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

Optimal. Leaf size=564 \[ \frac{i b^2 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right ) \left (a+b \tan ^{-1}(c+d x)\right )}{d^3}+\frac{b^3 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (3,1-\frac{2}{1+i (c+d x)}\right )}{2 d^3}-\frac{3 i b^3 f (d e-c f) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{d^3}-\frac{6 b^2 f (d e-c f) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \tan ^{-1}(c+d x)\right )}{d^3}+\frac{a b^2 f^2 x}{d^2}+\frac{i \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^3}{3 d^3}-\frac{(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac{b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d^3}-\frac{3 i b f (d e-c f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d^3}-\frac{3 b f (c+d x) (d e-c f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d^3}-\frac{b f^2 \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d^3}-\frac{b f^2 (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d^3}+\frac{(e+f x)^3 \left (a+b \tan ^{-1}(c+d x)\right )^3}{3 f}-\frac{b^3 f^2 \log \left ((c+d x)^2+1\right )}{2 d^3}+\frac{b^3 f^2 (c+d x) \tan ^{-1}(c+d x)}{d^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.937082, antiderivative size = 564, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 14, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5047, 4864, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 4984, 4994, 6610} \[ \frac{i b^2 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right ) \left (a+b \tan ^{-1}(c+d x)\right )}{d^3}+\frac{b^3 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (3,1-\frac{2}{1+i (c+d x)}\right )}{2 d^3}-\frac{3 i b^3 f (d e-c f) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{d^3}-\frac{6 b^2 f (d e-c f) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \tan ^{-1}(c+d x)\right )}{d^3}+\frac{a b^2 f^2 x}{d^2}+\frac{i \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^3}{3 d^3}-\frac{(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \tan ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac{b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d^3}-\frac{3 i b f (d e-c f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d^3}-\frac{3 b f (c+d x) (d e-c f) \left (a+b \tan ^{-1}(c+d x)\right )^2}{d^3}-\frac{b f^2 \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d^3}-\frac{b f^2 (c+d x)^2 \left (a+b \tan ^{-1}(c+d x)\right )^2}{2 d^3}+\frac{(e+f x)^3 \left (a+b \tan ^{-1}(c+d x)\right )^3}{3 f}-\frac{b^3 f^2 \log \left ((c+d x)^2+1\right )}{2 d^3}+\frac{b^3 f^2 (c+d x) \tan ^{-1}(c+d x)}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5047

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

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

Mathematica [B]  time = 8.96162, size = 1844, normalized size = 3.27 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*(a*d^2*e^2 - 3*b*d*e*f + 2*b*c*f^2)*x)/d^2 - (a^2*f*(-2*a*d*e + b*f)*x^2)/(2*d) + (a^3*f^2*x^3)/3 + ((3*a
^2*b*c*d^2*e^2 + 3*a^2*b*d*e*f - 3*a^2*b*c^2*d*e*f - 3*a^2*b*c*f^2 + a^2*b*c^3*f^2)*ArcTan[c + d*x])/d^3 + a^2
*b*x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcTan[c + d*x] + ((-3*a^2*b*d^2*e^2 + 6*a^2*b*c*d*e*f + a^2*b*f^2 - 3*a^2*b*
c^2*f^2)*Log[1 + c^2 + 2*c*d*x + d^2*x^2])/(2*d^3) + (3*a*b^2*e^2*((-I)*ArcTan[c + d*x]^2 + (c + d*x)*ArcTan[c
+ d*x]^2 + 2*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])] - I*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])]))/
d + 6*a*b^2*e*f*(-(((c + d*x)*ArcTan[c + d*x])/d^2) + (I*c*ArcTan[c + d*x]^2)/d^2 - (c*(c + d*x)*ArcTan[c + d*
x]^2)/d^2 + ((1 + (c + d*x)^2)*ArcTan[c + d*x]^2)/(2*d^2) - (2*c*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d
*x])])/d^2 - Log[1/Sqrt[1 + (c + d*x)^2]]/d^2 + (I*c*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])])/d^2) + (b^3*e^2*(
(-I)*ArcTan[c + d*x]^3 + (c + d*x)*ArcTan[c + d*x]^3 + 3*ArcTan[c + d*x]^2*Log[1 + E^((2*I)*ArcTan[c + d*x])]
- (3*I)*ArcTan[c + d*x]*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] + (3*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])])/2)
)/d + (b^3*e*f*(ArcTan[c + d*x]*((3*I)*ArcTan[c + d*x] + (2*I)*c*ArcTan[c + d*x]^2 + (1 + (c + d*x)^2)*ArcTan[
c + d*x]^2 - (c + d*x)*ArcTan[c + d*x]*(3 + 2*c*ArcTan[c + d*x]) - 6*Log[1 + E^((2*I)*ArcTan[c + d*x])] - 6*c*
ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])]) + (3*I)*(1 + 2*c*ArcTan[c + d*x])*PolyLog[2, -E^((2*I)*Arc
Tan[c + d*x])] - 3*c*PolyLog[3, -E^((2*I)*ArcTan[c + d*x])]))/d^2 + (a*b^2*f^2*(1 + (c + d*x)^2)^(3/2)*((c + d
*x)/Sqrt[1 + (c + d*x)^2] + (6*c*(c + d*x)*ArcTan[c + d*x])/Sqrt[1 + (c + d*x)^2] + (3*(c + d*x)*ArcTan[c + d*
x]^2)/Sqrt[1 + (c + d*x)^2] + (3*c^2*(c + d*x)*ArcTan[c + d*x]^2)/Sqrt[1 + (c + d*x)^2] + I*ArcTan[c + d*x]^2*
Cos[3*ArcTan[c + d*x]] - (3*I)*c^2*ArcTan[c + d*x]^2*Cos[3*ArcTan[c + d*x]] - 2*ArcTan[c + d*x]*Cos[3*ArcTan[c
+ d*x]]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 6*c^2*ArcTan[c + d*x]*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*Ar
cTan[c + d*x])] + 6*c*Cos[3*ArcTan[c + d*x]]*Log[1/Sqrt[1 + (c + d*x)^2]] + (ArcTan[c + d*x]*(-4 + (3*I - 12*c
- (9*I)*c^2)*ArcTan[c + d*x]) + 6*(-1 + 3*c^2)*ArcTan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 18*c*Log[
1/Sqrt[1 + (c + d*x)^2]])/Sqrt[1 + (c + d*x)^2] - ((4*I)*(-1 + 3*c^2)*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])])/
(1 + (c + d*x)^2)^(3/2) + Sin[3*ArcTan[c + d*x]] + 6*c*ArcTan[c + d*x]*Sin[3*ArcTan[c + d*x]] - ArcTan[c + d*x
]^2*Sin[3*ArcTan[c + d*x]] + 3*c^2*ArcTan[c + d*x]^2*Sin[3*ArcTan[c + d*x]]))/(4*d^3) + (b^3*f^2*((-I)*(3*c -
ArcTan[c + d*x] + 3*c^2*ArcTan[c + d*x])*PolyLog[2, -E^((2*I)*ArcTan[c + d*x])] + ((1 + (c + d*x)^2)^(3/2)*((3
*(c + d*x)*ArcTan[c + d*x])/Sqrt[1 + (c + d*x)^2] + (9*c*(c + d*x)*ArcTan[c + d*x]^2)/Sqrt[1 + (c + d*x)^2] +
(3*(c + d*x)*ArcTan[c + d*x]^3)/Sqrt[1 + (c + d*x)^2] + (3*c^2*(c + d*x)*ArcTan[c + d*x]^3)/Sqrt[1 + (c + d*x)
^2] - (9*I)*c*ArcTan[c + d*x]^2*Cos[3*ArcTan[c + d*x]] + I*ArcTan[c + d*x]^3*Cos[3*ArcTan[c + d*x]] - (3*I)*c^
2*ArcTan[c + d*x]^3*Cos[3*ArcTan[c + d*x]] + 18*c*ArcTan[c + d*x]*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*ArcT
an[c + d*x])] - 3*ArcTan[c + d*x]^2*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 9*c^2*ArcTan[c
+ d*x]^2*Cos[3*ArcTan[c + d*x]]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + 3*Cos[3*ArcTan[c + d*x]]*Log[1/Sqrt[1 +
(c + d*x)^2]] + (3*(ArcTan[c + d*x]^2*(-2 - (9*I)*c + I*ArcTan[c + d*x] - 4*c*ArcTan[c + d*x] - (3*I)*c^2*ArcT
an[c + d*x]) + 3*ArcTan[c + d*x]*(6*c - ArcTan[c + d*x] + 3*c^2*ArcTan[c + d*x])*Log[1 + E^((2*I)*ArcTan[c + d
*x])] + 3*Log[1/Sqrt[1 + (c + d*x)^2]]))/Sqrt[1 + (c + d*x)^2] + (6*(-1 + 3*c^2)*PolyLog[3, -E^((2*I)*ArcTan[c
+ d*x])])/(1 + (c + d*x)^2)^(3/2) + 3*ArcTan[c + d*x]*Sin[3*ArcTan[c + d*x]] + 9*c*ArcTan[c + d*x]^2*Sin[3*Ar
cTan[c + d*x]] - ArcTan[c + d*x]^3*Sin[3*ArcTan[c + d*x]] + 3*c^2*ArcTan[c + d*x]^3*Sin[3*ArcTan[c + d*x]]))/1
2))/d^3

________________________________________________________________________________________

Maple [C]  time = 2.95, size = 6682, normalized size = 11.9 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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

[Out]

7/8*b^3*c^2*e^2*arctan(d*x + c)^3*arctan((d^2*x + c*d)/d)/d + 3*a*b^2*c^2*e^2*arctan(d*x + c)^2*arctan((d^2*x
+ c*d)/d)/d - (3*arctan(d*x + c)*arctan((d^2*x + c*d)/d)^2/d - arctan((d^2*x + c*d)/d)^3/d)*a*b^2*c^2*e^2 - 7/
32*(6*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)^2/d - 4*arctan(d*x + c)*arctan((d^2*x + c*d)/d)^3/d + arctan((
d^2*x + c*d)/d)^4/d)*b^3*c^2*e^2 + 1/3*a^3*f^2*x^3 + 7/8*b^3*e^2*arctan(d*x + c)^3*arctan((d^2*x + c*d)/d)/d +
28*b^3*d^2*f^2*integrate(1/32*x^4*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*d^2*f^2*integra
te(1/32*x^4*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 96*a*b^2*d^
2*f^2*integrate(1/32*x^4*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*d^2*e*f*integrate(1/32*x
^3*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*c*d*f^2*integrate(1/32*x^3*arctan(d*x + c)^3/(
d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 4*b^3*d^2*f^2*integrate(1/32*x^4*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^
2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 6*b^3*d^2*e*f*integrate(1/32*x^3*arctan(d*x + c)*log(d^2*x^2 + 2*c*
d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 6*b^3*c*d*f^2*integrate(1/32*x^3*arctan(d*x + c)*log(d^2*
x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 192*a*b^2*d^2*e*f*integrate(1/32*x^3*arctan(d*x
+ c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 192*a*b^2*c*d*f^2*integrate(1/32*x^3*arctan(d*x + c)^2/(d^2*x^2 +
2*c*d*x + c^2 + 1), x) + 28*b^3*d^2*e^2*integrate(1/32*x^2*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x)
+ 112*b^3*c*d*e*f*integrate(1/32*x^2*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 28*b^3*c^2*f^2*int
egrate(1/32*x^2*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^3*d^2*e*f*integrate(1/32*x^3*arctan
(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 4*b^3*c*d*f^2*integrate(1/32*x^
3*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*d^2*e^2*integrate
(1/32*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^3*c*d*e*
f*integrate(1/32*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*
b^3*c^2*f^2*integrate(1/32*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1
), x) + 96*a*b^2*d^2*e^2*integrate(1/32*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 384*a*b^2*c*
d*e*f*integrate(1/32*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 96*a*b^2*c^2*f^2*integrate(1/32
*x^2*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*c*d*e^2*integrate(1/32*x*arctan(d*x + c)^3/(
d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*c^2*e*f*integrate(1/32*x*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2
+ 1), x) + 12*b^3*d^2*e^2*integrate(1/32*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d
*x + c^2 + 1), x) + 12*b^3*c*d*e*f*integrate(1/32*x^2*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^
2 + 2*c*d*x + c^2 + 1), x) + 6*b^3*c*d*e^2*integrate(1/32*x*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2
/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 6*b^3*c^2*e*f*integrate(1/32*x*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^
2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 192*a*b^2*c*d*e^2*integrate(1/32*x*arctan(d*x + c)^2/(d^2*x^2 + 2
*c*d*x + c^2 + 1), x) + 192*a*b^2*c^2*e*f*integrate(1/32*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x)
+ 12*b^3*c*d*e^2*integrate(1/32*x*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 +
1), x) + 3*b^3*c^2*e^2*integrate(1/32*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x +
c^2 + 1), x) + a^3*e*f*x^2 + 3*a*b^2*e^2*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)/d - 4*b^3*d*f^2*integrate(
1/32*x^3*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + b^3*d*f^2*integrate(1/32*x^3*log(d^2*x^2 + 2*c*
d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) - 12*b^3*d*e*f*integrate(1/32*x^2*arctan(d*x + c)^2/(d^2*x^
2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*d*e*f*integrate(1/32*x^2*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*
d*x + c^2 + 1), x) - 12*b^3*d*e^2*integrate(1/32*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3
*d*e^2*integrate(1/32*x*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) - (3*arctan(d*x +
c)*arctan((d^2*x + c*d)/d)^2/d - arctan((d^2*x + c*d)/d)^3/d)*a*b^2*e^2 - 7/32*(6*arctan(d*x + c)^2*arctan((d
^2*x + c*d)/d)^2/d - 4*arctan(d*x + c)*arctan((d^2*x + c*d)/d)^3/d + arctan((d^2*x + c*d)/d)^4/d)*b^3*e^2 + 3*
(x^2*arctan(d*x + c) - d*(x/d^2 + (c^2 - 1)*arctan((d^2*x + c*d)/d)/d^3 - c*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/d
^3))*a^2*b*e*f + 1/2*(2*x^3*arctan(d*x + c) - d*((d*x^2 - 4*c*x)/d^3 - 2*(c^3 - 3*c)*arctan((d^2*x + c*d)/d)/d
^4 + (3*c^2 - 1)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/d^4))*a^2*b*f^2 + a^3*e^2*x + 28*b^3*f^2*integrate(1/32*x^2*
arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*f^2*integrate(1/32*x^2*arctan(d*x + c)*log(d^2*x^2
+ 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 96*a*b^2*f^2*integrate(1/32*x^2*arctan(d*x + c)^2/
(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 56*b^3*e*f*integrate(1/32*x*arctan(d*x + c)^3/(d^2*x^2 + 2*c*d*x + c^2 + 1
), x) + 6*b^3*e*f*integrate(1/32*x*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2
+ 1), x) + 192*a*b^2*e*f*integrate(1/32*x*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^3*e^2*int
egrate(1/32*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3/2*(2*(d*x
+ c)*arctan(d*x + c) - log((d*x + c)^2 + 1))*a^2*b*e^2/d + 1/24*(b^3*f^2*x^3 + 3*b^3*e*f*x^2 + 3*b^3*e^2*x)*a
rctan(d*x + c)^3 - 1/32*(b^3*f^2*x^3 + 3*b^3*e*f*x^2 + 3*b^3*e^2*x)*arctan(d*x + c)*log(d^2*x^2 + 2*c*d*x + c^
2 + 1)^2

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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