3.279 \(\int (a+b \coth ^{-1}(c x)) (d+e \log (f+g x^2)) \, dx\)

Optimal. Leaf size=546 \[ \frac{b e \text{PolyLog}\left (2,\frac{c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}-\frac{i b e \sqrt{f} \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} (1-c x)}{\left (-\sqrt{g}+i c \sqrt{f}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{g}}+\frac{i b e \sqrt{f} \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} (c x+1)}{\left (\sqrt{g}+i c \sqrt{f}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{g}}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{2 a e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}-2 a e x+\frac{b \log \left (\frac{g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}-\frac{b e \log \left (1-c^2 x^2\right )}{c}-\frac{b e \sqrt{f} \log \left (1-\frac{1}{c x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}+\frac{b e \sqrt{f} \log \left (\frac{1}{c x}+1\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} (1-c x)}{\left (-\sqrt{g}+i c \sqrt{f}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{g}}-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} (c x+1)}{\left (\sqrt{g}+i c \sqrt{f}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{g}}-2 b e x \coth ^{-1}(c x) \]

[Out]

-2*a*e*x - 2*b*e*x*ArcCoth[c*x] + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] - (b*e*Sqrt[f]*ArcTan[(S
qrt[g]*x)/Sqrt[f]]*Log[1 - 1/(c*x)])/Sqrt[g] + (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[1 + 1/(c*x)])/Sqrt
[g] + (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(-2*Sqrt[f]*Sqrt[g]*(1 - c*x))/((I*c*Sqrt[f] - Sqrt[g])*(Sq
rt[f] - I*Sqrt[g]*x))])/Sqrt[g] - (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(1 + c*x))/(
(I*c*Sqrt[f] + Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/Sqrt[g] - (b*e*Log[1 - c^2*x^2])/c + x*(a + b*ArcCoth[c*x])
*(d + e*Log[f + g*x^2]) + (b*Log[(g*(1 - c^2*x^2))/(c^2*f + g)]*(d + e*Log[f + g*x^2]))/(2*c) + (b*e*PolyLog[2
, (c^2*(f + g*x^2))/(c^2*f + g)])/(2*c) - ((I/2)*b*e*Sqrt[f]*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(1 - c*x))/((I*
c*Sqrt[f] - Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/Sqrt[g] + ((I/2)*b*e*Sqrt[f]*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]
*(1 + c*x))/((I*c*Sqrt[f] + Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/Sqrt[g]

________________________________________________________________________________________

Rubi [A]  time = 1.38741, antiderivative size = 546, normalized size of antiderivative = 1., number of steps used = 38, number of rules used = 20, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.952, Rules used = {6074, 2475, 2394, 2393, 2391, 5981, 5911, 260, 5975, 205, 5973, 2470, 12, 6688, 4876, 4848, 4856, 2402, 2315, 2447} \[ \frac{b e \text{PolyLog}\left (2,\frac{c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}-\frac{i b e \sqrt{f} \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} (1-c x)}{\left (-\sqrt{g}+i c \sqrt{f}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{g}}+\frac{i b e \sqrt{f} \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} (c x+1)}{\left (\sqrt{g}+i c \sqrt{f}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{g}}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{2 a e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}-2 a e x+\frac{b \log \left (\frac{g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}-\frac{b e \log \left (1-c^2 x^2\right )}{c}-\frac{b e \sqrt{f} \log \left (1-\frac{1}{c x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}+\frac{b e \sqrt{f} \log \left (\frac{1}{c x}+1\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} (1-c x)}{\left (-\sqrt{g}+i c \sqrt{f}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{g}}-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} (c x+1)}{\left (\sqrt{g}+i c \sqrt{f}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{g}}-2 b e x \coth ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*a*e*x - 2*b*e*x*ArcCoth[c*x] + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] - (b*e*Sqrt[f]*ArcTan[(S
qrt[g]*x)/Sqrt[f]]*Log[1 - 1/(c*x)])/Sqrt[g] + (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[1 + 1/(c*x)])/Sqrt
[g] + (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(-2*Sqrt[f]*Sqrt[g]*(1 - c*x))/((I*c*Sqrt[f] - Sqrt[g])*(Sq
rt[f] - I*Sqrt[g]*x))])/Sqrt[g] - (b*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(1 + c*x))/(
(I*c*Sqrt[f] + Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/Sqrt[g] - (b*e*Log[1 - c^2*x^2])/c + x*(a + b*ArcCoth[c*x])
*(d + e*Log[f + g*x^2]) + (b*Log[(g*(1 - c^2*x^2))/(c^2*f + g)]*(d + e*Log[f + g*x^2]))/(2*c) + (b*e*PolyLog[2
, (c^2*(f + g*x^2))/(c^2*f + g)])/(2*c) - ((I/2)*b*e*Sqrt[f]*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(1 - c*x))/((I*
c*Sqrt[f] - Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/Sqrt[g] + ((I/2)*b*e*Sqrt[f]*PolyLog[2, 1 - (2*Sqrt[f]*Sqrt[g]
*(1 + c*x))/((I*c*Sqrt[f] + Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/Sqrt[g]

Rule 6074

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

Rule 2475

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q,
x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && IntegerQ[r] && IntegerQ[s/n] && Intege
rQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0])

Rule 2394

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

Rule 2393

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

Rule 2391

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

Rule 5981

Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2
/e, Int[(f*x)^(m - 2)*(a + b*ArcCoth[c*x])^p, x], x] - Dist[(d*f^2)/e, Int[((f*x)^(m - 2)*(a + b*ArcCoth[c*x])
^p)/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]

Rule 5911

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*ArcCoth[c*x])^p, x] - Dist[b*c*p, In
t[(x*(a + b*ArcCoth[c*x])^(p - 1))/(1 - c^2*x^2), x], x] /; FreeQ[{a, b, c}, x] && IGtQ[p, 0]

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 5975

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

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 5973

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

Rule 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rule 4876

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Int[Ex
pandIntegrand[(a + b*ArcTan[c*x])^p, (f*x)^m*(d + e*x)^q, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[p,
0] && IntegerQ[q] && (GtQ[q, 0] || NeQ[a, 0] || IntegerQ[m])

Rule 4848

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

Rule 4856

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])*Log[2/(1 -
I*c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d
+ e*x))/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x], x] + Simp[((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d
+ I*e)*(1 - I*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 \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx &=x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-(b c) \int \frac{x \left (d+e \log \left (f+g x^2\right )\right )}{1-c^2 x^2} \, dx-(2 e g) \int \frac{x^2 \left (a+b \coth ^{-1}(c x)\right )}{f+g x^2} \, dx\\ &=x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{d+e \log (f+g x)}{1-c^2 x} \, dx,x,x^2\right )-(2 e) \int \left (a+b \coth ^{-1}(c x)\right ) \, dx+(2 e f) \int \frac{a+b \coth ^{-1}(c x)}{f+g x^2} \, dx\\ &=-2 a e x+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b \log \left (\frac{g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}-(2 b e) \int \coth ^{-1}(c x) \, dx+(2 a e f) \int \frac{1}{f+g x^2} \, dx+(2 b e f) \int \frac{\coth ^{-1}(c x)}{f+g x^2} \, dx-\frac{(b e g) \operatorname{Subst}\left (\int \frac{\log \left (\frac{g \left (1-c^2 x\right )}{c^2 f+g}\right )}{f+g x} \, dx,x,x^2\right )}{2 c}\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac{2 a e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b \log \left (\frac{g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}-\frac{(b e) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{c^2 x}{c^2 f+g}\right )}{x} \, dx,x,f+g x^2\right )}{2 c}+(2 b c e) \int \frac{x}{1-c^2 x^2} \, dx-(b e f) \int \frac{\log \left (1-\frac{1}{c x}\right )}{f+g x^2} \, dx+(b e f) \int \frac{\log \left (1+\frac{1}{c x}\right )}{f+g x^2} \, dx\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac{2 a e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1-\frac{1}{c x}\right )}{\sqrt{g}}+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1+\frac{1}{c x}\right )}{\sqrt{g}}-\frac{b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b \log \left (\frac{g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac{b e \text{Li}_2\left (\frac{c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}+\frac{(b e f) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g} \left (1-\frac{1}{c x}\right ) x^2} \, dx}{c}+\frac{(b e f) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g} \left (1+\frac{1}{c x}\right ) x^2} \, dx}{c}\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac{2 a e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1-\frac{1}{c x}\right )}{\sqrt{g}}+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1+\frac{1}{c x}\right )}{\sqrt{g}}-\frac{b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b \log \left (\frac{g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac{b e \text{Li}_2\left (\frac{c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}+\frac{\left (b e \sqrt{f}\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\left (1-\frac{1}{c x}\right ) x^2} \, dx}{c \sqrt{g}}+\frac{\left (b e \sqrt{f}\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\left (1+\frac{1}{c x}\right ) x^2} \, dx}{c \sqrt{g}}\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac{2 a e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1-\frac{1}{c x}\right )}{\sqrt{g}}+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1+\frac{1}{c x}\right )}{\sqrt{g}}-\frac{b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b \log \left (\frac{g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac{b e \text{Li}_2\left (\frac{c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}+\frac{\left (b e \sqrt{f}\right ) \int \frac{c \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{x (-1+c x)} \, dx}{c \sqrt{g}}+\frac{\left (b e \sqrt{f}\right ) \int \frac{c \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{x (1+c x)} \, dx}{c \sqrt{g}}\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac{2 a e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1-\frac{1}{c x}\right )}{\sqrt{g}}+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1+\frac{1}{c x}\right )}{\sqrt{g}}-\frac{b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b \log \left (\frac{g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac{b e \text{Li}_2\left (\frac{c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}+\frac{\left (b e \sqrt{f}\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{x (-1+c x)} \, dx}{\sqrt{g}}+\frac{\left (b e \sqrt{f}\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{x (1+c x)} \, dx}{\sqrt{g}}\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac{2 a e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1-\frac{1}{c x}\right )}{\sqrt{g}}+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1+\frac{1}{c x}\right )}{\sqrt{g}}-\frac{b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b \log \left (\frac{g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac{b e \text{Li}_2\left (\frac{c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}+\frac{\left (b e \sqrt{f}\right ) \int \left (-\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{x}+\frac{c \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{-1+c x}\right ) \, dx}{\sqrt{g}}+\frac{\left (b e \sqrt{f}\right ) \int \left (\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{x}-\frac{c \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{1+c x}\right ) \, dx}{\sqrt{g}}\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac{2 a e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1-\frac{1}{c x}\right )}{\sqrt{g}}+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1+\frac{1}{c x}\right )}{\sqrt{g}}-\frac{b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b \log \left (\frac{g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac{b e \text{Li}_2\left (\frac{c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}+\frac{\left (b c e \sqrt{f}\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{-1+c x} \, dx}{\sqrt{g}}-\frac{\left (b c e \sqrt{f}\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{1+c x} \, dx}{\sqrt{g}}\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac{2 a e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1-\frac{1}{c x}\right )}{\sqrt{g}}+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1+\frac{1}{c x}\right )}{\sqrt{g}}+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} (1-c x)}{\left (i c \sqrt{f}-\sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{g}}-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} (1+c x)}{\left (i c \sqrt{f}+\sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{g}}-\frac{b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b \log \left (\frac{g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac{b e \text{Li}_2\left (\frac{c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}-(b e) \int \frac{\log \left (\frac{2 \sqrt{g} (-1+c x)}{\sqrt{f} \left (i c-\frac{\sqrt{g}}{\sqrt{f}}\right ) \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}\right )}{1+\frac{g x^2}{f}} \, dx+(b e) \int \frac{\log \left (\frac{2 \sqrt{g} (1+c x)}{\sqrt{f} \left (i c+\frac{\sqrt{g}}{\sqrt{f}}\right ) \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}\right )}{1+\frac{g x^2}{f}} \, dx\\ &=-2 a e x-2 b e x \coth ^{-1}(c x)+\frac{2 a e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1-\frac{1}{c x}\right )}{\sqrt{g}}+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (1+\frac{1}{c x}\right )}{\sqrt{g}}+\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} (1-c x)}{\left (i c \sqrt{f}-\sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{g}}-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} (1+c x)}{\left (i c \sqrt{f}+\sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{\sqrt{g}}-\frac{b e \log \left (1-c^2 x^2\right )}{c}+x \left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{b \log \left (\frac{g \left (1-c^2 x^2\right )}{c^2 f+g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac{b e \text{Li}_2\left (\frac{c^2 \left (f+g x^2\right )}{c^2 f+g}\right )}{2 c}-\frac{i b e \sqrt{f} \text{Li}_2\left (1+\frac{2 \sqrt{f} \sqrt{g} (1-c x)}{\left (i c \sqrt{f}-\sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{g}}+\frac{i b e \sqrt{f} \text{Li}_2\left (1-\frac{2 \sqrt{f} \sqrt{g} (1+c x)}{\left (i c \sqrt{f}+\sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 \sqrt{g}}\\ \end{align*}

Mathematica [B]  time = 3.21188, size = 1287, normalized size = 2.36 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

a*d*x - 2*a*e*x + b*d*x*ArcCoth[c*x] + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] + (b*d*Log[1 - c^2*
x^2])/(2*c) + a*e*x*Log[f + g*x^2] + b*e*(x*ArcCoth[c*x] + Log[1 - c^2*x^2]/(2*c))*Log[f + g*x^2] + (b*e*(-4*c
*x*ArcCoth[c*x] + 4*Log[1/(c*Sqrt[1 - 1/(c^2*x^2)]*x)] + (Sqrt[c^2*f*g]*((-2*I)*ArcCos[(c^2*f - g)/(c^2*f + g)
]*ArcTan[Sqrt[c^2*f*g]/(c*g*x)] + 4*ArcCoth[c*x]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] - (ArcCos[(c^2*f - g)/(c^2*f +
g)] + 2*ArcTan[Sqrt[c^2*f*g]/(c*g*x)])*Log[((2*I)*g*(I*c^2*f + Sqrt[c^2*f*g])*(-1 + 1/(c*x)))/((c^2*f + g)*(g
+ (I*Sqrt[c^2*f*g])/(c*x)))] - (ArcCos[(c^2*f - g)/(c^2*f + g)] - 2*ArcTan[Sqrt[c^2*f*g]/(c*g*x)])*Log[(2*g*(c
^2*f + I*Sqrt[c^2*f*g])*(1 + 1/(c*x)))/((c^2*f + g)*(g + (I*Sqrt[c^2*f*g])/(c*x)))] + (ArcCos[(c^2*f - g)/(c^2
*f + g)] + 2*(ArcTan[Sqrt[c^2*f*g]/(c*g*x)] + ArcTan[(c*g*x)/Sqrt[c^2*f*g]]))*Log[(Sqrt[2]*Sqrt[c^2*f*g])/(E^A
rcCoth[c*x]*Sqrt[c^2*f + g]*Sqrt[-(c^2*f) + g + (c^2*f + g)*Cosh[2*ArcCoth[c*x]]])] + (ArcCos[(c^2*f - g)/(c^2
*f + g)] - 2*(ArcTan[Sqrt[c^2*f*g]/(c*g*x)] + ArcTan[(c*g*x)/Sqrt[c^2*f*g]]))*Log[(Sqrt[2]*E^ArcCoth[c*x]*Sqrt
[c^2*f*g])/(Sqrt[c^2*f + g]*Sqrt[-(c^2*f) + g + (c^2*f + g)*Cosh[2*ArcCoth[c*x]]])] + I*(-PolyLog[2, ((-(c^2*f
) + g + (2*I)*Sqrt[c^2*f*g])*(g - (I*Sqrt[c^2*f*g])/(c*x)))/((c^2*f + g)*(g + (I*Sqrt[c^2*f*g])/(c*x)))] + Pol
yLog[2, ((c^2*f - g + (2*I)*Sqrt[c^2*f*g])*(I*g + Sqrt[c^2*f*g]/(c*x)))/((c^2*f + g)*((-I)*g + Sqrt[c^2*f*g]/(
c*x)))])))/g))/(2*c) - (b*e*g*(((-Log[-c^(-1) + x] - Log[c^(-1) + x] + Log[1 - c^2*x^2])*Log[f + g*x^2])/(2*g)
+ (Log[-c^(-1) + x]*Log[1 - (Sqrt[g]*(-c^(-1) + x))/((-I)*Sqrt[f] - Sqrt[g]/c)] + PolyLog[2, (Sqrt[g]*(-c^(-1
) + x))/((-I)*Sqrt[f] - Sqrt[g]/c)])/(2*g) + (Log[-c^(-1) + x]*Log[1 - (Sqrt[g]*(-c^(-1) + x))/(I*Sqrt[f] - Sq
rt[g]/c)] + PolyLog[2, (Sqrt[g]*(-c^(-1) + x))/(I*Sqrt[f] - Sqrt[g]/c)])/(2*g) + (Log[c^(-1) + x]*Log[1 - (Sqr
t[g]*(c^(-1) + x))/((-I)*Sqrt[f] + Sqrt[g]/c)] + PolyLog[2, (Sqrt[g]*(c^(-1) + x))/((-I)*Sqrt[f] + Sqrt[g]/c)]
)/(2*g) + (Log[c^(-1) + x]*Log[1 - (Sqrt[g]*(c^(-1) + x))/(I*Sqrt[f] + Sqrt[g]/c)] + PolyLog[2, (Sqrt[g]*(c^(-
1) + x))/(I*Sqrt[f] + Sqrt[g]/c)])/(2*g)))/c

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Maple [F]  time = 1.994, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{\rm arccoth} \left (cx\right ) \right ) \left ( d+e\ln \left ( g{x}^{2}+f \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*arccoth(c*x))*(d+e*log(g*x^2+f)),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 (b d \operatorname{arcoth}\left (c x\right ) + a d +{\left (b e \operatorname{arcoth}\left (c x\right ) + a e\right )} \log \left (g x^{2} + f\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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*acoth(c*x))*(d+e*ln(g*x**2+f)),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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