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

Optimal. Leaf size=337 \[ \frac{3 i b^2 (d e-c f) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}-\frac{3 b^3 (d e-c f) \text{PolyLog}\left (3,1-\frac{2}{1+i (c+d x)}\right )}{2 d^2}+\frac{3 i b^3 f \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{2 d^2}-\frac{3 b^2 f \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}+\frac{i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d^2}-\frac{(-c f+d e+f) (d e-(c+1) f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 d^2 f}-\frac{3 b (d e-c f) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}+\frac{3 i b f \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{3 b f (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.66385, antiderivative size = 337, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {5048, 4865, 4847, 4921, 4855, 2402, 2315, 4985, 4885, 4995, 6610} \[ \frac{3 i b^2 (d e-c f) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}-\frac{3 b^3 (d e-c f) \text{PolyLog}\left (3,1-\frac{2}{1+i (c+d x)}\right )}{2 d^2}+\frac{3 i b^3 f \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{2 d^2}-\frac{3 b^2 f \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{d^2}+\frac{i (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{d^2}-\frac{(-c f+d e+f) (d e-(c+1) f) \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 d^2 f}-\frac{3 b (d e-c f) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}+\frac{3 i b f \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{3 b f (c+d x) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 d^2}+\frac{(e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^3}{2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 1.22149, size = 630, normalized size = 1.87 \[ \frac{6 a b^2 d e \left (i \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )+\cot ^{-1}(c+d x) \left ((c+d x+i) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )\right )-6 a b^2 c f \left (i \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )+\cot ^{-1}(c+d x) \left ((c+d x+i) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )\right )+2 b^3 d e \left (-3 i \cot ^{-1}(c+d x) \text{PolyLog}\left (2,e^{-2 i \cot ^{-1}(c+d x)}\right )-\frac{3}{2} \text{PolyLog}\left (3,e^{-2 i \cot ^{-1}(c+d x)}\right )+(c+d x) \cot ^{-1}(c+d x)^3-i \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left (1-e^{-2 i \cot ^{-1}(c+d x)}\right )+\frac{i \pi ^3}{8}\right )+b^3 f \left (3 i \left (\cot ^{-1}(c+d x)^2+\text{PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )\right )+\left ((c+d x)^2+1\right ) \cot ^{-1}(c+d x)^3+3 (c+d x) \cot ^{-1}(c+d x)^2-6 \cot ^{-1}(c+d x) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )-2 b^3 c f \left (-3 i \cot ^{-1}(c+d x) \text{PolyLog}\left (2,e^{-2 i \cot ^{-1}(c+d x)}\right )-\frac{3}{2} \text{PolyLog}\left (3,e^{-2 i \cot ^{-1}(c+d x)}\right )+(c+d x) \cot ^{-1}(c+d x)^3-i \cot ^{-1}(c+d x)^3-3 \cot ^{-1}(c+d x)^2 \log \left (1-e^{-2 i \cot ^{-1}(c+d x)}\right )+\frac{i \pi ^3}{8}\right )+a^2 (c+d x) (-2 a c f+2 a d e+3 b f)+3 a^2 b (d e-c f) \log \left ((c+d x)^2+1\right )-3 a^2 b (c+d x) \cot ^{-1}(c+d x) (c f-d (2 e+f x))-3 a^2 b f \tan ^{-1}(c+d x)+a^3 f (c+d x)^2+6 a b^2 f \left (-\log \left (\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right )+\frac{1}{2} \left ((c+d x)^2+1\right ) \cot ^{-1}(c+d x)^2+(c+d x) \cot ^{-1}(c+d x)\right )}{2 d^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(a^2*(2*a*d*e + 3*b*f - 2*a*c*f)*(c + d*x) + a^3*f*(c + d*x)^2 - 3*a^2*b*(c + d*x)*(c*f - d*(2*e + f*x))*ArcCo
t[c + d*x] - 3*a^2*b*f*ArcTan[c + d*x] + 6*a*b^2*f*((c + d*x)*ArcCot[c + d*x] + ((1 + (c + d*x)^2)*ArcCot[c +
d*x]^2)/2 - Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])]) + 3*a^2*b*(d*e - c*f)*Log[1 + (c + d*x)^2] + 6*a*b^2*
d*e*(ArcCot[c + d*x]*((I + c + d*x)*ArcCot[c + d*x] - 2*Log[1 - E^((2*I)*ArcCot[c + d*x])]) + I*PolyLog[2, E^(
(2*I)*ArcCot[c + d*x])]) - 6*a*b^2*c*f*(ArcCot[c + d*x]*((I + c + d*x)*ArcCot[c + d*x] - 2*Log[1 - E^((2*I)*Ar
cCot[c + d*x])]) + I*PolyLog[2, E^((2*I)*ArcCot[c + d*x])]) + b^3*f*(3*(c + d*x)*ArcCot[c + d*x]^2 + (1 + (c +
d*x)^2)*ArcCot[c + d*x]^3 - 6*ArcCot[c + d*x]*Log[1 - E^((2*I)*ArcCot[c + d*x])] + (3*I)*(ArcCot[c + d*x]^2 +
PolyLog[2, E^((2*I)*ArcCot[c + d*x])])) + 2*b^3*d*e*((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)*ArcCot[c + d*x])] - (3*I)*ArcCot[c + d*x]*PolyLog[2, E^((-2*I)*
ArcCot[c + d*x])] - (3*PolyLog[3, E^((-2*I)*ArcCot[c + d*x])])/2) - 2*b^3*c*f*((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)*ArcCot[c + d*x])] - (3*I)*ArcCot[c + d
*x]*PolyLog[2, E^((-2*I)*ArcCot[c + d*x])] - (3*PolyLog[3, E^((-2*I)*ArcCot[c + d*x])])/2))/(2*d^2)

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Maple [B]  time = 0.517, size = 1570, normalized size = 4.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

6/d^2*b^3*c*f*polylog(3,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+6/d^2*b^3*c*f*polylog(3,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))
-3/d*b^3*e*arccot(d*x+c)^2*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-3/d*b^3*e*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-3/2/d^2*a^2*b*f*arctan(d*x+c)+3*I/d^2*b^3*f*polylog(2,(d*x+c+I)/(1+
(d*x+c)^2)^(1/2))+3/2*I/d^2*b^3*f*arccot(d*x+c)^2+3*I/d^2*b^3*f*polylog(2,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+3/2*
a*b^2*arccot(d*x+c)^2*f*x^2+I/d*b^3*arccot(d*x+c)^3*e+3/2/d*a^2*b*ln(1+(d*x+c)^2)*e+3/2/d^2*a*b^2*f*ln(1+(d*x+
c)^2)-3/2*I/d^2*a*b^2*ln(d*x+c+I)*ln(1+(d*x+c)^2)*c*f-3/2*I/d^2*a*b^2*ln(d*x+c-I)*ln(-1/2*I*(d*x+c+I))*c*f+3/2
*I/d^2*a*b^2*ln(d*x+c-I)*ln(1+(d*x+c)^2)*c*f+3/2*I/d^2*a*b^2*ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))*c*f-3/4*I/d*a*b^2
*ln(d*x+c+I)^2*e-3/2*I/d*a*b^2*dilog(1/2*I*(d*x+c-I))*e+3/d*a*b^2*arccot(d*x+c)*f*x+3/d^2*a*b^2*arccot(d*x+c)*
f*c+3/d*arccot(d*x+c)*a^2*b*c*e+3/d*arccot(d*x+c)^2*a*b^2*c*e-3/2/d^2*a*b^2*arccot(d*x+c)^2*c^2*f-3/2/d^2*a^2*
b*arccot(d*x+c)*c^2*f-3/2/d^2*a^2*b*ln(1+(d*x+c)^2)*c*f+3/d^2*b^3*c*f*arccot(d*x+c)^2*ln(1+(d*x+c+I)/(1+(d*x+c
)^2)^(1/2))+3/d^2*b^3*c*f*arccot(d*x+c)^2*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-3/d^2*a*b^2*arctan(d*x+c)*arccot
(d*x+c)*f+3/2/d^2*a^2*b*c*f+3/d*a*b^2*ln(1+(d*x+c)^2)*arccot(d*x+c)*e+6*I/d*b^3*e*arccot(d*x+c)*polylog(2,-(d*
x+c+I)/(1+(d*x+c)^2)^(1/2))+6*I/d*b^3*e*arccot(d*x+c)*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-I/d^2*b^3*arcco
t(d*x+c)^3*c*f+3/4*I/d*a*b^2*ln(d*x+c-I)^2*e+3/2*I/d*a*b^2*dilog(-1/2*I*(d*x+c+I))*e+1/2*a^3*x^2*f+a^3*x*e-3/2
/d^2*a*b^2*f*arctan(d*x+c)^2-3/d^2*b^3*f*arccot(d*x+c)*ln(1-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-3/d^2*b^3*f*arccot(
d*x+c)*ln(1+(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-1/2/d^2*b^3*arccot(d*x+c)^3*c^2*f+3/2/d*b^3*arccot(d*x+c)^2*f*x+3*a
rccot(d*x+c)*x*a^2*b*e+3*arccot(d*x+c)^2*x*a*b^2*e+3/2*a^2*b*arccot(d*x+c)*f*x^2+3/2*a^2*b/d*f*x+3/2*I/d*a*b^2
*ln(d*x+c-I)*ln(-1/2*I*(d*x+c+I))*e-3/d^2*a*b^2*ln(1+(d*x+c)^2)*arccot(d*x+c)*c*f-6*I/d^2*b^3*c*f*arccot(d*x+c
)*polylog(2,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-6*I/d^2*b^3*c*f*arccot(d*x+c)*polylog(2,-(d*x+c+I)/(1+(d*x+c)^2)^(1
/2))-3/4*I/d^2*a*b^2*ln(d*x+c-I)^2*c*f-3/2*I/d^2*a*b^2*dilog(-1/2*I*(d*x+c+I))*c*f+3/4*I/d^2*a*b^2*ln(d*x+c+I)
^2*c*f+3/2*I/d^2*a*b^2*dilog(1/2*I*(d*x+c-I))*c*f-3/2*I/d*a*b^2*ln(d*x+c-I)*ln(1+(d*x+c)^2)*e-3/2*I/d*a*b^2*ln
(d*x+c+I)*ln(1/2*I*(d*x+c-I))*e+3/2*I/d*a*b^2*ln(d*x+c+I)*ln(1+(d*x+c)^2)*e-1/2/d^2*a^3*f*c^2+1/d*a^3*c*e+1/2*
b^3*arccot(d*x+c)^3*f*x^2+arccot(d*x+c)^3*x*b^3*e-6/d*b^3*e*polylog(3,-(d*x+c+I)/(1+(d*x+c)^2)^(1/2))-6/d*b^3*
e*polylog(3,(d*x+c+I)/(1+(d*x+c)^2)^(1/2))+1/2/d^2*b^3*arccot(d*x+c)^3*f+3/2/d^2*b^3*arccot(d*x+c)^2*f*c

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

[Out]

1/16*b^3*f*x^2*arctan2(1, d*x + c)^3 + 1/8*b^3*e*x*arctan2(1, d*x + c)^3 + 1/2*a^3*f*x^2 + 3/2*(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*f +
a^3*e*x + 3/2*(2*(d*x + c)*arccot(d*x + c) + log((d*x + c)^2 + 1))*a^2*b*e/d - 3/64*(b^3*f*x^2*arctan2(1, d*x
+ c) + 2*b^3*e*x*arctan2(1, d*x + c))*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + integrate(1/64*(8*(7*b^3*arctan2(1,
d*x + c)^3 + 24*a*b^2*arctan2(1, d*x + c)^2)*d^2*f*x^3 + 4*(2*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2
(1, d*x + c)^2)*d^2*e + (3*b^3*arctan2(1, d*x + c)^2 + 4*(7*b^3*arctan2(1, d*x + c)^3 + 24*a*b^2*arctan2(1, d*
x + c)^2)*c)*d*f)*x^2 + 3*(2*b^3*d^2*f*x^3*arctan2(1, d*x + c) + (2*b^3*d^2*e*arctan2(1, d*x + c) + (4*b^3*c*a
rctan2(1, d*x + c) - b^3)*d*f)*x^2 + 2*(b^3*c^2*arctan2(1, d*x + c) + b^3*arctan2(1, d*x + c))*e + 2*((2*b^3*c
*arctan2(1, d*x + c) - b^3)*d*e + (b^3*c^2*arctan2(1, d*x + c) + b^3*arctan2(1, d*x + c))*f)*x)*log(d^2*x^2 +
2*c*d*x + c^2 + 1)^2 + 8*(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 + 8*((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 + (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)*f)*x + 12*(b^3*d^2*f*x^3*arctan2(1, d*x
+ c) + 2*b^3*c*d*e*x*arctan2(1, d*x + c) + (2*b^3*d^2*e*arctan2(1, d*x + c) + b^3*c*d*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 x + a^{3} e +{\left (b^{3} f x + b^{3} e\right )} \operatorname{arccot}\left (d x + c\right )^{3} + 3 \,{\left (a b^{2} f x + a b^{2} e\right )} \operatorname{arccot}\left (d x + c\right )^{2} + 3 \,{\left (a^{2} b f x + a^{2} b e\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)*(a+b*arccot(d*x+c))^3,x, algorithm="fricas")

[Out]

integral(a^3*f*x + a^3*e + (b^3*f*x + b^3*e)*arccot(d*x + c)^3 + 3*(a*b^2*f*x + a*b^2*e)*arccot(d*x + c)^2 + 3
*(a^2*b*f*x + a^2*b*e)*arccot(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 )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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