3.16 \(\int (d+e x)^2 (a+b \tan ^{-1}(c x))^3 \, dx\)

Optimal. Leaf size=411 \[ \frac{i b^2 \left (3 c^2 d^2-e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3}+\frac{b^3 \left (3 c^2 d^2-e^2\right ) \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{2 c^3}-\frac{3 i b^3 d e \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{c^2}-\frac{6 b^2 d e \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2}+\frac{a b^2 e^2 x}{c^2}+\frac{i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{3 c^3}-\frac{d \left (d^2-\frac{3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}+\frac{b \left (3 c^2 d^2-e^2\right ) \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3}-\frac{3 i b d e \left (a+b \tan ^{-1}(c x)\right )^2}{c^2}-\frac{b e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}-\frac{3 b d e x \left (a+b \tan ^{-1}(c x)\right )^2}{c}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}-\frac{b^3 e^2 \log \left (c^2 x^2+1\right )}{2 c^3}+\frac{b^3 e^2 x \tan ^{-1}(c x)}{c^2} \]

[Out]

(a*b^2*e^2*x)/c^2 + (b^3*e^2*x*ArcTan[c*x])/c^2 - ((3*I)*b*d*e*(a + b*ArcTan[c*x])^2)/c^2 - (b*e^2*(a + b*ArcT
an[c*x])^2)/(2*c^3) - (3*b*d*e*x*(a + b*ArcTan[c*x])^2)/c - (b*e^2*x^2*(a + b*ArcTan[c*x])^2)/(2*c) + ((I/3)*(
3*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])^3)/c^3 - (d*(d^2 - (3*e^2)/c^2)*(a + b*ArcTan[c*x])^3)/(3*e) + ((d + e*x)
^3*(a + b*ArcTan[c*x])^3)/(3*e) - (6*b^2*d*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^2 + (b*(3*c^2*d^2 - e^2
)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c^3 - (b^3*e^2*Log[1 + c^2*x^2])/(2*c^3) - ((3*I)*b^3*d*e*PolyLog[
2, 1 - 2/(1 + I*c*x)])/c^2 + (I*b^2*(3*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3 +
(b^3*(3*c^2*d^2 - e^2)*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^3)

________________________________________________________________________________________

Rubi [A]  time = 0.774111, antiderivative size = 411, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 13, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.722, Rules used = {4864, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 4984, 4994, 6610} \[ \frac{i b^2 \left (3 c^2 d^2-e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3}+\frac{b^3 \left (3 c^2 d^2-e^2\right ) \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{2 c^3}-\frac{3 i b^3 d e \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{c^2}-\frac{6 b^2 d e \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2}+\frac{a b^2 e^2 x}{c^2}+\frac{i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{3 c^3}-\frac{d \left (d^2-\frac{3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}+\frac{b \left (3 c^2 d^2-e^2\right ) \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3}-\frac{3 i b d e \left (a+b \tan ^{-1}(c x)\right )^2}{c^2}-\frac{b e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}-\frac{3 b d e x \left (a+b \tan ^{-1}(c x)\right )^2}{c}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}-\frac{b^3 e^2 \log \left (c^2 x^2+1\right )}{2 c^3}+\frac{b^3 e^2 x \tan ^{-1}(c x)}{c^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^2*(a + b*ArcTan[c*x])^3,x]

[Out]

(a*b^2*e^2*x)/c^2 + (b^3*e^2*x*ArcTan[c*x])/c^2 - ((3*I)*b*d*e*(a + b*ArcTan[c*x])^2)/c^2 - (b*e^2*(a + b*ArcT
an[c*x])^2)/(2*c^3) - (3*b*d*e*x*(a + b*ArcTan[c*x])^2)/c - (b*e^2*x^2*(a + b*ArcTan[c*x])^2)/(2*c) + ((I/3)*(
3*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])^3)/c^3 - (d*(d^2 - (3*e^2)/c^2)*(a + b*ArcTan[c*x])^3)/(3*e) + ((d + e*x)
^3*(a + b*ArcTan[c*x])^3)/(3*e) - (6*b^2*d*e*(a + b*ArcTan[c*x])*Log[2/(1 + I*c*x)])/c^2 + (b*(3*c^2*d^2 - e^2
)*(a + b*ArcTan[c*x])^2*Log[2/(1 + I*c*x)])/c^3 - (b^3*e^2*Log[1 + c^2*x^2])/(2*c^3) - ((3*I)*b^3*d*e*PolyLog[
2, 1 - 2/(1 + I*c*x)])/c^2 + (I*b^2*(3*c^2*d^2 - e^2)*(a + b*ArcTan[c*x])*PolyLog[2, 1 - 2/(1 + I*c*x)])/c^3 +
(b^3*(3*c^2*d^2 - e^2)*PolyLog[3, 1 - 2/(1 + I*c*x)])/(2*c^3)

Rule 4864

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1)*(a
+ b*ArcTan[c*x])^p)/(e*(q + 1)), x] - Dist[(b*c*p)/(e*(q + 1)), Int[ExpandIntegrand[(a + b*ArcTan[c*x])^(p -
1), (d + e*x)^(q + 1)/(1 + c^2*x^2), x], x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 1] && IntegerQ[q] && N
eQ[q, -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 4920

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

Rule 4854

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] + Dist[(b*c*p)/e, Int[((a + b*ArcTan[c*x])^(p - 1)*Log[2/(1 + (e*x)/d)])/(1 + c^2*x^
2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[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 4852

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

Rule 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 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 4884

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> Simp[(a + b*ArcTan[c*x])^(p +
1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]

Rule 4984

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> I
nt[ExpandIntegrand[(a + b*ArcTan[c*x])^p/(d + e*x^2), (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& IGtQ[p, 0] && EqQ[e, c^2*d] && IGtQ[m, 0]

Rule 4994

Int[(Log[u_]*((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Tan[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] + Dist[(b*p*I)/2, Int[((a + b*ArcTan[c*x])^(p - 1)*PolyLog[2, 1 - u
])/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && EqQ[e, c^2*d] && EqQ[(1 - u)^2 - (1 - (2*
I)/(I - c*x))^2, 0]

Rule 6610

Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
;  !FalseQ[w]] /; FreeQ[n, x]

Rubi steps

\begin{align*} \int (d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^3 \, dx &=\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}-\frac{(b c) \int \left (\frac{3 d e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^2}+\frac{e^3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^2}+\frac{\left (c^2 d^3-3 d e^2+e \left (3 c^2 d^2-e^2\right ) x\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 \left (1+c^2 x^2\right )}\right ) \, dx}{e}\\ &=\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}-\frac{b \int \frac{\left (c^2 d^3-3 d e^2+e \left (3 c^2 d^2-e^2\right ) x\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{c e}-\frac{(3 b d e) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{c}-\frac{\left (b e^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{c}\\ &=-\frac{3 b d e x \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}-\frac{b \int \left (\frac{c^2 d^3 \left (1-\frac{3 e^2}{c^2 d^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2}-\frac{e \left (-3 c^2 d^2+e^2\right ) x \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2}\right ) \, dx}{c e}+\left (6 b^2 d e\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\left (b^2 e^2\right ) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-\frac{3 i b d e \left (a+b \tan ^{-1}(c x)\right )^2}{c^2}-\frac{3 b d e x \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}-\frac{\left (6 b^2 d e\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{c}+\frac{\left (b^2 e^2\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^2}-\frac{\left (b^2 e^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^2}-\left (b d \left (\frac{c d^2}{e}-\frac{3 e}{c}\right )\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx-\frac{\left (b \left (3 c^2 d^2-e^2\right )\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{c}\\ &=\frac{a b^2 e^2 x}{c^2}-\frac{3 i b d e \left (a+b \tan ^{-1}(c x)\right )^2}{c^2}-\frac{b e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}-\frac{3 b d e x \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac{i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{3 c^3}-\frac{d \left (d^2-\frac{3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}-\frac{6 b^2 d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^2}+\frac{\left (6 b^3 d e\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c}+\frac{\left (b^3 e^2\right ) \int \tan ^{-1}(c x) \, dx}{c^2}+\frac{\left (b \left (3 c^2 d^2-e^2\right )\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{i-c x} \, dx}{c^2}\\ &=\frac{a b^2 e^2 x}{c^2}+\frac{b^3 e^2 x \tan ^{-1}(c x)}{c^2}-\frac{3 i b d e \left (a+b \tan ^{-1}(c x)\right )^2}{c^2}-\frac{b e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}-\frac{3 b d e x \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac{i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{3 c^3}-\frac{d \left (d^2-\frac{3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}-\frac{6 b^2 d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^2}+\frac{b \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^3}-\frac{\left (6 i b^3 d e\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{c^2}-\frac{\left (b^3 e^2\right ) \int \frac{x}{1+c^2 x^2} \, dx}{c}-\frac{\left (2 b^2 \left (3 c^2 d^2-e^2\right )\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2}\\ &=\frac{a b^2 e^2 x}{c^2}+\frac{b^3 e^2 x \tan ^{-1}(c x)}{c^2}-\frac{3 i b d e \left (a+b \tan ^{-1}(c x)\right )^2}{c^2}-\frac{b e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}-\frac{3 b d e x \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac{i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{3 c^3}-\frac{d \left (d^2-\frac{3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}-\frac{6 b^2 d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^2}+\frac{b \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^3}-\frac{b^3 e^2 \log \left (1+c^2 x^2\right )}{2 c^3}-\frac{3 i b^3 d e \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^2}+\frac{i b^2 \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^3}-\frac{\left (i b^3 \left (3 c^2 d^2-e^2\right )\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2}\\ &=\frac{a b^2 e^2 x}{c^2}+\frac{b^3 e^2 x \tan ^{-1}(c x)}{c^2}-\frac{3 i b d e \left (a+b \tan ^{-1}(c x)\right )^2}{c^2}-\frac{b e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}-\frac{3 b d e x \left (a+b \tan ^{-1}(c x)\right )^2}{c}-\frac{b e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac{i \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{3 c^3}-\frac{d \left (d^2-\frac{3 e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}+\frac{(d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^3}{3 e}-\frac{6 b^2 d e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^2}+\frac{b \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^3}-\frac{b^3 e^2 \log \left (1+c^2 x^2\right )}{2 c^3}-\frac{3 i b^3 d e \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^2}+\frac{i b^2 \left (3 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^3}+\frac{b^3 \left (3 c^2 d^2-e^2\right ) \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{2 c^3}\\ \end{align*}

Mathematica [A]  time = 1.11731, size = 621, normalized size = 1.51 \[ \frac{18 a b^2 c^2 d^2 \left (\tan ^{-1}(c x) \left ((c x-i) \tan ^{-1}(c x)+2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )\right )+6 a b^2 e^2 \left (i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+\left (c^3 x^3+i\right ) \tan ^{-1}(c x)^2-\tan ^{-1}(c x) \left (c^2 x^2+2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+1\right )+c x\right )+3 b^3 c^2 d^2 \left (-6 i \tan ^{-1}(c x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+3 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(c x)}\right )+2 \tan ^{-1}(c x)^2 \left ((c x-i) \tan ^{-1}(c x)+3 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )\right )+6 b^3 c d e \left (3 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+\tan ^{-1}(c x) \left (\left (c^2 x^2+1\right ) \tan ^{-1}(c x)^2+(-3 c x+3 i) \tan ^{-1}(c x)-6 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )\right )+b^3 e^2 \left (6 i \tan ^{-1}(c x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )-3 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(c x)}\right )-3 \log \left (c^2 x^2+1\right )+2 c^3 x^3 \tan ^{-1}(c x)^3-3 c^2 x^2 \tan ^{-1}(c x)^2+2 i \tan ^{-1}(c x)^3-3 \tan ^{-1}(c x)^2+6 c x \tan ^{-1}(c x)-6 \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-3 a^2 b \left (3 c^2 d^2-e^2\right ) \log \left (c^2 x^2+1\right )+6 a^2 b c^3 x \tan ^{-1}(c x) \left (3 d^2+3 d e x+e^2 x^2\right )+3 a^2 c^2 e x^2 (2 a c d-b e)+6 a^2 c^2 d x (a c d-3 b e)+18 a^2 b c d e \tan ^{-1}(c x)+2 a^3 c^3 e^2 x^3+18 a b^2 c d e \left (\log \left (c^2 x^2+1\right )+\left (c^2 x^2+1\right ) \tan ^{-1}(c x)^2-2 c x \tan ^{-1}(c x)\right )}{6 c^3} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(d + e*x)^2*(a + b*ArcTan[c*x])^3,x]

[Out]

(6*a^2*c^2*d*(a*c*d - 3*b*e)*x + 3*a^2*c^2*e*(2*a*c*d - b*e)*x^2 + 2*a^3*c^3*e^2*x^3 + 18*a^2*b*c*d*e*ArcTan[c
*x] + 6*a^2*b*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)*ArcTan[c*x] - 3*a^2*b*(3*c^2*d^2 - e^2)*Log[1 + c^2*x^2] + 18*
a*b^2*c*d*e*(-2*c*x*ArcTan[c*x] + (1 + c^2*x^2)*ArcTan[c*x]^2 + Log[1 + c^2*x^2]) + 18*a*b^2*c^2*d^2*(ArcTan[c
*x]*((-I + c*x)*ArcTan[c*x] + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) - I*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 6*a*
b^2*e^2*(c*x + (I + c^3*x^3)*ArcTan[c*x]^2 - ArcTan[c*x]*(1 + c^2*x^2 + 2*Log[1 + E^((2*I)*ArcTan[c*x])]) + I*
PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + 6*b^3*c*d*e*(ArcTan[c*x]*((3*I - 3*c*x)*ArcTan[c*x] + (1 + c^2*x^2)*ArcT
an[c*x]^2 - 6*Log[1 + E^((2*I)*ArcTan[c*x])]) + (3*I)*PolyLog[2, -E^((2*I)*ArcTan[c*x])]) + b^3*e^2*(6*c*x*Arc
Tan[c*x] - 3*ArcTan[c*x]^2 - 3*c^2*x^2*ArcTan[c*x]^2 + (2*I)*ArcTan[c*x]^3 + 2*c^3*x^3*ArcTan[c*x]^3 - 6*ArcTa
n[c*x]^2*Log[1 + E^((2*I)*ArcTan[c*x])] - 3*Log[1 + c^2*x^2] + (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c
*x])] - 3*PolyLog[3, -E^((2*I)*ArcTan[c*x])]) + 3*b^3*c^2*d^2*(2*ArcTan[c*x]^2*((-I + c*x)*ArcTan[c*x] + 3*Log
[1 + E^((2*I)*ArcTan[c*x])]) - (6*I)*ArcTan[c*x]*PolyLog[2, -E^((2*I)*ArcTan[c*x])] + 3*PolyLog[3, -E^((2*I)*A
rcTan[c*x])]))/(6*c^3)

________________________________________________________________________________________

Maple [C]  time = 2.606, size = 3022, normalized size = 7.4 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^2*(a+b*arctan(c*x))^3,x)

[Out]

1/3*a^3/e*d^3-1/2/c^3*b^3*e^2*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))+1/c^3*b^3*e^2*ln((1+I*c*x)^2/(c^2*x^2+1)+1)-
1/2/c^3*b^3*e^2*arctan(c*x)^2+3/2/c*b^3*polylog(3,-(1+I*c*x)^2/(c^2*x^2+1))*d^2+b^3*arctan(c*x)^3*x*d^2+1/3*b^
3*e^2*arctan(c*x)^3*x^3+a*b^2*e^2*x/c^2+b^3*e^2*x*arctan(c*x)/c^2-6/c^2*b^3*e*d*arctan(c*x)*ln(1+I*(1+I*c*x)/(
c^2*x^2+1)^(1/2))-6/c^2*b^3*e*d*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+I/c^3*b^3*e^2*arctan(c*x)*poly
log(2,-(1+I*c*x)^2/(c^2*x^2+1))+1/c^3*a*b^2*e^2*arctan(c*x)*ln(c^2*x^2+1)+3*a^2*b*e*arctan(c*x)*x^2*d+3*a*b^2*
e*arctan(c*x)^2*x^2*d-3/c*a*b^2*arctan(c*x)*ln(c^2*x^2+1)*d^2+3/c^2*a*b^2*e*ln(c^2*x^2+1)*d-3/c*b^3*e*arctan(c
*x)^2*d*x-1/c*a*b^2*e^2*arctan(c*x)*x^2+6*I/c^2*b^3*e*d*dilog(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))+3/4*I/c*a*b^2*l
n(c*x-I)^2*d^2+3/2*I/c*a*b^2*dilog(-1/2*I*(c*x+I))*d^2-3/4*I/c*a*b^2*ln(c*x+I)^2*d^2-3/2*I/c*a*b^2*dilog(1/2*I
*(c*x-I))*d^2-1/4*I/c^3*a*b^2*ln(c*x-I)^2*e^2-1/2*I/c^3*a*b^2*dilog(-1/2*I*(c*x+I))*e^2+1/4*I/c^3*a*b^2*ln(c*x
+I)^2*e^2+1/2*I/c^3*a*b^2*dilog(1/2*I*(c*x-I))*e^2-3*I/c*b^3*arctan(c*x)*polylog(2,-(1+I*c*x)^2/(c^2*x^2+1))*d
^2+3*I/c^2*b^3*e*d*arctan(c*x)^2+6*I/c^2*b^3*e*d*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-3*a^2*b/c*x*d*e+a^3*e*
x^2*d+3/c^2*a^2*b*e*arctan(c*x)*d+3/c^2*a*b^2*e*d*arctan(c*x)^2+3*a^2*b*arctan(c*x)*x*d^2+3*a*b^2*arctan(c*x)^
2*x*d^2+3/c*b^3*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))*arctan(c*x)^2*d^2-3/2/c*b^3*arctan(c*x)^2*ln(c^2*x^2+1)*d^2-1/
2/c*b^3*e^2*arctan(c*x)^2*x^2+1/2/c^3*a^2*b*ln(c^2*x^2+1)*e^2-1/c^3*b^3*e^2*ln((1+I*c*x)/(c^2*x^2+1)^(1/2))*ar
ctan(c*x)^2+1/2/c^3*b^3*e^2*arctan(c*x)^2*ln(c^2*x^2+1)-1/c^3*a*b^2*e^2*arctan(c*x)+1/c^2*b^3*e*arctan(c*x)^3*
d-1/c^3*b^3*e^2*ln(2)*arctan(c*x)^2+3/c*b^3*d^2*ln(2)*arctan(c*x)^2-3/2/c*a^2*b*ln(c^2*x^2+1)*d^2-I/c^3*b^3*e^
2*arctan(c*x)+1/3*I/c^3*b^3*e^2*arctan(c*x)^3-I/c*b^3*arctan(c*x)^3*d^2-1/2/c*a^2*b*x^2*e^2+b^3*e*arctan(c*x)^
3*x^2*d+a^2*b*e^2*arctan(c*x)*x^3+a*b^2*e^2*arctan(c*x)^2*x^3+a^3*x*d^2+1/3*a^3*e^2*x^3+1/2*I/c^3*b^3*e^2*Pi*c
sgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*arctan(c*x)^2+1/4*I/c^3*b^3*e^2*Pi*
csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*arctan(c*x)^2-3/4*I/c*b^3*d^2*Pi*csgn(I*
(1+I*c*x)/(c^2*x^2+1)^(1/2))^2*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*arctan(c*x)^2+3/4*I/c*b^3*d^2*Pi*csgn(I*(1+I*c*
x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*arctan(c*x)^2+3/4*I/c*b^3*d^
2*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1))^2*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*arctan(c*x)^2+3/4*I/c*b^3*d^2
*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*arct
an(c*x)^2+3/2*I/c*b^3*d^2*Pi*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^2*arctan(c*x)
^2-3/2*I/c*b^3*d^2*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1))*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*arctan(c*x)^
2-1/4*I/c^3*b^3*e^2*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1))^2*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*arctan(c*x)
^2-1/2*I/c^3*b^3*e^2*Pi*csgn(I*(1+I*c*x)/(c^2*x^2+1)^(1/2))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^2*arctan(c*x)^2-1/
4*I/c^3*b^3*e^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+
1)+1)^2)^2*arctan(c*x)^2-1/4*I/c^3*b^3*e^2*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/(
(1+I*c*x)^2/(c^2*x^2+1)+1)^2)^2*arctan(c*x)^2+1/2*I/c^3*a*b^2*ln(c*x+I)*ln(1/2*I*(c*x-I))*e^2-1/4*I/c^3*b^3*e^
2*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*arctan(c*x)^2+1/4*I/c^3*b^3*e^2*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)
)^3*arctan(c*x)^2-6/c*a*b^2*e*arctan(c*x)*d*x+1/4*I/c^3*b^3*e^2*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2
/(c^2*x^2+1)+1)^2)^3*arctan(c*x)^2-3/4*I/c*b^3*d^2*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+
1)^2)^3*arctan(c*x)^2+3/4*I/c*b^3*d^2*Pi*csgn(I*((1+I*c*x)^2/(c^2*x^2+1)+1)^2)^3*arctan(c*x)^2-3/4*I/c*b^3*d^2
*Pi*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))^3*arctan(c*x)^2-3/2*I/c*a*b^2*ln(c^2*x^2+1)*ln(c*x-I)*d^2+3/2*I/c*a*b^2*ln
(c*x-I)*ln(-1/2*I*(c*x+I))*d^2+3/2*I/c*a*b^2*ln(c^2*x^2+1)*ln(c*x+I)*d^2-3/2*I/c*a*b^2*ln(c*x+I)*ln(1/2*I*(c*x
-I))*d^2+1/2*I/c^3*a*b^2*ln(c^2*x^2+1)*ln(c*x-I)*e^2-1/2*I/c^3*a*b^2*ln(c*x-I)*ln(-1/2*I*(c*x+I))*e^2-1/2*I/c^
3*a*b^2*ln(c^2*x^2+1)*ln(c*x+I)*e^2-3/4*I/c*b^3*d^2*Pi*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*csgn(I*(1+I*c*x)^
2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*arctan(c*x)^2+1/4*I/c^3*b^3*e^2*P
i*csgn(I/((1+I*c*x)^2/(c^2*x^2+1)+1)^2)*csgn(I*(1+I*c*x)^2/(c^2*x^2+1))*csgn(I*(1+I*c*x)^2/(c^2*x^2+1)/((1+I*c
*x)^2/(c^2*x^2+1)+1)^2)*arctan(c*x)^2

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a+b*arctan(c*x))^3,x, algorithm="maxima")

[Out]

1/3*a^3*e^2*x^3 + 7/32*b^3*d^2*arctan(c*x)^4/c + 28*b^3*c^2*e^2*integrate(1/32*x^4*arctan(c*x)^3/(c^2*x^2 + 1)
, x) + 3*b^3*c^2*e^2*integrate(1/32*x^4*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 96*a*b^2*c^2*e^2*in
tegrate(1/32*x^4*arctan(c*x)^2/(c^2*x^2 + 1), x) + 56*b^3*c^2*d*e*integrate(1/32*x^3*arctan(c*x)^3/(c^2*x^2 +
1), x) + 4*b^3*c^2*e^2*integrate(1/32*x^4*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 6*b^3*c^2*d*e*integ
rate(1/32*x^3*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 192*a*b^2*c^2*d*e*integrate(1/32*x^3*arctan(c
*x)^2/(c^2*x^2 + 1), x) + 28*b^3*c^2*d^2*integrate(1/32*x^2*arctan(c*x)^3/(c^2*x^2 + 1), x) + 12*b^3*c^2*d*e*i
ntegrate(1/32*x^3*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + 3*b^3*c^2*d^2*integrate(1/32*x^2*arctan(c*x
)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 96*a*b^2*c^2*d^2*integrate(1/32*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) +
12*b^3*c^2*d^2*integrate(1/32*x^2*arctan(c*x)*log(c^2*x^2 + 1)/(c^2*x^2 + 1), x) + a^3*d*e*x^2 + a*b^2*d^2*ar
ctan(c*x)^3/c - 4*b^3*c*e^2*integrate(1/32*x^3*arctan(c*x)^2/(c^2*x^2 + 1), x) + b^3*c*e^2*integrate(1/32*x^3*
log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 12*b^3*c*d*e*integrate(1/32*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^3
*c*d*e*integrate(1/32*x^2*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) - 12*b^3*c*d^2*integrate(1/32*x*arctan(c*x)^2/(
c^2*x^2 + 1), x) + 3*b^3*c*d^2*integrate(1/32*x*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 3*(x^2*arctan(c*x) - c*
(x/c^2 - arctan(c*x)/c^3))*a^2*b*d*e + 1/2*(2*x^3*arctan(c*x) - c*(x^2/c^2 - log(c^2*x^2 + 1)/c^4))*a^2*b*e^2
+ a^3*d^2*x + 28*b^3*e^2*integrate(1/32*x^2*arctan(c*x)^3/(c^2*x^2 + 1), x) + 3*b^3*e^2*integrate(1/32*x^2*arc
tan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 96*a*b^2*e^2*integrate(1/32*x^2*arctan(c*x)^2/(c^2*x^2 + 1), x
) + 56*b^3*d*e*integrate(1/32*x*arctan(c*x)^3/(c^2*x^2 + 1), x) + 6*b^3*d*e*integrate(1/32*x*arctan(c*x)*log(c
^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 192*a*b^2*d*e*integrate(1/32*x*arctan(c*x)^2/(c^2*x^2 + 1), x) + 3*b^3*d^2*i
ntegrate(1/32*arctan(c*x)*log(c^2*x^2 + 1)^2/(c^2*x^2 + 1), x) + 3/2*(2*c*x*arctan(c*x) - log(c^2*x^2 + 1))*a^
2*b*d^2/c + 1/24*(b^3*e^2*x^3 + 3*b^3*d*e*x^2 + 3*b^3*d^2*x)*arctan(c*x)^3 - 1/32*(b^3*e^2*x^3 + 3*b^3*d*e*x^2
+ 3*b^3*d^2*x)*arctan(c*x)*log(c^2*x^2 + 1)^2

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a+b*arctan(c*x))^3,x, algorithm="fricas")

[Out]

integral(a^3*e^2*x^2 + 2*a^3*d*e*x + a^3*d^2 + (b^3*e^2*x^2 + 2*b^3*d*e*x + b^3*d^2)*arctan(c*x)^3 + 3*(a*b^2*
e^2*x^2 + 2*a*b^2*d*e*x + a*b^2*d^2)*arctan(c*x)^2 + 3*(a^2*b*e^2*x^2 + 2*a^2*b*d*e*x + a^2*b*d^2)*arctan(c*x)
, x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{atan}{\left (c x \right )}\right )^{3} \left (d + e x\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**2*(a+b*atan(c*x))**3,x)

[Out]

Integral((a + b*atan(c*x))**3*(d + e*x)**2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^2*(a+b*arctan(c*x))^3,x, algorithm="giac")

[Out]

integrate((e*x + d)^2*(b*arctan(c*x) + a)^3, x)