3.1300 \(\int \frac{(a+b \tan ^{-1}(c x)) (d+e \log (f+g x^2))}{x^2} \, dx\)

Optimal. Leaf size=672 \[ -\frac{1}{2} b c e \text{PolyLog}\left (2,\frac{c^2 \left (f+g x^2\right )}{c^2 f-g}\right )+\frac{1}{2} b c e \text{PolyLog}\left (2,\frac{g x^2}{f}+1\right )+\frac{i b e \sqrt{g} \text{PolyLog}\left (2,\frac{\sqrt{g} (-c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \text{PolyLog}\left (2,\frac{\sqrt{g} (1-i c x)}{\sqrt{g}+i c \sqrt{-f}}\right )}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \text{PolyLog}\left (2,\frac{\sqrt{g} (1+i c x)}{\sqrt{g}+i c \sqrt{-f}}\right )}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \text{PolyLog}\left (2,\frac{\sqrt{g} (c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac{2 a e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}-\frac{1}{2} b c \log \left (-\frac{g \left (c^2 x^2+1\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{1}{2} b c \log \left (-\frac{g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac{i b e \sqrt{g} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.756769, antiderivative size = 672, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 14, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {5017, 2475, 36, 29, 31, 2416, 2394, 2315, 2393, 2391, 4910, 205, 4908, 2409} \[ -\frac{1}{2} b c e \text{PolyLog}\left (2,\frac{c^2 \left (f+g x^2\right )}{c^2 f-g}\right )+\frac{1}{2} b c e \text{PolyLog}\left (2,\frac{g x^2}{f}+1\right )+\frac{i b e \sqrt{g} \text{PolyLog}\left (2,\frac{\sqrt{g} (-c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \text{PolyLog}\left (2,\frac{\sqrt{g} (1-i c x)}{\sqrt{g}+i c \sqrt{-f}}\right )}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \text{PolyLog}\left (2,\frac{\sqrt{g} (1+i c x)}{\sqrt{g}+i c \sqrt{-f}}\right )}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \text{PolyLog}\left (2,\frac{\sqrt{g} (c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac{2 a e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}-\frac{1}{2} b c \log \left (-\frac{g \left (c^2 x^2+1\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{1}{2} b c \log \left (-\frac{g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac{i b e \sqrt{g} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5017

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

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 36

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

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 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 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 4910

Int[(ArcTan[(c_.)*(x_)]*(b_.) + (a_))/((d_.) + (e_.)*(x_)^2), x_Symbol] :> Dist[a, Int[1/(d + e*x^2), x], x] +
Dist[b, Int[ArcTan[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 4908

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

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rubi steps

\begin{align*} \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x^2} \, dx &=-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+(b c) \int \frac{d+e \log \left (f+g x^2\right )}{x \left (1+c^2 x^2\right )} \, dx+(2 e g) \int \frac{a+b \tan ^{-1}(c x)}{f+g x^2} \, dx\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{d+e \log (f+g x)}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )+(2 a e g) \int \frac{1}{f+g x^2} \, dx+(2 b e g) \int \frac{\tan ^{-1}(c x)}{f+g x^2} \, dx\\ &=\frac{2 a e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac{1}{2} (b c) \operatorname{Subst}\left (\int \left (\frac{d+e \log (f+g x)}{x}-\frac{c^2 (d+e \log (f+g x))}{1+c^2 x}\right ) \, dx,x,x^2\right )+(i b e g) \int \frac{\log (1-i c x)}{f+g x^2} \, dx-(i b e g) \int \frac{\log (1+i c x)}{f+g x^2} \, dx\\ &=\frac{2 a e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{d+e \log (f+g x)}{x} \, dx,x,x^2\right )-\frac{1}{2} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{d+e \log (f+g x)}{1+c^2 x} \, dx,x,x^2\right )+(i b e g) \int \left (\frac{\sqrt{-f} \log (1-i c x)}{2 f \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\sqrt{-f} \log (1-i c x)}{2 f \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx-(i b e g) \int \left (\frac{\sqrt{-f} \log (1+i c x)}{2 f \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\sqrt{-f} \log (1+i c x)}{2 f \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx\\ &=\frac{2 a e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac{1}{2} b c \log \left (-\frac{g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac{1}{2} b c \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 )-\frac{1}{2} (b c e g) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{g x}{f}\right )}{f+g x} \, dx,x,x^2\right )+\frac{1}{2} (b c 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 )-\frac{(i b e g) \int \frac{\log (1-i c x)}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 \sqrt{-f}}-\frac{(i b e g) \int \frac{\log (1-i c x)}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 \sqrt{-f}}+\frac{(i b e g) \int \frac{\log (1+i c x)}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 \sqrt{-f}}+\frac{(i b e g) \int \frac{\log (1+i c x)}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 \sqrt{-f}}\\ &=\frac{2 a e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}-\frac{i b e \sqrt{g} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac{1}{2} b c \log \left (-\frac{g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac{1}{2} b c \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 )+\frac{1}{2} b c e \text{Li}_2\left (1+\frac{g x^2}{f}\right )+\frac{1}{2} (b c 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 )-\frac{\left (b c e \sqrt{g}\right ) \int \frac{\log \left (-\frac{i c \left (\sqrt{-f}-\sqrt{g} x\right )}{-i c \sqrt{-f}+\sqrt{g}}\right )}{1-i c x} \, dx}{2 \sqrt{-f}}-\frac{\left (b c e \sqrt{g}\right ) \int \frac{\log \left (\frac{i c \left (\sqrt{-f}-\sqrt{g} x\right )}{i c \sqrt{-f}+\sqrt{g}}\right )}{1+i c x} \, dx}{2 \sqrt{-f}}+\frac{\left (b c e \sqrt{g}\right ) \int \frac{\log \left (-\frac{i c \left (\sqrt{-f}+\sqrt{g} x\right )}{-i c \sqrt{-f}-\sqrt{g}}\right )}{1-i c x} \, dx}{2 \sqrt{-f}}+\frac{\left (b c e \sqrt{g}\right ) \int \frac{\log \left (\frac{i c \left (\sqrt{-f}+\sqrt{g} x\right )}{i c \sqrt{-f}-\sqrt{g}}\right )}{1+i c x} \, dx}{2 \sqrt{-f}}\\ &=\frac{2 a e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}-\frac{i b e \sqrt{g} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac{1}{2} b c \log \left (-\frac{g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac{1}{2} b c \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 )-\frac{1}{2} b c e \text{Li}_2\left (\frac{c^2 \left (f+g x^2\right )}{c^2 f-g}\right )+\frac{1}{2} b c e \text{Li}_2\left (1+\frac{g x^2}{f}\right )+\frac{\left (i b e \sqrt{g}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{g} x}{-i c \sqrt{-f}-\sqrt{g}}\right )}{x} \, dx,x,1-i c x\right )}{2 \sqrt{-f}}-\frac{\left (i b e \sqrt{g}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{g} x}{i c \sqrt{-f}-\sqrt{g}}\right )}{x} \, dx,x,1+i c x\right )}{2 \sqrt{-f}}-\frac{\left (i b e \sqrt{g}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{g} x}{-i c \sqrt{-f}+\sqrt{g}}\right )}{x} \, dx,x,1-i c x\right )}{2 \sqrt{-f}}+\frac{\left (i b e \sqrt{g}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{g} x}{i c \sqrt{-f}+\sqrt{g}}\right )}{x} \, dx,x,1+i c x\right )}{2 \sqrt{-f}}\\ &=\frac{2 a e \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f}}-\frac{i b e \sqrt{g} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{\left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}+\frac{1}{2} b c \log \left (-\frac{g x^2}{f}\right ) \left (d+e \log \left (f+g x^2\right )\right )-\frac{1}{2} b c \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 )+\frac{i b e \sqrt{g} \text{Li}_2\left (\frac{\sqrt{g} (i-c x)}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \text{Li}_2\left (\frac{\sqrt{g} (1-i c x)}{i c \sqrt{-f}+\sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{i b e \sqrt{g} \text{Li}_2\left (\frac{\sqrt{g} (1+i c x)}{i c \sqrt{-f}+\sqrt{g}}\right )}{2 \sqrt{-f}}+\frac{i b e \sqrt{g} \text{Li}_2\left (\frac{\sqrt{g} (i+c x)}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{-f}}-\frac{1}{2} b c e \text{Li}_2\left (\frac{c^2 \left (f+g x^2\right )}{c^2 f-g}\right )+\frac{1}{2} b c e \text{Li}_2\left (1+\frac{g x^2}{f}\right )\\ \end{align*}

Mathematica [A]  time = 0.902962, size = 552, normalized size = 0.82 \[ \frac{1}{2} \left (\frac{e \sqrt{g} \left (i b \sqrt{f} \left (\text{PolyLog}\left (2,\frac{\sqrt{g} (-c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right )+\log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )\right )-i b \sqrt{f} \left (\text{PolyLog}\left (2,\frac{\sqrt{g} (1-i c x)}{\sqrt{g}+i c \sqrt{-f}}\right )+\log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )\right )-i b \sqrt{f} \left (\text{PolyLog}\left (2,\frac{\sqrt{g} (1+i c x)}{\sqrt{g}+i c \sqrt{-f}}\right )+\log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )\right )+i b \sqrt{f} \left (\text{PolyLog}\left (2,\frac{\sqrt{g} (c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right )+\log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )\right )+4 a \sqrt{-f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )\right )}{\sqrt{-f^2}}+b c \left (-e \text{PolyLog}\left (2,\frac{c^2 \left (f+g x^2\right )}{c^2 f-g}\right )+e \text{PolyLog}\left (2,\frac{g x^2}{f}+1\right )+\left (\log \left (-\frac{g x^2}{f}\right )-\log \left (-\frac{g \left (c^2 x^2+1\right )}{c^2 f-g}\right )\right ) \left (d+e \log \left (f+g x^2\right )\right )\right )-\frac{2 \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x}\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 1.717, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arctan \left ( cx \right ) \right ) \left ( d+e\ln \left ( g{x}^{2}+f \right ) \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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*arctan(c*x))*(d+e*log(g*x^2+f))/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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