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

Optimal. Leaf size=470 \[ -\frac{b e \left (c^2 f+g\right ) \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 f}+\frac{b e \left (c^2 f+g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )}{4 f}+\frac{b e \left (c^2 f+g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )}{4 f}-\frac{b e g \text{PolyLog}(2,-c x)}{2 f}+\frac{b e g \text{PolyLog}(2,c x)}{2 f}-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac{a e g \log \left (f+g x^2\right )}{2 f}+\frac{a e g \log (x)}{f}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b e \left (c^2 f+g\right ) \log \left (\frac{2}{c x+1}\right ) \tanh ^{-1}(c x)}{f}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )}{2 f}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )}{2 f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.746135, antiderivative size = 470, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 17, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.708, Rules used = {5916, 325, 206, 6085, 801, 635, 205, 260, 446, 72, 6725, 5912, 5992, 5920, 2402, 2315, 2447} \[ -\frac{b e \left (c^2 f+g\right ) \text{PolyLog}\left (2,1-\frac{2}{c x+1}\right )}{2 f}+\frac{b e \left (c^2 f+g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )}{4 f}+\frac{b e \left (c^2 f+g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )}{4 f}-\frac{b e g \text{PolyLog}(2,-c x)}{2 f}+\frac{b e g \text{PolyLog}(2,c x)}{2 f}-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac{a e g \log \left (f+g x^2\right )}{2 f}+\frac{a e g \log (x)}{f}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b e \left (c^2 f+g\right ) \log \left (\frac{2}{c x+1}\right ) \tanh ^{-1}(c x)}{f}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}-\sqrt{g}\right )}\right )}{2 f}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(c x+1) \left (c \sqrt{-f}+\sqrt{g}\right )}\right )}{2 f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}} \]

Antiderivative was successfully verified.

[In]

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

Rule 5916

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

Rule 325

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 206

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

Rule 6085

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*(e_.))*(x_)^(m_.), x_Symbol] :> Wit
h[{u = IntHide[x^m*(a + b*ArcTanh[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 801

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 635

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 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 260

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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 6725

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]

Rule 5912

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

Rule 5992

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 5920

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 2402

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 2315

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

Rule 2447

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]]

Rubi steps

\begin{align*} \int \frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^3} \, dx &=-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-(2 e g) \int \left (\frac{-a-b c x}{2 x \left (f+g x^2\right )}+\frac{b \left (-1+c^2 x^2\right ) \tanh ^{-1}(c x)}{2 x \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{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-(e g) \int \frac{-a-b c x}{x \left (f+g x^2\right )} \, dx-(b e g) \int \frac{\left (-1+c^2 x^2\right ) \tanh ^{-1}(c x)}{x \left (f+g x^2\right )} \, dx\\ &=-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-(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{\tanh ^{-1}(c x)}{f x}+\frac{\left (c^2 f+g\right ) x \tanh ^{-1}(c x)}{f \left (f+g x^2\right )}\right ) \, dx\\ &=\frac{a e g \log (x)}{f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac{(e g) \int \frac{-b c f+a g x}{f+g x^2} \, dx}{f}+\frac{(b e g) \int \frac{\tanh ^{-1}(c x)}{x} \, dx}{f}-\frac{\left (b e g \left (c^2 f+g\right )\right ) \int \frac{x \tanh ^{-1}(c x)}{f+g x^2} \, dx}{f}\\ &=\frac{a e g \log (x)}{f}-\frac{b c \left (d+e \log \left (f+g x^2\right )\right )}{2 x}+\frac{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac{b e g \text{Li}_2(-c x)}{2 f}+\frac{b e g \text{Li}_2(c x)}{2 f}+(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 \left (c^2 f+g\right )\right ) \int \left (-\frac{\tanh ^{-1}(c x)}{2 \sqrt{g} \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\tanh ^{-1}(c x)}{2 \sqrt{g} \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{f}\\ &=\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}+\frac{a e g \log (x)}{f}-\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{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac{b e g \text{Li}_2(-c x)}{2 f}+\frac{b e g \text{Li}_2(c x)}{2 f}+\frac{\left (b e \sqrt{g} \left (c^2 f+g\right )\right ) \int \frac{\tanh ^{-1}(c x)}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 f}-\frac{\left (b e \sqrt{g} \left (c^2 f+g\right )\right ) \int \frac{\tanh ^{-1}(c x)}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 f}\\ &=\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}+\frac{a e g \log (x)}{f}+\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )}{f}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-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{b e \left (c^2 f+g\right ) \tanh ^{-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{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{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac{b e g \text{Li}_2(-c x)}{2 f}+\frac{b e g \text{Li}_2(c x)}{2 f}-2 \frac{\left (b c e \left (c^2 f+g\right )\right ) \int \frac{\log \left (\frac{2}{1+c x}\right )}{1-c^2 x^2} \, dx}{2 f}+\frac{\left (b c e \left (c^2 f+g\right )\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}{2 f}+\frac{\left (b c e \left (c^2 f+g\right )\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}{2 f}\\ &=\frac{b c e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}+\frac{a e g \log (x)}{f}+\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )}{f}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-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{b e \left (c^2 f+g\right ) \tanh ^{-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{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{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac{b e g \text{Li}_2(-c x)}{2 f}+\frac{b e g \text{Li}_2(c x)}{2 f}+\frac{b e \left (c^2 f+g\right ) \text{Li}_2\left (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{b e \left (c^2 f+g\right ) \text{Li}_2\left (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{\left (b e \left (c^2 f+g\right )\right ) \operatorname{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} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}+\frac{a e g \log (x)}{f}+\frac{b e \left (c^2 f+g\right ) \tanh ^{-1}(c x) \log \left (\frac{2}{1+c x}\right )}{f}-\frac{b e \left (c^2 f+g\right ) \tanh ^{-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{b e \left (c^2 f+g\right ) \tanh ^{-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{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{1}{2} b c^2 \tanh ^{-1}(c x) \left (d+e \log \left (f+g x^2\right )\right )-\frac{\left (a+b \tanh ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 x^2}-\frac{b e g \text{Li}_2(-c x)}{2 f}+\frac{b e g \text{Li}_2(c x)}{2 f}-\frac{b e \left (c^2 f+g\right ) \text{Li}_2\left (1-\frac{2}{1+c x}\right )}{2 f}+\frac{b e \left (c^2 f+g\right ) \text{Li}_2\left (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{b e \left (c^2 f+g\right ) \text{Li}_2\left (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]  time = 4.54415, size = 982, normalized size = 2.09 \[ -\frac{-4 b e g \tanh ^{-1}(c x)^2 x^2-4 b c e \sqrt{f} \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) x^2-2 b c^2 d f \tanh ^{-1}(c x) x^2-4 i b c^2 e f \sin ^{-1}\left (\sqrt{\frac{c^2 f}{f c^2+g}}\right ) \tanh ^{-1}\left (\frac{c g x}{\sqrt{-c^2 f g}}\right ) x^2-4 b e g \tanh ^{-1}(c x) \log \left (1-e^{-2 \tanh ^{-1}(c x)}\right ) x^2-4 b c^2 e f \tanh ^{-1}(c x) \log \left (1+e^{-2 \tanh ^{-1}(c x)}\right ) x^2-2 i b c^2 e f \sin ^{-1}\left (\sqrt{\frac{c^2 f}{f c^2+g}}\right ) \log \left (\frac{e^{-2 \tanh ^{-1}(c x)} \left (\left (1+e^{2 \tanh ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 \tanh ^{-1}(c x)}\right ) g-2 \sqrt{-c^2 f g}\right )}{f c^2+g}\right ) x^2+2 b c^2 e f \tanh ^{-1}(c x) \log \left (\frac{e^{-2 \tanh ^{-1}(c x)} \left (\left (1+e^{2 \tanh ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 \tanh ^{-1}(c x)}\right ) g-2 \sqrt{-c^2 f g}\right )}{f c^2+g}\right ) x^2+2 i b c^2 e f \sin ^{-1}\left (\sqrt{\frac{c^2 f}{f c^2+g}}\right ) \log \left (\frac{e^{-2 \tanh ^{-1}(c x)} \left (\left (1+e^{2 \tanh ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 \tanh ^{-1}(c x)}\right ) g+2 \sqrt{-c^2 f g}\right )}{f c^2+g}\right ) x^2+2 b c^2 e f \tanh ^{-1}(c x) \log \left (\frac{e^{-2 \tanh ^{-1}(c x)} \left (\left (1+e^{2 \tanh ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 \tanh ^{-1}(c x)}\right ) g+2 \sqrt{-c^2 f g}\right )}{f c^2+g}\right ) x^2+2 b e g \tanh ^{-1}(c x) \log \left (\frac{e^{2 \tanh ^{-1}(c x)} \left (f c^2+g\right )}{f c^2-2 \sqrt{-f} \sqrt{g} c-g}+1\right ) x^2+2 b e g \tanh ^{-1}(c x) \log \left (\frac{e^{2 \tanh ^{-1}(c x)} \left (f c^2+g\right )}{f c^2+2 \sqrt{-f} \sqrt{g} c-g}+1\right ) x^2-4 a e g \log (x) x^2+2 a e g \log \left (g x^2+f\right ) x^2-2 b c^2 e f \tanh ^{-1}(c x) \log \left (g x^2+f\right ) x^2+2 b c^2 e f \text{PolyLog}\left (2,-e^{-2 \tanh ^{-1}(c x)}\right ) x^2+2 b e g \text{PolyLog}\left (2,e^{-2 \tanh ^{-1}(c x)}\right ) x^2+b e g \text{PolyLog}\left (2,-\frac{e^{2 \tanh ^{-1}(c x)} \left (f c^2+g\right )}{f c^2-2 \sqrt{-f} \sqrt{g} c-g}\right ) x^2+b e g \text{PolyLog}\left (2,-\frac{e^{2 \tanh ^{-1}(c x)} \left (f c^2+g\right )}{f c^2+2 \sqrt{-f} \sqrt{g} c-g}\right ) x^2-b c^2 e f \text{PolyLog}\left (2,\frac{e^{-2 \tanh ^{-1}(c x)} \left (-f c^2+g-2 \sqrt{-c^2 f g}\right )}{f c^2+g}\right ) x^2-b c^2 e f \text{PolyLog}\left (2,\frac{e^{-2 \tanh ^{-1}(c x)} \left (-f c^2+g+2 \sqrt{-c^2 f g}\right )}{f c^2+g}\right ) x^2+2 b c d f x+2 b c e f \log \left (g x^2+f\right ) x+2 a d f+2 b d f \tanh ^{-1}(c x)+2 a e f \log \left (g x^2+f\right )+2 b e f \tanh ^{-1}(c x) \log \left (g x^2+f\right )}{4 f x^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(2*a*d*f + 2*b*c*d*f*x - 4*b*c*e*Sqrt[f]*Sqrt[g]*x^2*ArcTan[(Sqrt[g]*x)/Sqrt[f]] + 2*b*d*f*ArcTanh[c*x] - 2*b
*c^2*d*f*x^2*ArcTanh[c*x] - 4*b*e*g*x^2*ArcTanh[c*x]^2 - (4*I)*b*c^2*e*f*x^2*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]]
*ArcTanh[(c*g*x)/Sqrt[-(c^2*f*g)]] - 4*b*e*g*x^2*ArcTanh[c*x]*Log[1 - E^(-2*ArcTanh[c*x])] - 4*b*c^2*e*f*x^2*A
rcTanh[c*x]*Log[1 + E^(-2*ArcTanh[c*x])] - (2*I)*b*c^2*e*f*x^2*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]]*Log[(c^2*(1 +
E^(2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g - 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] +
2*b*c^2*e*f*x^2*ArcTanh[c*x]*Log[(c^2*(1 + E^(2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g - 2*Sqrt[-(c^2
*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] + (2*I)*b*c^2*e*f*x^2*ArcSin[Sqrt[(c^2*f)/(c^2*f + g)]]*Log[(c^2*(1
+ E^(2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g + 2*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))]
+ 2*b*c^2*e*f*x^2*ArcTanh[c*x]*Log[(c^2*(1 + E^(2*ArcTanh[c*x]))*f + (-1 + E^(2*ArcTanh[c*x]))*g + 2*Sqrt[-(c^
2*f*g)])/(E^(2*ArcTanh[c*x])*(c^2*f + g))] + 2*b*e*g*x^2*ArcTanh[c*x]*Log[1 + (E^(2*ArcTanh[c*x])*(c^2*f + g))
/(c^2*f - 2*c*Sqrt[-f]*Sqrt[g] - g)] + 2*b*e*g*x^2*ArcTanh[c*x]*Log[1 + (E^(2*ArcTanh[c*x])*(c^2*f + g))/(c^2*
f + 2*c*Sqrt[-f]*Sqrt[g] - 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*ArcTanh[c*x]*Log[f + g*x^2] - 2*b*c^2*e*f*x^2*ArcTanh[c*x]*Log[f + g*x^2] +
2*b*c^2*e*f*x^2*PolyLog[2, -E^(-2*ArcTanh[c*x])] + 2*b*e*g*x^2*PolyLog[2, E^(-2*ArcTanh[c*x])] + b*e*g*x^2*Po
lyLog[2, -((E^(2*ArcTanh[c*x])*(c^2*f + g))/(c^2*f - 2*c*Sqrt[-f]*Sqrt[g] - g))] + b*e*g*x^2*PolyLog[2, -((E^(
2*ArcTanh[c*x])*(c^2*f + g))/(c^2*f + 2*c*Sqrt[-f]*Sqrt[g] - g))] - b*c^2*e*f*x^2*PolyLog[2, (-(c^2*f) + g - 2
*Sqrt[-(c^2*f*g)])/(E^(2*ArcTanh[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*ArcTanh[c*x])*(c^2*f + g))])/(4*f*x^2)

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Maple [B]  time = 2.897, size = 961, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b d \operatorname{artanh}\left (c x\right ) + a d +{\left (b e \operatorname{artanh}\left (c x\right ) + a e\right )} \log \left (g x^{2} + f\right )}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (c x\right ) + a\right )}{\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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