3.1298 \(\int (a+b \tan ^{-1}(c x)) (d+e \log (f+g x^2)) \, dx\)

Optimal. Leaf size=656 \[ -\frac{b e \text{PolyLog}\left (2,\frac{c^2 \left (f+g x^2\right )}{c^2 f-g}\right )}{2 c}-\frac{i b e \sqrt{-f} \text{PolyLog}\left (2,\frac{\sqrt{g} (-c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{g}}+\frac{i b e \sqrt{-f} \text{PolyLog}\left (2,\frac{\sqrt{g} (1-i c x)}{\sqrt{g}+i c \sqrt{-f}}\right )}{2 \sqrt{g}}+\frac{i b e \sqrt{-f} \text{PolyLog}\left (2,\frac{\sqrt{g} (1+i c x)}{\sqrt{g}+i c \sqrt{-f}}\right )}{2 \sqrt{g}}-\frac{i b e \sqrt{-f} \text{PolyLog}\left (2,\frac{\sqrt{g} (c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{g}}+x \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{2 a e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}-2 a e x-\frac{b \log \left (-\frac{g \left (c^2 x^2+1\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac{b e \log \left (c^2 x^2+1\right )}{c}+\frac{i b e \sqrt{-f} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )}{2 \sqrt{g}}-\frac{i b e \sqrt{-f} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{g}}+\frac{i b e \sqrt{-f} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )}{2 \sqrt{g}}-\frac{i b e \sqrt{-f} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{g}}-2 b e x \tan ^{-1}(c x) \]

[Out]

-2*a*e*x - 2*b*e*x*ArcTan[c*x] + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] + ((I/2)*b*e*Sqrt[-f]*Log
[1 + I*c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])])/Sqrt[g] - ((I/2)*b*e*Sqrt[-f]*Log[1 - I*
c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[g] + ((I/2)*b*e*Sqrt[-f]*Log[1 - I*c*x]*Lo
g[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])])/Sqrt[g] - ((I/2)*b*e*Sqrt[-f]*Log[1 + I*c*x]*Log[(c*(S
qrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[g] + (b*e*Log[1 + c^2*x^2])/c + x*(a + b*ArcTan[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) - ((I/2)*b*e*Sqr
t[-f]*PolyLog[2, (Sqrt[g]*(I - c*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[g] + ((I/2)*b*e*Sqrt[-f]*PolyLog[2, (Sqrt
[g]*(1 - I*c*x))/(I*c*Sqrt[-f] + Sqrt[g])])/Sqrt[g] + ((I/2)*b*e*Sqrt[-f]*PolyLog[2, (Sqrt[g]*(1 + I*c*x))/(I*
c*Sqrt[-f] + Sqrt[g])])/Sqrt[g] - ((I/2)*b*e*Sqrt[-f]*PolyLog[2, (Sqrt[g]*(I + c*x))/(c*Sqrt[-f] + I*Sqrt[g])]
)/Sqrt[g] - (b*e*PolyLog[2, (c^2*(f + g*x^2))/(c^2*f - g)])/(2*c)

________________________________________________________________________________________

Rubi [A]  time = 0.830916, antiderivative size = 656, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {5009, 2475, 2394, 2393, 2391, 4916, 4846, 260, 4910, 205, 4908, 2409} \[ -\frac{b e \text{PolyLog}\left (2,\frac{c^2 \left (f+g x^2\right )}{c^2 f-g}\right )}{2 c}-\frac{i b e \sqrt{-f} \text{PolyLog}\left (2,\frac{\sqrt{g} (-c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{g}}+\frac{i b e \sqrt{-f} \text{PolyLog}\left (2,\frac{\sqrt{g} (1-i c x)}{\sqrt{g}+i c \sqrt{-f}}\right )}{2 \sqrt{g}}+\frac{i b e \sqrt{-f} \text{PolyLog}\left (2,\frac{\sqrt{g} (1+i c x)}{\sqrt{g}+i c \sqrt{-f}}\right )}{2 \sqrt{g}}-\frac{i b e \sqrt{-f} \text{PolyLog}\left (2,\frac{\sqrt{g} (c x+i)}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{g}}+x \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )+\frac{2 a e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{g}}-2 a e x-\frac{b \log \left (-\frac{g \left (c^2 x^2+1\right )}{c^2 f-g}\right ) \left (d+e \log \left (f+g x^2\right )\right )}{2 c}+\frac{b e \log \left (c^2 x^2+1\right )}{c}+\frac{i b e \sqrt{-f} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )}{2 \sqrt{g}}-\frac{i b e \sqrt{-f} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}-\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{g}}+\frac{i b e \sqrt{-f} \log (1-i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}-i \sqrt{g}}\right )}{2 \sqrt{g}}-\frac{i b e \sqrt{-f} \log (1+i c x) \log \left (\frac{c \left (\sqrt{-f}+\sqrt{g} x\right )}{c \sqrt{-f}+i \sqrt{g}}\right )}{2 \sqrt{g}}-2 b e x \tan ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*a*e*x - 2*b*e*x*ArcTan[c*x] + (2*a*e*Sqrt[f]*ArcTan[(Sqrt[g]*x)/Sqrt[f]])/Sqrt[g] + ((I/2)*b*e*Sqrt[-f]*Log
[1 + I*c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])])/Sqrt[g] - ((I/2)*b*e*Sqrt[-f]*Log[1 - I*
c*x]*Log[(c*(Sqrt[-f] - Sqrt[g]*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[g] + ((I/2)*b*e*Sqrt[-f]*Log[1 - I*c*x]*Lo
g[(c*(Sqrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] - I*Sqrt[g])])/Sqrt[g] - ((I/2)*b*e*Sqrt[-f]*Log[1 + I*c*x]*Log[(c*(S
qrt[-f] + Sqrt[g]*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[g] + (b*e*Log[1 + c^2*x^2])/c + x*(a + b*ArcTan[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) - ((I/2)*b*e*Sqr
t[-f]*PolyLog[2, (Sqrt[g]*(I - c*x))/(c*Sqrt[-f] + I*Sqrt[g])])/Sqrt[g] + ((I/2)*b*e*Sqrt[-f]*PolyLog[2, (Sqrt
[g]*(1 - I*c*x))/(I*c*Sqrt[-f] + Sqrt[g])])/Sqrt[g] + ((I/2)*b*e*Sqrt[-f]*PolyLog[2, (Sqrt[g]*(1 + I*c*x))/(I*
c*Sqrt[-f] + Sqrt[g])])/Sqrt[g] - ((I/2)*b*e*Sqrt[-f]*PolyLog[2, (Sqrt[g]*(I + c*x))/(c*Sqrt[-f] + I*Sqrt[g])]
)/Sqrt[g] - (b*e*PolyLog[2, (c^2*(f + g*x^2))/(c^2*f - g)])/(2*c)

Rule 5009

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*(e_.)), x_Symbol] :> Simp[x*(d + e*L
og[f + g*x^2])*(a + b*ArcTan[c*x]), x] + (-Dist[b*c, Int[(x*(d + e*Log[f + g*x^2]))/(1 + c^2*x^2), x], x] - Di
st[2*e*g, Int[(x^2*(a + b*ArcTan[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 4916

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

Rule 4846

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

Mathematica [B]  time = 3.48472, size = 1362, normalized size = 2.08 \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 3.373, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\arctan \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*arctan(c*x))*(d+e*ln(g*x^2+f)),x)

[Out]

int((a+b*arctan(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*arctan(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 \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\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, algorithm="fricas")

[Out]

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

[Out]

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