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

Optimal. Leaf size=382 \[ \frac{i b^2 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{3 d^3}+\frac{i \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3}-\frac{(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3 f}-\frac{2 b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac{2 a b f x (d e-c f)}{d^2}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}+\frac{b^2 f (d e-c f) \log \left ((c+d x)^2+1\right )}{d^3}+\frac{2 b^2 f (c+d x) (d e-c f) \cot ^{-1}(c+d x)}{d^3}-\frac{b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}+\frac{b^2 f^2 x}{3 d^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.582392, antiderivative size = 382, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 13, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.65, Rules used = {5048, 4865, 4847, 260, 4853, 321, 203, 4985, 4885, 4921, 4855, 2402, 2315} \[ \frac{i b^2 \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1+i (c+d x)}\right )}{3 d^3}+\frac{i \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3}-\frac{(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3 f}-\frac{2 b \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac{2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac{2 a b f x (d e-c f)}{d^2}+\frac{b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac{(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}+\frac{b^2 f (d e-c f) \log \left ((c+d x)^2+1\right )}{d^3}+\frac{2 b^2 f (c+d x) (d e-c f) \cot ^{-1}(c+d x)}{d^3}-\frac{b^2 f^2 \tan ^{-1}(c+d x)}{3 d^3}+\frac{b^2 f^2 x}{3 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*f^2*x)/(3*d^2) + (2*a*b*f*(d*e - c*f)*x)/d^2 + (2*b^2*f*(d*e - c*f)*(c + d*x)*ArcCot[c + d*x])/d^3 + (b*f
^2*(c + d*x)^2*(a + b*ArcCot[c + d*x]))/(3*d^3) + ((I/3)*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*(a + b*ArcC
ot[c + d*x])^2)/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])^2)/(3*d^3*f)
+ ((e + f*x)^3*(a + b*ArcCot[c + d*x])^2)/(3*f) - (b^2*f^2*ArcTan[c + d*x])/(3*d^3) - (2*b*(3*d^2*e^2 - 6*c*d*
e*f - (1 - 3*c^2)*f^2)*(a + b*ArcCot[c + d*x])*Log[2/(1 + I*(c + d*x))])/(3*d^3) + (b^2*f*(d*e - c*f)*Log[1 +
(c + d*x)^2])/d^3 + ((I/3)*b^2*(3*d^2*e^2 - 6*c*d*e*f - (1 - 3*c^2)*f^2)*PolyLog[2, 1 - 2/(1 + I*(c + d*x))])/
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 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 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 321

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

Rule 203

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

Rule 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 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]

Rubi steps

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

Mathematica [A]  time = 4.98049, size = 665, normalized size = 1.74 \[ \frac{b^2 e f \left (-2 i c \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )+\left (-c^2-2 i c+d^2 x^2+1\right ) \cot ^{-1}(c+d x)^2-2 \log \left (\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right )+2 \cot ^{-1}(c+d x) \left (2 c \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )+c+d x\right )\right )}{d^2}+\frac{b^2 f^2 \left (4 i \left (3 c^2-1\right ) \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )+(c+d x) \left ((c+d x)^2+1\right ) \left (3 \left (c^2+1\right ) \cot ^{-1}(c+d x)^2-6 c \cot ^{-1}(c+d x)+1\right )-(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1} \left ((c+d x)^2+1\right ) \left (\left (3 c^2-1\right ) \cot ^{-1}(c+d x)^2-6 c \cot ^{-1}(c+d x)+1\right ) \cos \left (3 \cot ^{-1}(c+d x)\right )+2 \left ((c+d x)^2+1\right ) \left (-i \cot ^{-1}(c+d x)^2 \left (\left (3 c^2-1\right ) \cos \left (2 \cot ^{-1}(c+d x)\right )-3 c^2-6 i c+1\right )+2 \cot ^{-1}(c+d x) \left (\left (1-3 c^2\right ) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )+\left (3 c^2-1\right ) \cos \left (2 \cot ^{-1}(c+d x)\right ) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )+1\right )-6 c \log \left (\frac{1}{(c+d x) \sqrt{\frac{1}{(c+d x)^2}+1}}\right ) \left (\cos \left (2 \cot ^{-1}(c+d x)\right )-1\right )\right )\right )}{12 d^3}+\frac{b^2 e^2 \left (i \text{PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )+\cot ^{-1}(c+d x) \left ((c+d x+i) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )\right )}{d}+a^2 e^2 x+a^2 e f x^2+\frac{1}{3} a^2 f^2 x^3+\frac{a b \left (\left (\left (3 c^2-1\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (c^2+2 c d x+d^2 x^2+1\right )-2 \left (-3 c^2 d e f+c^3 f^2+3 c d^2 e^2-3 c f^2+3 d e f\right ) \tan ^{-1}(c+d x)+2 d^3 x \left (3 e^2+3 e f x+f^2 x^2\right ) \cot ^{-1}(c+d x)+d f x (-4 c f+6 d e+d f x)\right )}{3 d^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

a^2*e^2*x + a^2*e*f*x^2 + (a^2*f^2*x^3)/3 + (a*b*(d*f*x*(6*d*e - 4*c*f + d*f*x) + 2*d^3*x*(3*e^2 + 3*e*f*x + f
^2*x^2)*ArcCot[c + d*x] - 2*(3*c*d^2*e^2 + 3*d*e*f - 3*c^2*d*e*f - 3*c*f^2 + c^3*f^2)*ArcTan[c + d*x] + (3*d^2
*e^2 - 6*c*d*e*f + (-1 + 3*c^2)*f^2)*Log[1 + c^2 + 2*c*d*x + d^2*x^2]))/(3*d^3) + (b^2*e^2*(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])]
))/d + (b^2*e*f*((1 - (2*I)*c - c^2 + d^2*x^2)*ArcCot[c + d*x]^2 + 2*ArcCot[c + d*x]*(c + d*x + 2*c*Log[1 - E^
((2*I)*ArcCot[c + d*x])]) - 2*Log[1/((c + d*x)*Sqrt[1 + (c + d*x)^(-2)])] - (2*I)*c*PolyLog[2, E^((2*I)*ArcCot
[c + d*x])]))/d^2 + (b^2*f^2*((c + d*x)*(1 + (c + d*x)^2)*(1 - 6*c*ArcCot[c + d*x] + 3*(1 + c^2)*ArcCot[c + d*
x]^2) - (c + d*x)*Sqrt[1 + (c + d*x)^(-2)]*(1 + (c + d*x)^2)*(1 - 6*c*ArcCot[c + d*x] + (-1 + 3*c^2)*ArcCot[c
+ d*x]^2)*Cos[3*ArcCot[c + d*x]] + 2*(1 + (c + d*x)^2)*((-I)*ArcCot[c + d*x]^2*(1 - (6*I)*c - 3*c^2 + (-1 + 3*
c^2)*Cos[2*ArcCot[c + d*x]]) + 2*ArcCot[c + d*x]*(1 + (1 - 3*c^2)*Log[1 - E^((2*I)*ArcCot[c + d*x])] + (-1 + 3
*c^2)*Cos[2*ArcCot[c + d*x]]*Log[1 - E^((2*I)*ArcCot[c + d*x])]) - 6*c*(-1 + Cos[2*ArcCot[c + d*x]])*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])]))/(12*d^3)

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Maple [B]  time = 0.153, size = 1832, normalized size = 4.8 \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)^2*(a+b*arccot(d*x+c))^2,x)

[Out]

a^2*x*e^2+1/3*a^2*f^2*x^3+2/3*a*b/f*arctan(d*x+c)*e^3+2/3*a*b/f*arccot(d*x+c)*e^3+2*arccot(d*x+c)*x*a*b*e^2+2/
3*a*b*f^2*arccot(d*x+c)*x^3+b^2*f*arccot(d*x+c)^2*e*x^2+2/3*b^2/f*arccot(d*x+c)*arctan(d*x+c)*e^3+1/6*I/d^3*b^
2*dilog(1/2*I*(d*x+c-I))*f^2-1/2*I/d*b^2*dilog(1/2*I*(d*x+c-I))*e^2+1/4*I/d*b^2*ln(d*x+c-I)^2*e^2-1/4*I/d*b^2*
ln(d*x+c+I)^2*e^2+1/2*I/d*b^2*dilog(-1/2*I*(d*x+c+I))*e^2+1/12*I/d^3*b^2*ln(d*x+c+I)^2*f^2+1/d^2*b^2*f*ln(1+(d
*x+c)^2)*e-1/d^2*b^2*f*arctan(d*x+c)^2*e-1/3/d^3*b^2*f^2*arccot(d*x+c)*ln(1+(d*x+c)^2)-1/d*b^2*arctan(d*x+c)^2
*c*e^2+1/d*b^2*arccot(d*x+c)*ln(1+(d*x+c)^2)*e^2-1/3/d^3*b^2*f^2*arctan(d*x+c)^2*c^3+1/3/d*a*b*f^2*x^2-1/6*I/d
^3*b^2*dilog(-1/2*I*(d*x+c+I))*f^2-1/d^3*b^2*f^2*ln(1+(d*x+c)^2)*c+1/d^3*b^2*f^2*arctan(d*x+c)^2*c-1/3/d^3*a*b
*f^2*ln(1+(d*x+c)^2)-5/3/d^3*b^2*f^2*arccot(d*x+c)*c^2+1/3/d*b^2*f^2*arccot(d*x+c)*x^2+1/d*a*b*ln(1+(d*x+c)^2)
*e^2-5/3/d^3*a*b*f^2*c^2+I/d^2*b^2*ln(d*x+c-I)*ln(1+(d*x+c)^2)*c*e*f-I/d^2*b^2*ln(d*x+c-I)*ln(-1/2*I*(d*x+c+I)
)*c*e*f-I/d^2*b^2*ln(1+(d*x+c)^2)*ln(d*x+c+I)*c*e*f+I/d^2*b^2*ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))*c*e*f+1/d^3*a*b*
f^2*ln(1+(d*x+c)^2)*c^2-2/d^2*a*b*f*arctan(d*x+c)*e-2/d*b^2*arccot(d*x+c)*arctan(d*x+c)*c*e^2-2/d*a*b*arctan(d
*x+c)*c*e^2+2/d^3*b^2*f^2*arccot(d*x+c)*arctan(d*x+c)*c-4/3/d^2*b^2*f^2*arccot(d*x+c)*c*x+2/d*b^2*f*arccot(d*x
+c)*e*x+2/d^2*b^2*f*arccot(d*x+c)*e*c-2/3/d^3*a*b*f^2*arctan(d*x+c)*c^3+2/d^2*a*b*f*c*e-1/6*I/d^3*b^2*ln(d*x+c
-I)*ln(-1/2*I*(d*x+c+I))*f^2+1/6*I/d^3*b^2*ln(d*x+c-I)*ln(1+(d*x+c)^2)*f^2-1/2*I/d^3*b^2*dilog(1/2*I*(d*x+c-I)
)*c^2*f^2-1/6*I/d^3*b^2*ln(1+(d*x+c)^2)*ln(d*x+c+I)*f^2+1/2*I/d*b^2*ln(1+(d*x+c)^2)*ln(d*x+c+I)*e^2-1/2*I/d*b^
2*ln(d*x+c-I)*ln(1+(d*x+c)^2)*e^2+1/6*I/d^3*b^2*ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))*f^2-1/4*I/d^3*b^2*ln(d*x+c+I)^
2*c^2*f^2+1/4*I/d^3*b^2*ln(d*x+c-I)^2*c^2*f^2-2/d^2*b^2*f*arccot(d*x+c)*arctan(d*x+c)*e-2/3/d^3*b^2*f^2*arccot
(d*x+c)*arctan(d*x+c)*c^3+1/d^3*b^2*f^2*arccot(d*x+c)*ln(1+(d*x+c)^2)*c^2-1/2*I/d*b^2*ln(d*x+c+I)*ln(1/2*I*(d*
x+c-I))*e^2+1/2*I/d*b^2*ln(d*x+c-I)*ln(-1/2*I*(d*x+c+I))*e^2+1/2*I/d^3*b^2*dilog(-1/2*I*(d*x+c+I))*c^2*f^2-4/3
*a*b/d^2*f^2*c*x+2*a*b/d*f*e*x+2*a*b*f*arccot(d*x+c)*e*x^2+1/d^2*b^2*f*arctan(d*x+c)^2*c^2*e+2/d^3*a*b*f^2*arc
tan(d*x+c)*c-1/12*I/d^3*b^2*ln(d*x+c-I)^2*f^2+2/d^2*b^2*f*arccot(d*x+c)*arctan(d*x+c)*c^2*e-I/d^2*b^2*dilog(-1
/2*I*(d*x+c+I))*c*e*f+1/2*I/d^2*b^2*ln(d*x+c+I)^2*c*e*f+1/3*b^2*f^2*arccot(d*x+c)^2*x^3+arccot(d*x+c)^2*x*b^2*
e^2+a^2*f*x^2*e+1/3*b^2/f*arctan(d*x+c)^2*e^3+1/3*b^2/f*arccot(d*x+c)^2*e^3+1/3/d^3*b^2*f^2*c+1/3*a^2/f*e^3+1/
2*I/d^3*b^2*ln(1+(d*x+c)^2)*ln(d*x+c+I)*c^2*f^2-1/2*I/d^3*b^2*ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))*c^2*f^2-1/2*I/d^
2*b^2*ln(d*x+c-I)^2*c*e*f+1/2*I/d^3*b^2*ln(d*x+c-I)*ln(-1/2*I*(d*x+c+I))*c^2*f^2-1/2*I/d^3*b^2*ln(d*x+c-I)*ln(
1+(d*x+c)^2)*c^2*f^2+I/d^2*b^2*dilog(1/2*I*(d*x+c-I))*c*e*f+2/d^2*a*b*f*arctan(d*x+c)*c^2*e-2/d^2*b^2*f*arccot
(d*x+c)*ln(1+(d*x+c)^2)*c*e-2/d^2*a*b*f*ln(1+(d*x+c)^2)*c*e+1/3*b^2*f^2*x/d^2-1/3*b^2*f^2*arctan(d*x+c)/d^3

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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))^2,x, algorithm="maxima")

[Out]

1/12*b^2*f^2*x^3*arctan2(1, d*x + c)^2 + 1/4*b^2*e*f*x^2*arctan2(1, d*x + c)^2 + 1/3*a^2*f^2*x^3 + 1/4*b^2*e^2
*x*arctan2(1, d*x + c)^2 + a^2*e*f*x^2 + 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*b*e*f + 1/3*(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*b*f^2 + a^2
*e^2*x + (2*(d*x + c)*arccot(d*x + c) + log((d*x + c)^2 + 1))*a*b*e^2/d - 1/48*(b^2*f^2*x^3 + 3*b^2*e*f*x^2 +
3*b^2*e^2*x)*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2 + integrate(1/48*(36*b^2*d^2*f^2*x^4*arctan2(1, d*x + c)^2 + 8
*(9*b^2*d^2*e*f*arctan2(1, d*x + c)^2 + (9*b^2*c*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c))*d*f^2)*x^3 +
36*(b^2*c^2*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c)^2)*e^2 + 12*(3*b^2*d^2*e^2*arctan2(1, d*x + c)^2
+ 2*(6*b^2*c*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c))*d*e*f + 3*(b^2*c^2*arctan2(1, d*x + c)^2 + b^2*a
rctan2(1, d*x + c)^2)*f^2)*x^2 + 3*(b^2*d^2*f^2*x^4 + 2*(b^2*d^2*e*f + b^2*c*d*f^2)*x^3 + (b^2*c^2 + b^2)*e^2
+ (b^2*d^2*e^2 + 4*b^2*c*d*e*f + (b^2*c^2 + b^2)*f^2)*x^2 + 2*(b^2*c*d*e^2 + (b^2*c^2 + b^2)*e*f)*x)*log(d^2*x
^2 + 2*c*d*x + c^2 + 1)^2 + 24*((3*b^2*c*arctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c))*d*e^2 + 3*(b^2*c^2*a
rctan2(1, d*x + c)^2 + b^2*arctan2(1, d*x + c)^2)*e*f)*x + 4*(b^2*d^2*f^2*x^4 + 3*b^2*c*d*e^2*x + (3*b^2*d^2*e
*f + b^2*c*d*f^2)*x^3 + 3*(b^2*d^2*e^2 + b^2*c*d*e*f)*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^{2} f^{2} x^{2} + 2 \, a^{2} e f x + a^{2} e^{2} +{\left (b^{2} f^{2} x^{2} + 2 \, b^{2} e f x + b^{2} e^{2}\right )} \operatorname{arccot}\left (d x + c\right )^{2} + 2 \,{\left (a b f^{2} x^{2} + 2 \, a b e f x + a 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))^2,x, algorithm="fricas")

[Out]

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

[Out]

Integral((a + b*acot(c + d*x))**2*(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 )}^{2}\,{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))^2,x, algorithm="giac")

[Out]

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