[Log[2/(1 + c*x)]/(1 - c^2*x^2), x], x] - Dist[b*(c/e), Int[Log[2*c*((d
+ e*x)/((c*d + e)*(1 + c*x)))]/(1 - c^2*x^2), x], x] + Simp[(a + b*ArcCoth[c*x])*(Log[2*c*((d + e*x)/((c*d + e
)*(1 + c*x)))]/e), x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 - e^2, 0]

Rule 6139

Int[(((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTanh[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[
a, 0])

Rule 6140

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcCoth[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[
a, 0])

Rule 6233

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*(e_.))*(x_)^(m_.), x_Symbol] :> Wit
h[{u = IntHide[x^m*(a + b*ArcCoth[c*x]), x]}, Dist[d + e*Log[f + g*x^2], u, x] - Dist[2*e*g, Int[ExpandIntegra
nd[x*(u/(f + g*x^2)), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && IntegerQ[m] && NeQ[m, -1]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )-(2 e g) \int \left (\frac {-a-b c x-b \coth ^{-1}(c x)}{2 x \left (f+g x^2\right )}+\frac {b c^2 x \text {arctanh}(c x)}{2 \left (f+g x^2\right )}\right ) \, dx \\ & = -\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )-(e g) \int \frac {-a-b c x-b \coth ^{-1}(c x)}{x \left (f+g x^2\right )} \, dx-\left (b c^2 e g\right ) \int \frac {x \text {arctanh}(c x)}{f+g x^2} \, dx \\ & = -\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )-(e g) \int \left (\frac {-a-b c x}{x \left (f+g x^2\right )}-\frac {b \coth ^{-1}(c x)}{x \left (f+g x^2\right )}\right ) \, dx-\left (b c^2 e g\right ) \int \left (-\frac {\text {arctanh}(c x)}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\text {arctanh}(c x)}{2 \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx \\ & = -\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {1}{2} \left (b c^2 e \sqrt {g}\right ) \int \frac {\text {arctanh}(c x)}{\sqrt {-f}-\sqrt {g} x} \, dx-\frac {1}{2} \left (b c^2 e \sqrt {g}\right ) \int \frac {\text {arctanh}(c x)}{\sqrt {-f}+\sqrt {g} x} \, dx-(e g) \int \frac {-a-b c x}{x \left (f+g x^2\right )} \, dx+(b e g) \int \frac {\coth ^{-1}(c x)}{x \left (f+g x^2\right )} \, dx \\ & = b c^2 e \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )-2 \left (\frac {1}{2} \left (b c^3 e\right ) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx\right )+\frac {1}{2} \left (b c^3 e\right ) \int \frac {\log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx+\frac {1}{2} \left (b c^3 e\right ) \int \frac {\log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx-(e g) \int \left (-\frac {a}{f x}+\frac {-b c f+a g x}{f \left (f+g x^2\right )}\right ) \, dx+(b e g) \int \left (\frac {\coth ^{-1}(c x)}{f x}-\frac {g x \coth ^{-1}(c x)}{f \left (f+g x^2\right )}\right ) \, dx \\ & = \frac {a e g \log (x)}{f}+b c^2 e \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-2 \left (\frac {1}{2} \left (b c^2 e\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )\right )-\frac {(e g) \int \frac {-b c f+a g x}{f+g x^2} \, dx}{f}+\frac {(b e g) \int \frac {\coth ^{-1}(c x)}{x} \, dx}{f}-\frac {\left (b e g^2\right ) \int \frac {x \coth ^{-1}(c x)}{f+g x^2} \, dx}{f} \\ & = \frac {a e g \log (x)}{f}+b c^2 e \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b e g \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )}{2 f}-\frac {b e g \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )}{2 f}-\frac {1}{2} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )+(b c e g) \int \frac {1}{f+g x^2} \, dx-\frac {\left (a e g^2\right ) \int \frac {x}{f+g x^2} \, dx}{f}-\frac {\left (b e g^2\right ) \int \left (-\frac {\coth ^{-1}(c x)}{2 \sqrt {g} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {\coth ^{-1}(c x)}{2 \sqrt {g} \left (\sqrt {-f}+\sqrt {g} x\right )}\right ) \, dx}{f} \\ & = \frac {b c e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}+b c^2 e \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b e g \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )}{2 f}-\frac {b e g \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )}{2 f}-\frac {1}{2} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )+\frac {\left (b e g^{3/2}\right ) \int \frac {\coth ^{-1}(c x)}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 f}-\frac {\left (b e g^{3/2}\right ) \int \frac {\coth ^{-1}(c x)}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 f} \\ & = \frac {b c e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}+\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )}{f}+b c^2 e \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b e g \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )}{2 f}-\frac {b e g \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )}{2 f}-\frac {1}{2} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-2 \frac {(b c e g) \int \frac {\log \left (\frac {2}{1+c x}\right )}{1-c^2 x^2} \, dx}{2 f}+\frac {(b c e g) \int \frac {\log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx}{2 f}+\frac {(b c e g) \int \frac {\log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{1-c^2 x^2} \, dx}{2 f} \\ & = \frac {b c e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}+\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )}{f}+b c^2 e \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b e g \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )}{2 f}-\frac {b e g \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )}{2 f}-\frac {1}{2} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )+\frac {b e g \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{4 f}+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )+\frac {b e g \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{4 f}-2 \frac {(b e g) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+c x}\right )}{2 f} \\ & = \frac {b c e \sqrt {g} \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )}{\sqrt {f}}+\frac {a e g \log (x)}{f}+\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2}{1+c x}\right )}{f}+b c^2 e \text {arctanh}(c x) \log \left (\frac {2}{1+c x}\right )-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )-\frac {b e g \coth ^{-1}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{2 f}-\frac {1}{2} b c^2 e \text {arctanh}(c x) \log \left (\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )-\frac {a e g \log \left (f+g x^2\right )}{2 f}-\frac {b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}+\frac {1}{2} b c^2 \text {arctanh}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac {b e g \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )}{2 f}-\frac {b e g \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )}{2 f}-\frac {1}{2} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )-\frac {b e g \operatorname {PolyLog}\left (2,1-\frac {2}{1+c x}\right )}{2 f}+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )+\frac {b e g \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}-\sqrt {g} x\right )}{\left (c \sqrt {-f}-\sqrt {g}\right ) (1+c x)}\right )}{4 f}+\frac {1}{4} b c^2 e \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )+\frac {b e g \operatorname {PolyLog}\left (2,1-\frac {2 c \left (\sqrt {-f}+\sqrt {g} x\right )}{\left (c \sqrt {-f}+\sqrt {g}\right ) (1+c x)}\right )}{4 f} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.96 (sec) , antiderivative size = 1193, normalized size of antiderivative = 1.68 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\frac {-2 a d f-2 b c d f x-2 b d f \coth ^{-1}(c x)+2 b c^2 d f x^2 \coth ^{-1}(c x)+4 b c e \sqrt {f} \sqrt {g} x^2 \arctan \left (\frac {\sqrt {g} x}{\sqrt {f}}\right )+4 i b c^2 e f x^2 \arcsin \left (\sqrt {\frac {g}{c^2 f+g}}\right ) \text {arctanh}\left (\frac {c f}{\sqrt {-c^2 f g} x}\right )+4 i b e g x^2 \arcsin \left (\sqrt {\frac {g}{c^2 f+g}}\right ) \text {arctanh}\left (\frac {c f}{\sqrt {-c^2 f g} x}\right )+4 b c^2 e f x^2 \coth ^{-1}(c x) \log \left (1-e^{-2 \coth ^{-1}(c x)}\right )+4 b e g x^2 \coth ^{-1}(c x) \log \left (1+e^{-2 \coth ^{-1}(c x)}\right )-2 b c^2 e f x^2 \coth ^{-1}(c x) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g-2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )-2 b e g x^2 \coth ^{-1}(c x) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g-2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+2 i b c^2 e f x^2 \arcsin \left (\sqrt {\frac {g}{c^2 f+g}}\right ) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g-2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+2 i b e g x^2 \arcsin \left (\sqrt {\frac {g}{c^2 f+g}}\right ) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g-2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )-2 b c^2 e f x^2 \coth ^{-1}(c x) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )-2 b e g x^2 \coth ^{-1}(c x) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )-2 i b c^2 e f x^2 \arcsin \left (\sqrt {\frac {g}{c^2 f+g}}\right ) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )-2 i b e g x^2 \arcsin \left (\sqrt {\frac {g}{c^2 f+g}}\right ) \log \left (\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 \left (-1+e^{2 \coth ^{-1}(c x)}\right ) f+g+e^{2 \coth ^{-1}(c x)} g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+4 a e g x^2 \log (x)-2 a e f \log \left (f+g x^2\right )-2 b c e f x \log \left (f+g x^2\right )-2 a e g x^2 \log \left (f+g x^2\right )-2 b e f \coth ^{-1}(c x) \log \left (f+g x^2\right )+2 b c^2 e f x^2 \coth ^{-1}(c x) \log \left (f+g x^2\right )-2 b e g x^2 \operatorname {PolyLog}\left (2,-e^{-2 \coth ^{-1}(c x)}\right )-2 b c^2 e f x^2 \operatorname {PolyLog}\left (2,e^{-2 \coth ^{-1}(c x)}\right )+b c^2 e f x^2 \operatorname {PolyLog}\left (2,\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 f-g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+b e g x^2 \operatorname {PolyLog}\left (2,\frac {e^{-2 \coth ^{-1}(c x)} \left (c^2 f-g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+b c^2 e f x^2 \operatorname {PolyLog}\left (2,-\frac {e^{-2 \coth ^{-1}(c x)} \left (-c^2 f+g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )+b e g x^2 \operatorname {PolyLog}\left (2,-\frac {e^{-2 \coth ^{-1}(c x)} \left (-c^2 f+g+2 \sqrt {-c^2 f g}\right )}{c^2 f+g}\right )}{4 f x^2} \]

[In]

Integrate[((a + b*ArcCoth[c*x])*(d + e*Log[f + g*x^2]))/x^3,x]

[Out]

(-2*a*d*f - 2*b*c*d*f*x - 2*b*d*f*ArcCoth[c*x] + 2*b*c^2*d*f*x^2*ArcCoth[c*x] + 4*b*c*e*Sqrt[f]*Sqrt[g]*x^2*Ar
cTan[(Sqrt[g]*x)/Sqrt[f]] + (4*I)*b*c^2*e*f*x^2*ArcSin[Sqrt[g/(c^2*f + g)]]*ArcTanh[(c*f)/(Sqrt[-(c^2*f*g)]*x)
] + (4*I)*b*e*g*x^2*ArcSin[Sqrt[g/(c^2*f + g)]]*ArcTanh[(c*f)/(Sqrt[-(c^2*f*g)]*x)] + 4*b*c^2*e*f*x^2*ArcCoth[
c*x]*Log[1 - E^(-2*ArcCoth[c*x])] + 4*b*e*g*x^2*ArcCoth[c*x]*Log[1 + E^(-2*ArcCoth[c*x])] - 2*b*c^2*e*f*x^2*Ar
cCoth[c*x]*Log[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth
[c*x])*(c^2*f + g))] - 2*b*e*g*x^2*ArcCoth[c*x]*Log[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*
g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] + (2*I)*b*c^2*e*f*x^2*ArcSin[Sqrt[g/(c^2*f + g)]]*Lo
g[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f
+ g))] + (2*I)*b*e*g*x^2*ArcSin[Sqrt[g/(c^2*f + g)]]*Log[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[
c*x])*g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] - 2*b*c^2*e*f*x^2*ArcCoth[c*x]*Log[(c^2*(-1 +
E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] - 2*b
*e*g*x^2*ArcCoth[c*x]*Log[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g + 2*Sqrt[-(c^2*f*g)])/(E
^(2*ArcCoth[c*x])*(c^2*f + g))] - (2*I)*b*c^2*e*f*x^2*ArcSin[Sqrt[g/(c^2*f + g)]]*Log[(c^2*(-1 + E^(2*ArcCoth[
c*x]))*f + g + E^(2*ArcCoth[c*x])*g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] - (2*I)*b*e*g*x^2*
ArcSin[Sqrt[g/(c^2*f + g)]]*Log[(c^2*(-1 + E^(2*ArcCoth[c*x]))*f + g + E^(2*ArcCoth[c*x])*g + 2*Sqrt[-(c^2*f*g
)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] + 4*a*e*g*x^2*Log[x] - 2*a*e*f*Log[f + g*x^2] - 2*b*c*e*f*x*Log[f + g*x^
2] - 2*a*e*g*x^2*Log[f + g*x^2] - 2*b*e*f*ArcCoth[c*x]*Log[f + g*x^2] + 2*b*c^2*e*f*x^2*ArcCoth[c*x]*Log[f + g
*x^2] - 2*b*e*g*x^2*PolyLog[2, -E^(-2*ArcCoth[c*x])] - 2*b*c^2*e*f*x^2*PolyLog[2, E^(-2*ArcCoth[c*x])] + b*c^2
*e*f*x^2*PolyLog[2, (c^2*f - g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] + b*e*g*x^2*PolyLog[2,
(c^2*f - g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g))] + b*c^2*e*f*x^2*PolyLog[2, -((-(c^2*f) + g
+ 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g)))] + b*e*g*x^2*PolyLog[2, -((-(c^2*f) + g + 2*Sqrt[-(c^2
*f*g)])/(E^(2*ArcCoth[c*x])*(c^2*f + g)))])/(4*f*x^2)

Maple [A] (verified)

Time = 9.69 (sec) , antiderivative size = 937, normalized size of antiderivative = 1.32

method	result	size
risch	\(-\frac {a d}{2 x^{2}}-\frac {a e g \ln \left (g \,x^{2}+f \right )}{2 f}+\frac {a e g \ln \left (x \right )}{f}+\left (-\frac {b e \ln \left (c x +1\right )}{4 x^{2}}-\frac {e \left (b \,x^{2} \ln \left (c x -1\right ) c^{2}-b \,c^{2} \ln \left (c x +1\right ) x^{2}+2 b c x -b \ln \left (c x -1\right )+2 a \right )}{4 x^{2}}\right ) \ln \left (g \,x^{2}+f \right )-\frac {g b e \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 f}-\frac {g b e \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 f}+\frac {g e b c \arctan \left (\frac {x g}{\sqrt {f g}}\right )}{\sqrt {f g}}+\frac {g b e \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}-g \left (c x -1\right )-g}{c \sqrt {-f g}-g}\right )}{4 f}+\frac {g b e \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}+g \left (c x -1\right )+g}{c \sqrt {-f g}+g}\right )}{4 f}-\frac {g b e \ln \left (c x -1\right ) \ln \left (c x \right )}{2 f}-\frac {d b c}{2 x}+\frac {d b \,c^{2} \ln \left (c x +1\right )}{4}-\frac {d b \ln \left (c x +1\right )}{4 x^{2}}-\frac {d b \,c^{2} \ln \left (c x -1\right )}{4}+\frac {d b \ln \left (c x -1\right )}{4 x^{2}}-\frac {b e \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}-\frac {b e \ln \left (c x +1\right ) \ln \left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}-\frac {g b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right )}{4 f}-\frac {g b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right )}{4 f}+\frac {b e \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}-g \left (c x -1\right )-g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}+\frac {b e \ln \left (c x -1\right ) \ln \left (\frac {c \sqrt {-f g}+g \left (c x -1\right )+g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}+\frac {g b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}-g \left (c x -1\right )-g}{c \sqrt {-f g}-g}\right )}{4 f}+\frac {g b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}+g \left (c x -1\right )+g}{c \sqrt {-f g}+g}\right )}{4 f}-\frac {g b e \operatorname {dilog}\left (c x \right )}{2 f}-\frac {g b e \operatorname {dilog}\left (c x +1\right )}{2 f}-\frac {b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}-\left (c x +1\right ) g +g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}-\frac {b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}+\left (c x +1\right ) g -g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}+\frac {b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}-g \left (c x -1\right )-g}{c \sqrt {-f g}-g}\right ) c^{2}}{4}+\frac {b e \operatorname {dilog}\left (\frac {c \sqrt {-f g}+g \left (c x -1\right )+g}{c \sqrt {-f g}+g}\right ) c^{2}}{4}\)	\(937\)

[In]

int((a+b*arccoth(c*x))*(d+e*ln(g*x^2+f))/x^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*a*d/x^2-1/2*a*e*g*ln(g*x^2+f)/f+a*e*g*ln(x)/f+(-1/4*b*e/x^2*ln(c*x+1)-1/4*e*(b*x^2*ln(c*x-1)*c^2-b*c^2*ln
(c*x+1)*x^2+2*b*c*x-b*ln(c*x-1)+2*a)/x^2)*ln(g*x^2+f)-1/4*g*b*e/f*ln(c*x+1)*ln((c*(-f*g)^(1/2)-(c*x+1)*g+g)/(c
*(-f*g)^(1/2)+g))-1/4*g*b*e/f*ln(c*x+1)*ln((c*(-f*g)^(1/2)+(c*x+1)*g-g)/(c*(-f*g)^(1/2)-g))+g*e*b*c/(f*g)^(1/2
)*arctan(x*g/(f*g)^(1/2))+1/4*g*b*e/f*ln(c*x-1)*ln((c*(-f*g)^(1/2)-g*(c*x-1)-g)/(c*(-f*g)^(1/2)-g))+1/4*g*b*e/
f*ln(c*x-1)*ln((c*(-f*g)^(1/2)+g*(c*x-1)+g)/(c*(-f*g)^(1/2)+g))-1/2*g*b*e/f*ln(c*x-1)*ln(c*x)-1/2*d*b*c/x+1/4*
d*b*c^2*ln(c*x+1)-1/4*d*b*ln(c*x+1)/x^2-1/4*d*b*c^2*ln(c*x-1)+1/4*d*b*ln(c*x-1)/x^2-1/4*b*e*ln(c*x+1)*ln((c*(-
f*g)^(1/2)-(c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))*c^2-1/4*b*e*ln(c*x+1)*ln((c*(-f*g)^(1/2)+(c*x+1)*g-g)/(c*(-f*g)^(1
/2)-g))*c^2-1/4*g*b*e/f*dilog((c*(-f*g)^(1/2)-(c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))-1/4*g*b*e/f*dilog((c*(-f*g)^(1/
2)+(c*x+1)*g-g)/(c*(-f*g)^(1/2)-g))+1/4*b*e*ln(c*x-1)*ln((c*(-f*g)^(1/2)-g*(c*x-1)-g)/(c*(-f*g)^(1/2)-g))*c^2+
1/4*b*e*ln(c*x-1)*ln((c*(-f*g)^(1/2)+g*(c*x-1)+g)/(c*(-f*g)^(1/2)+g))*c^2+1/4*g*b*e/f*dilog((c*(-f*g)^(1/2)-g*
(c*x-1)-g)/(c*(-f*g)^(1/2)-g))+1/4*g*b*e/f*dilog((c*(-f*g)^(1/2)+g*(c*x-1)+g)/(c*(-f*g)^(1/2)+g))-1/2*g*b*e/f*
dilog(c*x)-1/2*g*b*e/f*dilog(c*x+1)-1/4*b*e*dilog((c*(-f*g)^(1/2)-(c*x+1)*g+g)/(c*(-f*g)^(1/2)+g))*c^2-1/4*b*e
*dilog((c*(-f*g)^(1/2)+(c*x+1)*g-g)/(c*(-f*g)^(1/2)-g))*c^2+1/4*b*e*dilog((c*(-f*g)^(1/2)-g*(c*x-1)-g)/(c*(-f*
g)^(1/2)-g))*c^2+1/4*b*e*dilog((c*(-f*g)^(1/2)+g*(c*x-1)+g)/(c*(-f*g)^(1/2)+g))*c^2

Fricas [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{3}} \,d x } \]

[In]

integrate((a+b*arccoth(c*x))*(d+e*log(g*x^2+f))/x^3,x, algorithm="fricas")

[Out]

integral((b*d*arccoth(c*x) + a*d + (b*e*arccoth(c*x) + a*e)*log(g*x^2 + f))/x^3, x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\text {Timed out} \]

[In]

integrate((a+b*acoth(c*x))*(d+e*ln(g*x**2+f))/x**3,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{3}} \,d x } \]

[In]

integrate((a+b*arccoth(c*x))*(d+e*log(g*x^2+f))/x^3,x, algorithm="maxima")

[Out]

1/4*((c*log(c*x + 1) - c*log(c*x - 1) - 2/x)*c - 2*arccoth(c*x)/x^2)*b*d - 1/2*(g*(log(g*x^2 + f)/f - log(x^2)
/f) + log(g*x^2 + f)/x^2)*a*e - 1/4*(2*c^2*g*integrate(x^2*log(c*x + 1)/(g*x^3 + f*x), x) - 2*c^2*g*integrate(
x^2*log(c*x - 1)/(g*x^3 + f*x), x) + 2*I*c*g*(log(I*g*x/sqrt(f*g) + 1) - log(-I*g*x/sqrt(f*g) + 1))/sqrt(f*g)
- 2*g*integrate(log(c*x + 1)/(g*x^3 + f*x), x) + 2*g*integrate(log(c*x - 1)/(g*x^3 + f*x), x) + (2*c*x - (c^2*
x^2 - 1)*log(c*x + 1) + (c^2*x^2 - 1)*log(c*x - 1))*log(g*x^2 + f)/x^2)*b*e - 1/2*a*d/x^2

Giac [F]

\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{3}} \,d x } \]

[In]

integrate((a+b*arccoth(c*x))*(d+e*log(g*x^2+f))/x^3,x, algorithm="giac")

[Out]

integrate((b*arccoth(c*x) + a)*(e*log(g*x^2 + f) + d)/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right )}{x^3} \,d x \]

[In]

int(((a + b*acoth(c*x))*(d + e*log(f + g*x^2)))/x^3,x)

[Out]

int(((a + b*acoth(c*x))*(d + e*log(f + g*x^2)))/x^3, x)