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

Optimal. Leaf size=565 \[ \frac{i b^2 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^3}-\frac{b^3 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (3,1-\frac{2}{1+i (c+d x)}\right )}{2 d^3}+\frac{3 i b^3 f (d e-c f) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{d^3}-\frac{6 b^2 f (d e-c f) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^3}+\frac{a b^2 f^2 x}{d^2}+\frac{i \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3}-\frac{(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3 f}-\frac{b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{3 b f (c+d x) (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}+\frac{b^3 f^2 \log \left ((c+d x)^2+1\right )}{2 d^3}+\frac{b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3} \]

[Out]

(a*b^2*f^2*x)/d^2 + (b^3*f^2*(c + d*x)*ArcCot[c + d*x])/d^3 + (b*f^2*(a + b*ArcCot[c + d*x])^2)/(2*d^3) + ((3*
I)*b*f*(d*e - c*f)*(a + b*ArcCot[c + d*x])^2)/d^3 + (3*b*f*(d*e - c*f)*(c + d*x)*(a + b*ArcCot[c + d*x])^2)/d^
3 + (b*f^2*(c + d*x)^2*(a + b*ArcCot[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*ArcCot[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*ArcCot[c + d*x])^3)/
(3*d^3*f) + ((e + f*x)^3*(a + b*ArcCot[c + d*x])^3)/(3*f) - (6*b^2*f*(d*e - c*f)*(a + b*ArcCot[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*ArcCot[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*ArcCot[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 = 0.963499, antiderivative size = 565, 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 = {5048, 4865, 4847, 4921, 4855, 2402, 2315, 4853, 4917, 260, 4885, 4985, 4995, 6610} \[ \frac{i b^2 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^3}-\frac{b^3 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (3,1-\frac{2}{1+i (c+d x)}\right )}{2 d^3}+\frac{3 i b^3 f (d e-c f) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{d^3}-\frac{6 b^2 f (d e-c f) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^3}+\frac{a b^2 f^2 x}{d^2}+\frac{i \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3}-\frac{(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 d^3 f}-\frac{b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{3 b f (c+d x) (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}+\frac{b^3 f^2 \log \left ((c+d x)^2+1\right )}{2 d^3}+\frac{b^3 f^2 (c+d x) \cot ^{-1}(c+d x)}{d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*b^2*f^2*x)/d^2 + (b^3*f^2*(c + d*x)*ArcCot[c + d*x])/d^3 + (b*f^2*(a + b*ArcCot[c + d*x])^2)/(2*d^3) + ((3*
I)*b*f*(d*e - c*f)*(a + b*ArcCot[c + d*x])^2)/d^3 + (3*b*f*(d*e - c*f)*(c + d*x)*(a + b*ArcCot[c + d*x])^2)/d^
3 + (b*f^2*(c + d*x)^2*(a + b*ArcCot[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*ArcCot[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*ArcCot[c + d*x])^3)/
(3*d^3*f) + ((e + f*x)^3*(a + b*ArcCot[c + d*x])^3)/(3*f) - (6*b^2*f*(d*e - c*f)*(a + b*ArcCot[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*ArcCot[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*ArcCot[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 5048

Int[((a_.) + ArcCot[(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*ArcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x]
&& IGtQ[p, 0]

Rule 4865

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1)*(a
+ b*ArcCot[c*x])^p)/(e*(q + 1)), x] + Dist[(b*c*p)/(e*(q + 1)), Int[ExpandIntegrand[(a + b*ArcCot[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 4847

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

Rule 4921

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

Rule 4855

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcCot[c*x])^p*Lo
g[2/(1 + (e*x)/d)])/e, x] - Dist[(b*c*p)/e, Int[((a + b*ArcCot[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 4853

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcCo
t[c*x])^p)/(d*(m + 1)), x] + Dist[(b*c*p)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcCot[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 4917

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[f^2/
e, Int[(f*x)^(m - 2)*(a + b*ArcCot[c*x])^p, x], x] - Dist[(d*f^2)/e, Int[((f*x)^(m - 2)*(a + b*ArcCot[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 4885

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(a + b*ArcCot[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 4985

Int[(((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> I
nt[ExpandIntegrand[(a + b*ArcCot[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 4995

Int[(Log[u_]*((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Simp[(I*(a + b*Arc
Cot[c*x])^p*PolyLog[2, 1 - u])/(2*c*d), x] - Dist[(b*p*I)/2, Int[((a + b*ArcCot[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 \cot ^{-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 \cot ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac{(e+f x)^3 \left (a+b \cot ^{-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 \cot ^{-1}(x)\right )^2}{d^3}+\frac{f^3 x \left (a+b \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac{(e+f x)^3 \left (a+b \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}+\frac{\left (b^2 f^2\right ) \operatorname{Subst}\left (\int \left (a+b \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-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 \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac{3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-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 \cot ^{-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 \cot ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac{6 b^2 f (d e-c f) \left (a+b \cot ^{-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 \cot ^{-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 \cot ^{-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) \cot ^{-1}(c+d x)}{d^3}+\frac{b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac{3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-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 \cot ^{-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 \cot ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac{6 b^2 f (d e-c f) \left (a+b \cot ^{-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 \cot ^{-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 \cot ^{-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) \cot ^{-1}(c+d x)}{d^3}+\frac{b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac{3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-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 \cot ^{-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 \cot ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac{6 b^2 f (d e-c f) \left (a+b \cot ^{-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 \cot ^{-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 \cot ^{-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) \cot ^{-1}(c+d x)}{d^3}+\frac{b f^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^3}+\frac{3 i b f (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{3 b f (d e-c f) (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^3}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-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 \cot ^{-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 \cot ^{-1}(c+d x)\right )^3}{3 d^3 f}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^3}{3 f}-\frac{6 b^2 f (d e-c f) \left (a+b \cot ^{-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 \cot ^{-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 \cot ^{-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 = 10.4507, size = 2336, normalized size = 4.13 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(e + f*x)^2*(a + b*ArcCot[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 + a^2*b*
x*(3*e^2 + 3*e*f*x + f^2*x^2)*ArcCot[c + d*x] + ((-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 + ((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) + (a*b^2*f^2*x^2*(1 + (c + d*x)^2)*((c + d*x)*(1 - 6*c*ArcCo
t[c + d*x] + 3*ArcCot[c + d*x]^2 + 3*c^2*ArcCot[c + d*x]^2) - (c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(1 - 6*c*ArcC
ot[c + d*x] - ArcCot[c + d*x]^2 + 3*c^2*ArcCot[c + d*x]^2)*Cos[3*ArcCot[c + d*x]] - 2*(-2*ArcCot[c + d*x] + I*
ArcCot[c + d*x]^2 + 6*c*ArcCot[c + d*x]^2 - (3*I)*c^2*ArcCot[c + d*x]^2 + 2*(-1 + 3*c^2)*ArcCot[c + d*x]*Log[1
- E^((2*I)*ArcCot[c + d*x])] - 6*c*Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])] + Cos[2*ArcCot[c + d*x]]*(I*(-
1 + 3*c^2)*ArcCot[c + d*x]^2 + (2 - 6*c^2)*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] + 6*c*Log[1/((c
+ d*x)*Sqrt[1 + (c + d*x)^(-2)])])) + ((4*I)*(-1 + 3*c^2)*PolyLog[2, E^((2*I)*ArcCot[c + d*x])])/((c + d*x)^2*
(1 + (c + d*x)^(-2)))))/(4*d*(c + d*x)^2*(1 + (c + d*x)^(-2))*(1/Sqrt[1 + (c + d*x)^(-2)] - c/((c + d*x)*Sqrt[
1 + (c + d*x)^(-2)]))^2) - (3*a*b^2*e^2*(1 + (c + d*x)^2)*(-((c + d*x)*ArcCot[c + d*x]^2) + 2*ArcCot[c + d*x]*
Log[1 - E^((2*I)*ArcCot[c + d*x])] - I*(ArcCot[c + d*x]^2 + PolyLog[2, E^((2*I)*ArcCot[c + d*x])])))/(d*(c + d
*x)^2*(1 + (c + d*x)^(-2))) + (6*a*b^2*e*f*(1 + (c + d*x)^2)*(((c + d*x)*ArcCot[c + d*x])/d^2 - (c*(c + d*x)*A
rcCot[c + d*x]^2)/d^2 + ((c + d*x)^2*(1 + (c + d*x)^(-2))*ArcCot[c + d*x]^2)/(2*d^2) - Log[1/((c + d*x)*Sqrt[1
+ (c + d*x)^(-2)])]/d^2 + (2*c*(ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] - (I/2)*(ArcCot[c + d*x]^2
+ PolyLog[2, E^((2*I)*ArcCot[c + d*x])])))/d^2))/((c + d*x)^2*(1 + (c + d*x)^(-2))) - (b^3*e^2*(1 + (c + d*x)
^2)*((-I/8)*Pi^3 + I*ArcCot[c + d*x]^3 - (c + d*x)*ArcCot[c + d*x]^3 + 3*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*A
rcCot[c + d*x])] + (3*I)*ArcCot[c + d*x]*PolyLog[2, E^((-2*I)*ArcCot[c + d*x])] + (3*PolyLog[3, E^((-2*I)*ArcC
ot[c + d*x])])/2))/(d*(c + d*x)^2*(1 + (c + d*x)^(-2))) + (b^3*e*f*(1 + (c + d*x)^2)*((-I)*c*Pi^3 + (12*I)*Arc
Cot[c + d*x]^2 + 12*(c + d*x)*ArcCot[c + d*x]^2 + (8*I)*c*ArcCot[c + d*x]^3 - 8*c*(c + d*x)*ArcCot[c + d*x]^3
+ 4*(c + d*x)^2*(1 + (c + d*x)^(-2))*ArcCot[c + d*x]^3 + 24*c*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d
*x])] - 24*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] + (24*I)*c*ArcCot[c + d*x]*PolyLog[2, E^((-2*I)*
ArcCot[c + d*x])] + (12*I)*PolyLog[2, E^((2*I)*ArcCot[c + d*x])] + 12*c*PolyLog[3, E^((-2*I)*ArcCot[c + d*x])]
))/(4*d^2*(c + d*x)^2*(1 + (c + d*x)^(-2))) - (b^3*f^2*(1 + (c + d*x)^2)*(I*(-1 + 3*c^2)*ArcCot[c + d*x]*PolyL
og[2, E^((-2*I)*ArcCot[c + d*x])] + ((c + d*x)^3*(1 + (c + d*x)^(-2))^(3/2)*(((3*I)*Pi^3)/((c + d*x)*Sqrt[1 +
(c + d*x)^(-2)]) - ((9*I)*c^2*Pi^3)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) - (24*ArcCot[c + d*x])/Sqrt[1 + (c +
d*x)^(-2)] + (72*c*ArcCot[c + d*x]^2)/Sqrt[1 + (c + d*x)^(-2)] - (48*ArcCot[c + d*x]^2)/((c + d*x)*Sqrt[1 + (c
+ d*x)^(-2)]) + ((216*I)*c*ArcCot[c + d*x]^2)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) - (24*ArcCot[c + d*x]^3)/S
qrt[1 + (c + d*x)^(-2)] - (24*c^2*ArcCot[c + d*x]^3)/Sqrt[1 + (c + d*x)^(-2)] - ((24*I)*ArcCot[c + d*x]^3)/((c
+ d*x)*Sqrt[1 + (c + d*x)^(-2)]) + (96*c*ArcCot[c + d*x]^3)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + ((72*I)*c^
2*ArcCot[c + d*x]^3)/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + 24*ArcCot[c + d*x]*Cos[3*ArcCot[c + d*x]] - 72*c*A
rcCot[c + d*x]^2*Cos[3*ArcCot[c + d*x]] - 8*ArcCot[c + d*x]^3*Cos[3*ArcCot[c + d*x]] + 24*c^2*ArcCot[c + d*x]^
3*Cos[3*ArcCot[c + d*x]] - (72*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])])/((c + d*x)*Sqrt[1 + (c +
d*x)^(-2)]) + (216*c^2*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])])/((c + d*x)*Sqrt[1 + (c + d*x)^(
-2)]) - (432*c*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])])/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + (72*
Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])])/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)]) + ((288*I)*c*PolyLog[2, E^((
2*I)*ArcCot[c + d*x])])/((c + d*x)^3*(1 + (c + d*x)^(-2))^(3/2)) + (48*(-1 + 3*c^2)*PolyLog[3, E^((-2*I)*ArcCo
t[c + d*x])])/((c + d*x)^3*(1 + (c + d*x)^(-2))^(3/2)) - I*Pi^3*Sin[3*ArcCot[c + d*x]] + (3*I)*c^2*Pi^3*Sin[3*
ArcCot[c + d*x]] - (72*I)*c*ArcCot[c + d*x]^2*Sin[3*ArcCot[c + d*x]] + (8*I)*ArcCot[c + d*x]^3*Sin[3*ArcCot[c
+ d*x]] - (24*I)*c^2*ArcCot[c + d*x]^3*Sin[3*ArcCot[c + d*x]] + 24*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[
c + d*x])]*Sin[3*ArcCot[c + d*x]] - 72*c^2*ArcCot[c + d*x]^2*Log[1 - E^((-2*I)*ArcCot[c + d*x])]*Sin[3*ArcCot[
c + d*x]] + 144*c*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])]*Sin[3*ArcCot[c + d*x]] - 24*Log[1/((c + d
*x)*Sqrt[1 + (c + d*x)^(-2)])]*Sin[3*ArcCot[c + d*x]]))/96))/(d^3*(c + d*x)^2*(1 + (c + d*x)^(-2)))

________________________________________________________________________________________

Maple [B]  time = 0.552, size = 3693, normalized size = 6.5 \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*arccot(d*x+c))^3,x)

[Out]

1/3*a^3*f^2*x^3+a^3*x*e^2+6/d^2*a*b^2*f*arccot(d*x+c)*e*c-3/d^2*a^2*b*f*ln(1+(d*x+c)^2)*c*e+3/d^2*a^2*b*f*arct
an(d*x+c)*c^2*e-6/d*a*b^2*arccot(d*x+c)*arctan(d*x+c)*c*e^2+3/d^2*a*b^2*f*arctan(d*x+c)^2*c^2*e+6/d^2*b^3*f*c*
e*arccot(d*x+c)^2*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-1/2*I/d^3*a*b^2*ln(d*x+c-I)*ln(-1/2*I*(d*x+c+I))*f^2-1/2
*I/d^3*a*b^2*ln(1+(d*x+c)^2)*ln(d*x+c+I)*f^2+1/2*I/d^3*a*b^2*ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))*f^2+1/2*I/d^3*a*b
^2*ln(1+(d*x+c)^2)*ln(d*x+c-I)*f^2+6*I/d^3*b^3*f^2*c^2*arccot(d*x+c)*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+
3/2*I/d^3*a*b^2*dilog(-1/2*I*(d*x+c+I))*c^2*f^2+6*I/d^3*b^3*f^2*c^2*arccot(d*x+c)*polylog(2,-(d*x+c+I)/(1+(d*x
+c)^2)^(1/2))-3/4*I/d^3*a*b^2*ln(d*x+c+I)^2*c^2*f^2+a*b^2*f^2*x/d^2+3/2/d^3*a^2*b*f^2*ln(1+(d*x+c)^2)*c^2+3/d^
2*a*b^2*f*ln(1+(d*x+c)^2)*e-3/d^2*a*b^2*f*arctan(d*x+c)^2*e-3*I/d^3*b^3*f^2*arccot(d*x+c)^2*c+3/4*I/d*a*b^2*ln
(d*x+c-I)^2*e^2+3/2*I/d*a*b^2*dilog(-1/2*I*(d*x+c+I))*e^2-3/4*I/d*a*b^2*ln(d*x+c+I)^2*e^2+3*I/d^2*b^3*f*arccot
(d*x+c)^2*e-1/2*I/d^3*a*b^2*dilog(-1/2*I*(d*x+c+I))*f^2-1/4*I/d^3*a*b^2*ln(d*x+c-I)^2*f^2+6*I/d^2*b^3*f*e*poly
log(2,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+6*I/d^2*b^3*f*e*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-6*I/d^3*b^3*f^2
*c*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+3*a^2*b*f*arccot(d*x+c)*e*x^2+3*a*b^2*f*arccot(d*x+c)^2*e*x^2-3/d*
a^2*b*arctan(d*x+c)*c*e^2-3/d*a*b^2*arctan(d*x+c)^2*c*e^2+3/2/d*e^2*a^2*b*ln(1+(d*x+c)^2)+1/d^3*a*b^2*f^2*c-5/
2/d^3*a^2*b*f^2*c^2-6/d^2*a*b^2*f*arccot(d*x+c)*ln(1+(d*x+c)^2)*c*e+6/d^2*a*b^2*f*arccot(d*x+c)*arctan(d*x+c)*
c^2*e+3*I/d^2*a*b^2*dilog(1/2*I*(d*x+c-I))*c*e*f-3/2*I/d^3*a*b^2*ln(1+(d*x+c)^2)*ln(d*x+c-I)*c^2*f^2-12*I/d^2*
b^3*f*c*e*arccot(d*x+c)*polylog(2,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-12*I/d^2*b^3*f*c*e*arccot(d*x+c)*polylog(2,(
d*x+c+I)/(1+(d*x+c)^2)^(1/2))+3/2*I/d^3*a*b^2*ln(d*x+c-I)*ln(-1/2*I*(d*x+c+I))*c^2*f^2-3/2*I/d^2*a*b^2*ln(d*x+
c-I)^2*c*e*f+1/3*a^3/f*e^3+a^3*f*x^2*e+3/2*I/d^2*a*b^2*ln(d*x+c+I)^2*c*e*f-3*I/d^2*a*b^2*dilog(-1/2*I*(d*x+c+I
))*c*e*f+3/2*I/d^3*a*b^2*ln(1+(d*x+c)^2)*ln(d*x+c+I)*c^2*f^2-3/2*I/d^3*a*b^2*ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))*c
^2*f^2+1/d^3*b^3*f^2*arccot(d*x+c)^2*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+1/d^3*b^3*f^2*arccot(d*x+c)^2*ln(1+(d
*x+c+I)/(1+(d*x+c)^2)^(1/2))-5/2/d^3*b^3*f^2*arccot(d*x+c)^2*c^2+1/3/d^3*b^3*f^2*arccot(d*x+c)^3*c^3+1/2/d*b^3
*f^2*arccot(d*x+c)^2*x^2+1/d^2*b^3*f^2*arccot(d*x+c)*x+1/d^3*b^3*f^2*arccot(d*x+c)*c-1/2/d^3*a^2*b*f^2*ln(1+(d
*x+c)^2)+a*b^2/f*arccot(d*x+c)^2*e^3-1/d^3*a*b^2*f^2*arctan(d*x+c)+1/d^2*b^3*f*arccot(d*x+c)^3*e-1/d^3*b^3*f^2
*arccot(d*x+c)^3*c+b^3*f*arccot(d*x+c)^3*e*x^2+a^2*b*f^2*arccot(d*x+c)*x^3+3*arccot(d*x+c)*x*a^2*b*e^2+3*arcco
t(d*x+c)^2*x*a*b^2*e^2+a*b^2*f^2*arccot(d*x+c)^2*x^3+a^2*b/f*arctan(d*x+c)*e^3+a^2*b/f*arccot(d*x+c)*e^3+a*b^2
/f*arctan(d*x+c)^2*e^3-1/3*I/d^3*b^3*f^2*arccot(d*x+c)^3-I/d^3*b^3*f^2*arccot(d*x+c)+1/3*b^3*f^2*arccot(d*x+c)
^3*x^3+arccot(d*x+c)^3*x*b^3*e^2-6/d*b^3*e^2*polylog(3,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-6/d*b^3*e^2*polylog(3,-(
d*x+c+I)/(1+(d*x+c)^2)^(1/2))+2/d^3*b^3*f^2*polylog(3,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+2/d^3*b^3*f^2*polylog(3,
(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+1/2/d^3*b^3*f^2*arccot(d*x+c)^2-1/d^3*b^3*f^2*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2
))-1/d^3*b^3*f^2*ln((d*x+c+I)/(1+(d*x+c)^2)^(1/2)-1)+2/d^3*b^3*f^2*ln((d*x+c+I)/(1+(d*x+c)^2)^(1/2))-2*a^2*b/d
^2*x*c*f^2+3*a^2*b/d*x*e*f+2*a*b^2/f*arccot(d*x+c)*arctan(d*x+c)*e^3-6/d^2*b^3*f*e*arccot(d*x+c)*ln(1+(d*x+c+I
)/(1+(d*x+c)^2)^(1/2))+3/d^3*a*b^2*f^2*arctan(d*x+c)^2*c-3/2*I/d*a*b^2*dilog(1/2*I*(d*x+c-I))*e^2+6*I/d*b^3*e^
2*arccot(d*x+c)*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-6/d^3*b^3*f^2*c^2*polylog(3,(d*x+c+I)/(1+(d*x+c)^2)^(
1/2))-6/d^3*b^3*f^2*c^2*polylog(3,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-3/d*b^3*e^2*arccot(d*x+c)^2*ln(1+(d*x+c+I)/(
1+(d*x+c)^2)^(1/2))-3/d*b^3*e^2*arccot(d*x+c)^2*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+1/d*arccot(d*x+c)^3*b^3*c*
e^2+I/d*b^3*arccot(d*x+c)^3*e^2-1/d^3*a^2*b*f^2*arctan(d*x+c)*c^3+12/d^2*b^3*f*c*e*polylog(3,-(d*x+c+I)/(1+(d*
x+c)^2)^(1/2))+12/d^2*b^3*f*c*e*polylog(3,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-1/d^2*b^3*f*arccot(d*x+c)^3*c^2*e-5/d
^3*a*b^2*f^2*arccot(d*x+c)*c^2-2/d^2*b^3*f^2*arccot(d*x+c)^2*c*x+3/d*b^3*f*arccot(d*x+c)^2*e*x+3/d^2*b^3*f*arc
cot(d*x+c)^2*e*c+6/d^3*b^3*f^2*c*arccot(d*x+c)*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+6/d^3*b^3*f^2*c*arccot(d*x+
c)*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-3/d^3*b^3*f^2*c^2*arccot(d*x+c)^2*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-3
/d^3*b^3*f^2*c^2*arccot(d*x+c)^2*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-6/d^2*b^3*f*e*arccot(d*x+c)*ln(1-(d*x+c+I
)/(1+(d*x+c)^2)^(1/2))-1/d^3*a*b^2*f^2*arccot(d*x+c)*ln(1+(d*x+c)^2)+1/d*a*b^2*f^2*arccot(d*x+c)*x^2+1/2/d*a^2
*b*f^2*x^2+6*I/d*b^3*e^2*arccot(d*x+c)*polylog(2,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+1/2*I/d^3*a*b^2*dilog(1/2*I*(
d*x+c-I))*f^2-2*I/d^3*b^3*f^2*arccot(d*x+c)*polylog(2,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-2*I/d^3*b^3*f^2*arccot(d
*x+c)*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-6*I/d^3*b^3*f^2*c*polylog(2,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+1/4
*I/d^3*a*b^2*ln(d*x+c+I)^2*f^2-1/d^3*a*b^2*f^2*arctan(d*x+c)^2*c^3+I/d^3*b^3*f^2*arccot(d*x+c)^3*c^2+3/d*a*b^2
*arccot(d*x+c)*ln(1+(d*x+c)^2)*e^2+3/d^3*a^2*b*f^2*arctan(d*x+c)*c-3/d^2*a^2*b*f*arctan(d*x+c)*e-3/d^3*a*b^2*f
^2*ln(1+(d*x+c)^2)*c-3*I/d^2*a*b^2*ln(1+(d*x+c)^2)*ln(d*x+c+I)*c*e*f-3*I/d^2*a*b^2*ln(d*x+c-I)*ln(-1/2*I*(d*x+
c+I))*c*e*f+3*I/d^2*a*b^2*ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))*c*e*f+3*I/d^2*a*b^2*ln(1+(d*x+c)^2)*ln(d*x+c-I)*c*e*
f-2*I/d^2*b^3*f*arccot(d*x+c)^3*c*e+3/4*I/d^3*a*b^2*ln(d*x+c-I)^2*c^2*f^2-3/2*I/d*a*b^2*ln(1+(d*x+c)^2)*ln(d*x
+c-I)*e^2+3/2*I/d*a*b^2*ln(d*x+c-I)*ln(-1/2*I*(d*x+c+I))*e^2+3/2*I/d*a*b^2*ln(1+(d*x+c)^2)*ln(d*x+c+I)*e^2-3/2
*I/d*a*b^2*ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))*e^2-3/2*I/d^3*a*b^2*dilog(1/2*I*(d*x+c-I))*c^2*f^2-6/d^2*a*b^2*f*ar
ccot(d*x+c)*arctan(d*x+c)*e+6/d^2*b^3*f*c*e*arccot(d*x+c)^2*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+3/d^3*a*b^2*f^
2*arccot(d*x+c)*ln(1+(d*x+c)^2)*c^2-4/d^2*a*b^2*f^2*arccot(d*x+c)*c*x-2/d^3*a*b^2*f^2*arccot(d*x+c)*arctan(d*x
+c)*c^3+6/d^3*a*b^2*f^2*arccot(d*x+c)*arctan(d*x+c)*c+6/d*a*b^2*f*arccot(d*x+c)*e*x+3/d^2*a^2*b*f*c*e

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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*arccot(d*x+c))^3,x, algorithm="maxima")

[Out]

1/24*b^3*f^2*x^3*arctan2(1, d*x + c)^3 + 1/8*b^3*e*f*x^2*arctan2(1, d*x + c)^3 + 1/8*b^3*e^2*x*arctan2(1, d*x
+ c)^3 + 1/3*a^3*f^2*x^3 + a^3*e*f*x^2 + 3*(x^2*arccot(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*arccot(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 + 3/2*(2*(d*x + c)*arccot(d*x + c) + log((d*x + c)^2 + 1))*a^2*b*e^2/d - 1/32*(b^3*f^2*x^3*arctan2(
1, d*x + c) + 3*b^3*e*f*x^2*arctan2(1, d*x + c) + 3*b^3*e^2*x*arctan2(1, d*x + c))*log(d^2*x^2 + 2*c*d*x + c^2
+ 1)^2 + integrate(1/32*(4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*d^2*f^2*x^4 + 4*(2*
(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*d^2*e*f + (b^3*arctan2(1, d*x + c)^2 + 2*(7*b^3
*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c)*d*f^2)*x^3 + 4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a
*b^2*arctan2(1, d*x + c)^2 + (7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c^2)*e^2 + 4*((7*b
^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*d^2*e^2 + (3*b^3*arctan2(1, d*x + c)^2 + 4*(7*b^3*a
rctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c)*d*e*f + (7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arct
an2(1, d*x + c)^2 + (7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c^2)*f^2)*x^2 + (3*b^3*d^2*
f^2*x^4*arctan2(1, d*x + c) + (6*b^3*d^2*e*f*arctan2(1, d*x + c) + (6*b^3*c*arctan2(1, d*x + c) - b^3)*d*f^2)*
x^3 + 3*(b^3*c^2*arctan2(1, d*x + c) + b^3*arctan2(1, d*x + c))*e^2 + 3*(b^3*d^2*e^2*arctan2(1, d*x + c) + (4*
b^3*c*arctan2(1, d*x + c) - b^3)*d*e*f + (b^3*c^2*arctan2(1, d*x + c) + b^3*arctan2(1, d*x + c))*f^2)*x^2 + 3*
((2*b^3*c*arctan2(1, d*x + c) - b^3)*d*e^2 + 2*(b^3*c^2*arctan2(1, d*x + c) + b^3*arctan2(1, d*x + c))*e*f)*x)
*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + 4*((3*b^3*arctan2(1, d*x + c)^2 + 2*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*
b^2*arctan2(1, d*x + c)^2)*c)*d*e^2 + 2*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2 + (7*b^3
*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*c^2)*e*f)*x + 4*(b^3*d^2*f^2*x^4*arctan2(1, d*x + c)
+ 3*b^3*c*d*e^2*x*arctan2(1, d*x + c) + (3*b^3*d^2*e*f*arctan2(1, d*x + c) + b^3*c*d*f^2*arctan2(1, d*x + c))*
x^3 + 3*(b^3*d^2*e^2*arctan2(1, d*x + c) + b^3*c*d*e*f*arctan2(1, d*x + c))*x^2)*log(d^2*x^2 + 2*c*d*x + c^2 +
1))/(d^2*x^2 + 2*c*d*x + c^2 + 1), x)

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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 )} \operatorname{arccot}\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 )} \operatorname{arccot}\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 )} \operatorname{arccot}\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*arccot(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)*arccot(d*x + c)^3 + 3*(a*
b^2*f^2*x^2 + 2*a*b^2*e*f*x + a*b^2*e^2)*arccot(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
cot(d*x + c), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \operatorname{acot}{\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*acot(d*x+c))**3,x)

[Out]

Integral((a + b*acot(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 \operatorname{arccot}\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*arccot(d*x+c))^3,x, algorithm="giac")

[Out]

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