3.1297 \(\int x (a+b \tan ^{-1}(c x)) (d+e \log (f+g x^2)) \, dx\)

Optimal. Leaf size=562 \[ \frac{i b e \left (c^2 f-g\right ) \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 c^2 g}-\frac{i b e \left (c^2 f-g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}-i \sqrt{g}\right )}\right )}{4 c^2 g}-\frac{i b e \left (c^2 f-g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}+i \sqrt{g}\right )}\right )}{4 c^2 g}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{e \left (f+g x^2\right ) \log \left (f+g x^2\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 g}-\frac{1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{b (d-e) \tan ^{-1}(c x)}{2 c^2}-\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}-\frac{b e \left (c^2 f-g\right ) \log \left (\frac{2}{1-i c x}\right ) \tan ^{-1}(c x)}{c^2 g}+\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}-i \sqrt{g}\right )}\right )}{2 c^2 g}+\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}+i \sqrt{g}\right )}\right )}{2 c^2 g}-\frac{b x (d-e)}{2 c}-\frac{b e x \log \left (f+g x^2\right )}{2 c}-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{c \sqrt{g}}+\frac{b e x}{c} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.711949, antiderivative size = 562, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 16, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.727, Rules used = {2454, 2389, 2295, 5019, 321, 203, 2528, 2448, 205, 2470, 12, 4928, 4856, 2402, 2315, 2447} \[ \frac{i b e \left (c^2 f-g\right ) \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right )}{2 c^2 g}-\frac{i b e \left (c^2 f-g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}-i \sqrt{g}\right )}\right )}{4 c^2 g}-\frac{i b e \left (c^2 f-g\right ) \text{PolyLog}\left (2,1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}+i \sqrt{g}\right )}\right )}{4 c^2 g}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{e \left (f+g x^2\right ) \log \left (f+g x^2\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 g}-\frac{1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{b (d-e) \tan ^{-1}(c x)}{2 c^2}-\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(c x) \log \left (f+g x^2\right )}{2 c^2 g}-\frac{b e \left (c^2 f-g\right ) \log \left (\frac{2}{1-i c x}\right ) \tan ^{-1}(c x)}{c^2 g}+\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}-i \sqrt{g}\right )}\right )}{2 c^2 g}+\frac{b e \left (c^2 f-g\right ) \tan ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{(1-i c x) \left (c \sqrt{-f}+i \sqrt{g}\right )}\right )}{2 c^2 g}-\frac{b x (d-e)}{2 c}-\frac{b e x \log \left (f+g x^2\right )}{2 c}-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{c \sqrt{g}}+\frac{b e x}{c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2454

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 2389

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

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 5019

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

Rule 321

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

Rule 203

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

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, 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 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 4928

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTan[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 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 x \left (a+b \tan ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right ) \, dx &=\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{e \left (f+g x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-(b c) \int \left (\frac{(d-e) x^2}{2 \left (1+c^2 x^2\right )}+\frac{e \left (f+g x^2\right ) \log \left (f+g x^2\right )}{2 g \left (1+c^2 x^2\right )}\right ) \, dx\\ &=\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{e \left (f+g x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-\frac{1}{2} (b c (d-e)) \int \frac{x^2}{1+c^2 x^2} \, dx-\frac{(b c e) \int \frac{\left (f+g x^2\right ) \log \left (f+g x^2\right )}{1+c^2 x^2} \, dx}{2 g}\\ &=-\frac{b (d-e) x}{2 c}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{e \left (f+g x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}+\frac{(b (d-e)) \int \frac{1}{1+c^2 x^2} \, dx}{2 c}-\frac{(b c e) \int \left (\frac{g \log \left (f+g x^2\right )}{c^2}+\frac{\left (c^2 f-g\right ) \log \left (f+g x^2\right )}{c^2 \left (1+c^2 x^2\right )}\right ) \, dx}{2 g}\\ &=-\frac{b (d-e) x}{2 c}+\frac{b (d-e) \tan ^{-1}(c x)}{2 c^2}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{e \left (f+g x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-\frac{(b e) \int \log \left (f+g x^2\right ) \, dx}{2 c}-\frac{\left (b c e \left (f-\frac{g}{c^2}\right )\right ) \int \frac{\log \left (f+g x^2\right )}{1+c^2 x^2} \, dx}{2 g}\\ &=-\frac{b (d-e) x}{2 c}+\frac{b (d-e) \tan ^{-1}(c x)}{2 c^2}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{b e x \log \left (f+g x^2\right )}{2 c}-\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (f+g x^2\right )}{2 g}+\frac{e \left (f+g x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}+\frac{(b e g) \int \frac{x^2}{f+g x^2} \, dx}{c}+\left (b c e \left (f-\frac{g}{c^2}\right )\right ) \int \frac{x \tan ^{-1}(c x)}{c \left (f+g x^2\right )} \, dx\\ &=-\frac{b (d-e) x}{2 c}+\frac{b e x}{c}+\frac{b (d-e) \tan ^{-1}(c x)}{2 c^2}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{b e x \log \left (f+g x^2\right )}{2 c}-\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (f+g x^2\right )}{2 g}+\frac{e \left (f+g x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-\frac{(b e f) \int \frac{1}{f+g x^2} \, dx}{c}+\left (b e \left (f-\frac{g}{c^2}\right )\right ) \int \frac{x \tan ^{-1}(c x)}{f+g x^2} \, dx\\ &=-\frac{b (d-e) x}{2 c}+\frac{b e x}{c}+\frac{b (d-e) \tan ^{-1}(c x)}{2 c^2}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{c \sqrt{g}}-\frac{b e x \log \left (f+g x^2\right )}{2 c}-\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (f+g x^2\right )}{2 g}+\frac{e \left (f+g x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}+\left (b e \left (f-\frac{g}{c^2}\right )\right ) \int \left (-\frac{\tan ^{-1}(c x)}{2 \sqrt{g} \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\tan ^{-1}(c x)}{2 \sqrt{g} \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx\\ &=-\frac{b (d-e) x}{2 c}+\frac{b e x}{c}+\frac{b (d-e) \tan ^{-1}(c x)}{2 c^2}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{c \sqrt{g}}-\frac{b e x \log \left (f+g x^2\right )}{2 c}-\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (f+g x^2\right )}{2 g}+\frac{e \left (f+g x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-\frac{\left (b e \left (f-\frac{g}{c^2}\right )\right ) \int \frac{\tan ^{-1}(c x)}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 \sqrt{g}}+\frac{\left (b e \left (f-\frac{g}{c^2}\right )\right ) \int \frac{\tan ^{-1}(c x)}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 \sqrt{g}}\\ &=-\frac{b (d-e) x}{2 c}+\frac{b e x}{c}+\frac{b (d-e) \tan ^{-1}(c x)}{2 c^2}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{c \sqrt{g}}-\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (\frac{2}{1-i c x}\right )}{g}+\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-i \sqrt{g}\right ) (1-i c x)}\right )}{2 g}+\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+i \sqrt{g}\right ) (1-i c x)}\right )}{2 g}-\frac{b e x \log \left (f+g x^2\right )}{2 c}-\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (f+g x^2\right )}{2 g}+\frac{e \left (f+g x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}+2 \frac{\left (b c e \left (f-\frac{g}{c^2}\right )\right ) \int \frac{\log \left (\frac{2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{2 g}-\frac{\left (b c e \left (f-\frac{g}{c^2}\right )\right ) \int \frac{\log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-i \sqrt{g}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 g}-\frac{\left (b c e \left (f-\frac{g}{c^2}\right )\right ) \int \frac{\log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+i \sqrt{g}\right ) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{2 g}\\ &=-\frac{b (d-e) x}{2 c}+\frac{b e x}{c}+\frac{b (d-e) \tan ^{-1}(c x)}{2 c^2}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{c \sqrt{g}}-\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (\frac{2}{1-i c x}\right )}{g}+\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-i \sqrt{g}\right ) (1-i c x)}\right )}{2 g}+\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+i \sqrt{g}\right ) (1-i c x)}\right )}{2 g}-\frac{b e x \log \left (f+g x^2\right )}{2 c}-\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (f+g x^2\right )}{2 g}+\frac{e \left (f+g x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}-\frac{i b e \left (f-\frac{g}{c^2}\right ) \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-i \sqrt{g}\right ) (1-i c x)}\right )}{4 g}-\frac{i b e \left (f-\frac{g}{c^2}\right ) \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+i \sqrt{g}\right ) (1-i c x)}\right )}{4 g}+2 \frac{\left (i b e \left (f-\frac{g}{c^2}\right )\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i c x}\right )}{2 g}\\ &=-\frac{b (d-e) x}{2 c}+\frac{b e x}{c}+\frac{b (d-e) \tan ^{-1}(c x)}{2 c^2}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{2} e x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{b e \sqrt{f} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{c \sqrt{g}}-\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (\frac{2}{1-i c x}\right )}{g}+\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-i \sqrt{g}\right ) (1-i c x)}\right )}{2 g}+\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+i \sqrt{g}\right ) (1-i c x)}\right )}{2 g}-\frac{b e x \log \left (f+g x^2\right )}{2 c}-\frac{b e \left (f-\frac{g}{c^2}\right ) \tan ^{-1}(c x) \log \left (f+g x^2\right )}{2 g}+\frac{e \left (f+g x^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{2 g}+\frac{i b e \left (f-\frac{g}{c^2}\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{2 g}-\frac{i b e \left (f-\frac{g}{c^2}\right ) \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}-\sqrt{g} x\right )}{\left (c \sqrt{-f}-i \sqrt{g}\right ) (1-i c x)}\right )}{4 g}-\frac{i b e \left (f-\frac{g}{c^2}\right ) \text{Li}_2\left (1-\frac{2 c \left (\sqrt{-f}+\sqrt{g} x\right )}{\left (c \sqrt{-f}+i \sqrt{g}\right ) (1-i c x)}\right )}{4 g}\\ \end{align*}

Mathematica [B]  time = 6.82376, size = 1138, normalized size = 2.02 \[ \frac{2 a d g x^2 c^2-2 a e g x^2 c^2+2 b d g x^2 \tan ^{-1}(c x) c^2-2 b e g x^2 \tan ^{-1}(c x) c^2+4 i b e f \sin ^{-1}\left (\sqrt{\frac{c^2 f}{c^2 f-g}}\right ) \tan ^{-1}\left (\frac{c g x}{\sqrt{c^2 f g}}\right ) c^2-4 b e f \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right ) c^2+2 b e f \sin ^{-1}\left (\sqrt{\frac{c^2 f}{c^2 f-g}}\right ) \log \left (\frac{\left (1+e^{2 i \tan ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 i \tan ^{-1}(c x)}\right ) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right ) c^2+2 b e f \tan ^{-1}(c x) \log \left (\frac{\left (1+e^{2 i \tan ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 i \tan ^{-1}(c x)}\right ) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right ) c^2-2 b e f \sin ^{-1}\left (\sqrt{\frac{c^2 f}{c^2 f-g}}\right ) \log \left (\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g+2 \sqrt{c^2 f g}\right )}{c^2 f-g}+1\right ) c^2+2 b e f \tan ^{-1}(c x) \log \left (\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g+2 \sqrt{c^2 f g}\right )}{c^2 f-g}+1\right ) c^2+2 a e g x^2 \log \left (g x^2+f\right ) c^2+2 a e f \log \left (g x^2+f\right ) c^2+2 b e g x^2 \tan ^{-1}(c x) \log \left (g x^2+f\right ) c^2-i b e f \text{PolyLog}\left (2,-\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g+2 \sqrt{c^2 f g}\right )}{c^2 f-g}\right ) c^2-2 b d g x c+6 b e g x c-4 b e \sqrt{f} \sqrt{g} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) c-2 b e g x \log \left (g x^2+f\right ) c+2 b d g \tan ^{-1}(c x)-2 b e g \tan ^{-1}(c x)-4 i b e g \sin ^{-1}\left (\sqrt{\frac{c^2 f}{c^2 f-g}}\right ) \tan ^{-1}\left (\frac{c g x}{\sqrt{c^2 f g}}\right )+4 b e g \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-2 b e g \sin ^{-1}\left (\sqrt{\frac{c^2 f}{c^2 f-g}}\right ) \log \left (\frac{\left (1+e^{2 i \tan ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 i \tan ^{-1}(c x)}\right ) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right )-2 b e g \tan ^{-1}(c x) \log \left (\frac{\left (1+e^{2 i \tan ^{-1}(c x)}\right ) f c^2+\left (-1+e^{2 i \tan ^{-1}(c x)}\right ) g-2 e^{2 i \tan ^{-1}(c x)} \sqrt{c^2 f g}}{c^2 f-g}\right )+2 b e g \sin ^{-1}\left (\sqrt{\frac{c^2 f}{c^2 f-g}}\right ) \log \left (\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g+2 \sqrt{c^2 f g}\right )}{c^2 f-g}+1\right )-2 b e g \tan ^{-1}(c x) \log \left (\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g+2 \sqrt{c^2 f g}\right )}{c^2 f-g}+1\right )+2 b e g \tan ^{-1}(c x) \log \left (g x^2+f\right )+2 i b e \left (c^2 f-g\right ) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )-i b e \left (c^2 f-g\right ) \text{PolyLog}\left (2,-\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g-2 \sqrt{c^2 f g}\right )}{c^2 f-g}\right )+i b e g \text{PolyLog}\left (2,-\frac{e^{2 i \tan ^{-1}(c x)} \left (f c^2+g+2 \sqrt{c^2 f g}\right )}{c^2 f-g}\right )}{4 c^2 g} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*b*c*d*g*x + 6*b*c*e*g*x + 2*a*c^2*d*g*x^2 - 2*a*c^2*e*g*x^2 + 2*b*d*g*ArcTan[c*x] - 2*b*e*g*ArcTan[c*x] +
2*b*c^2*d*g*x^2*ArcTan[c*x] - 2*b*c^2*e*g*x^2*ArcTan[c*x] - 4*b*c*e*Sqrt[f]*Sqrt[g]*ArcTan[(Sqrt[g]*x)/Sqrt[f]
] + (4*I)*b*c^2*e*f*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] - (4*I)*b*e*g*ArcSin[Sqrt[
(c^2*f)/(c^2*f - g)]]*ArcTan[(c*g*x)/Sqrt[c^2*f*g]] - 4*b*c^2*e*f*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] +
4*b*e*g*ArcTan[c*x]*Log[1 + E^((2*I)*ArcTan[c*x])] + 2*b*c^2*e*f*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[(c^2*(
1 + E^((2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcTan[c*x]))*g - 2*E^((2*I)*ArcTan[c*x])*Sqrt[c^2*f*g])/(c^2*f
- g)] - 2*b*e*g*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[(c^2*(1 + E^((2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcT
an[c*x]))*g - 2*E^((2*I)*ArcTan[c*x])*Sqrt[c^2*f*g])/(c^2*f - g)] + 2*b*c^2*e*f*ArcTan[c*x]*Log[(c^2*(1 + E^((
2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcTan[c*x]))*g - 2*E^((2*I)*ArcTan[c*x])*Sqrt[c^2*f*g])/(c^2*f - g)] -
2*b*e*g*ArcTan[c*x]*Log[(c^2*(1 + E^((2*I)*ArcTan[c*x]))*f + (-1 + E^((2*I)*ArcTan[c*x]))*g - 2*E^((2*I)*ArcTa
n[c*x])*Sqrt[c^2*f*g])/(c^2*f - g)] - 2*b*c^2*e*f*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[1 + (E^((2*I)*ArcTan[c
*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] + 2*b*e*g*ArcSin[Sqrt[(c^2*f)/(c^2*f - g)]]*Log[1 + (E^((2*I)
*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] + 2*b*c^2*e*f*ArcTan[c*x]*Log[1 + (E^((2*I)*ArcTan[c
*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] - 2*b*e*g*ArcTan[c*x]*Log[1 + (E^((2*I)*ArcTan[c*x])*(c^2*f +
g + 2*Sqrt[c^2*f*g]))/(c^2*f - g)] + 2*a*c^2*e*f*Log[f + g*x^2] - 2*b*c*e*g*x*Log[f + g*x^2] + 2*a*c^2*e*g*x^
2*Log[f + g*x^2] + 2*b*e*g*ArcTan[c*x]*Log[f + g*x^2] + 2*b*c^2*e*g*x^2*ArcTan[c*x]*Log[f + g*x^2] + (2*I)*b*e
*(c^2*f - g)*PolyLog[2, -E^((2*I)*ArcTan[c*x])] - I*b*e*(c^2*f - g)*PolyLog[2, -((E^((2*I)*ArcTan[c*x])*(c^2*f
+ g - 2*Sqrt[c^2*f*g]))/(c^2*f - g))] - I*b*c^2*e*f*PolyLog[2, -((E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c
^2*f*g]))/(c^2*f - g))] + I*b*e*g*PolyLog[2, -((E^((2*I)*ArcTan[c*x])*(c^2*f + g + 2*Sqrt[c^2*f*g]))/(c^2*f -
g))])/(4*c^2*g)

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Maple [C]  time = 3.667, size = 21442, normalized size = 38.2 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(b*d*x*arctan(c*x) + a*d*x + (b*e*x*arctan(c*x) + a*e*x)*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(x*(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 )} x\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(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, x)