\(\int (e+f x)^2 (a+b \cot ^{-1}(c+d x))^2 \, dx\) [23]

Optimal result

Integrand size = 20, antiderivative size = 382 \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\frac {b^2 f^2 x}{3 d^2}+\frac {2 a b f (d e-c f) x}{d^2}+\frac {2 b^2 f (d e-c f) (c+d x) \cot ^{-1}(c+d x)}{d^3}+\frac {b f^2 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )}{3 d^3}+\frac {i \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3}-\frac {(d e-c f) \left (d^2 e^2-2 c d e f-\left (3-c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 d^3 f}+\frac {(e+f x)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}-\frac {b^2 f^2 \arctan (c+d x)}{3 d^3}-\frac {2 b \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \left (a+b \cot ^{-1}(c+d x)\right ) \log \left (\frac {2}{1+i (c+d x)}\right )}{3 d^3}+\frac {b^2 f (d e-c f) \log \left (1+(c+d x)^2\right )}{d^3}+\frac {i b^2 \left (3 d^2 e^2-6 c d e f-\left (1-3 c^2\right ) f^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{3 d^3} \] Output:

1/3*b^2*f^2*x/d^2+2*a*b*f*(-c*f+d*e)*x/d^2+2*b^2*f*(-c*f+d*e)*(d*x+c)*arcc 
ot(d*x+c)/d^3+1/3*b*f^2*(d*x+c)^2*(a+b*arccot(d*x+c))/d^3+1/3*I*(3*d^2*e^2 
-6*c*d*e*f-(-3*c^2+1)*f^2)*(a+b*arccot(d*x+c))^2/d^3-1/3*(-c*f+d*e)*(d^2*e 
^2-2*c*d*e*f-(-c^2+3)*f^2)*(a+b*arccot(d*x+c))^2/d^3/f+1/3*(f*x+e)^3*(a+b* 
arccot(d*x+c))^2/f-1/3*b^2*f^2*arctan(d*x+c)/d^3-2/3*b*(3*d^2*e^2-6*c*d*e* 
f-(-3*c^2+1)*f^2)*(a+b*arccot(d*x+c))*ln(2/(1+I*(d*x+c)))/d^3+b^2*f*(-c*f+ 
d*e)*ln(1+(d*x+c)^2)/d^3+1/3*I*b^2*(3*d^2*e^2-6*c*d*e*f-(-3*c^2+1)*f^2)*po 
lylog(2,1-2/(1+I*(d*x+c)))/d^3
 

Mathematica [A] (warning: unable to verify)

Time = 8.25 (sec) , antiderivative size = 665, normalized size of antiderivative = 1.74 \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=a^2 e^2 x+a^2 e f x^2+\frac {1}{3} a^2 f^2 x^3+\frac {a b \left (d f x (6 d e-4 c f+d f x)+2 d^3 x \left (3 e^2+3 e f x+f^2 x^2\right ) \cot ^{-1}(c+d x)-2 \left (3 c d^2 e^2+3 d e f-3 c^2 d e f-3 c f^2+c^3 f^2\right ) \arctan (c+d x)+\left (3 d^2 e^2-6 c d e f+\left (-1+3 c^2\right ) f^2\right ) \log \left (1+c^2+2 c d x+d^2 x^2\right )\right )}{3 d^3}+\frac {b^2 e^2 \left (\cot ^{-1}(c+d x) \left ((i+c+d x) \cot ^{-1}(c+d x)-2 \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )+i \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )\right )}{d}+\frac {b^2 e f \left (\left (1-2 i c-c^2+d^2 x^2\right ) \cot ^{-1}(c+d x)^2+2 \cot ^{-1}(c+d x) \left (c+d x+2 c \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )-2 \log \left (\frac {1}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )-2 i c \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )\right )}{d^2}+\frac {b^2 f^2 \left ((c+d x) \left (1+(c+d x)^2\right ) \left (1-6 c \cot ^{-1}(c+d x)+3 \left (1+c^2\right ) \cot ^{-1}(c+d x)^2\right )-(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}} \left (1+(c+d x)^2\right ) \left (1-6 c \cot ^{-1}(c+d x)+\left (-1+3 c^2\right ) \cot ^{-1}(c+d x)^2\right ) \cos \left (3 \cot ^{-1}(c+d x)\right )+2 \left (1+(c+d x)^2\right ) \left (-i \cot ^{-1}(c+d x)^2 \left (1-6 i c-3 c^2+\left (-1+3 c^2\right ) \cos \left (2 \cot ^{-1}(c+d x)\right )\right )+2 \cot ^{-1}(c+d x) \left (1+\left (1-3 c^2\right ) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )+\left (-1+3 c^2\right ) \cos \left (2 \cot ^{-1}(c+d x)\right ) \log \left (1-e^{2 i \cot ^{-1}(c+d x)}\right )\right )-6 c \left (-1+\cos \left (2 \cot ^{-1}(c+d x)\right )\right ) \log \left (\frac {1}{(c+d x) \sqrt {1+\frac {1}{(c+d x)^2}}}\right )\right )+4 i \left (-1+3 c^2\right ) \operatorname {PolyLog}\left (2,e^{2 i \cot ^{-1}(c+d x)}\right )\right )}{12 d^3} \] Input:

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

Output:

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*A 
rcCot[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)
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5571, 27, 5390, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5571

\(\displaystyle \frac {\int \frac {\left (d \left (e-\frac {c f}{d}\right )+f (c+d x)\right )^2 \left (a+b \cot ^{-1}(c+d x)\right )^2}{d^2}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int (d e-c f+f (c+d x))^2 \left (a+b \cot ^{-1}(c+d x)\right )^2d(c+d x)}{d^3}\)

\(\Big \downarrow \) 5390

\(\displaystyle \frac {\frac {2 b \int \left ((c+d x) \left (a+b \cot ^{-1}(c+d x)\right ) f^3+3 (d e-c f) \left (a+b \cot ^{-1}(c+d x)\right ) f^2+\frac {\left ((d e-c f) \left (d^2 e^2-2 c d f e-\left (3-c^2\right ) f^2\right )+f \left (3 d^2 e^2-6 c d f e-\left (1-3 c^2\right ) f^2\right ) (c+d x)\right ) \left (a+b \cot ^{-1}(c+d x)\right )}{(c+d x)^2+1}\right )d(c+d x)}{3 f}+\frac {(f (c+d x)-c f+d e)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}}{d^3}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {(f (c+d x)-c f+d e)^3 \left (a+b \cot ^{-1}(c+d x)\right )^2}{3 f}+\frac {2 b \left (\frac {i f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 b}-\frac {(d e-c f) \left (-\left (3-c^2\right ) f^2-2 c d e f+d^2 e^2\right ) \left (a+b \cot ^{-1}(c+d x)\right )^2}{2 b}-f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \log \left (\frac {2}{1+i (c+d x)}\right ) \left (a+b \cot ^{-1}(c+d x)\right )+\frac {1}{2} f^3 (c+d x)^2 \left (a+b \cot ^{-1}(c+d x)\right )+3 a f^2 (c+d x) (d e-c f)-\frac {1}{2} b f^3 \arctan (c+d x)+\frac {1}{2} i b f \left (-\left (1-3 c^2\right ) f^2-6 c d e f+3 d^2 e^2\right ) \operatorname {PolyLog}\left (2,1-\frac {2}{i (c+d x)+1}\right )+\frac {3}{2} b f^2 (d e-c f) \log \left ((c+d x)^2+1\right )+3 b f^2 (c+d x) (d e-c f) \cot ^{-1}(c+d x)+\frac {1}{2} b f^3 (c+d x)\right )}{3 f}}{d^3}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5390

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_))^(q_.), x_Sy 
mbol] :> Simp[(d + e*x)^(q + 1)*((a + b*ArcCot[c*x])^p/(e*(q + 1))), x] + S 
imp[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] && NeQ[q, -1]
 

rule 5571

Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m 
_.), x_Symbol] :> Simp[1/d   Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b*A 
rcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && I 
GtQ[p, 0]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1071 vs. \(2 (362 ) = 724\).

Time = 2.13 (sec) , antiderivative size = 1072, normalized size of antiderivative = 2.81

method	result	size
parts	\(\text {Expression too large to display}\)	\(1072\)
derivativedivides	\(\text {Expression too large to display}\)	\(1087\)
default	\(\text {Expression too large to display}\)	\(1087\)
risch	\(\text {Expression too large to display}\)	\(3165\)

Input:

int((f*x+e)^2*(a+b*arccot(d*x+c))^2,x,method=_RETURNVERBOSE)
 

Output:

1/3*a^2*(f*x+e)^3/f+b^2/d*(1/3/d^2*f^2*arccot(d*x+c)^2*(d*x+c)^3-1/d^2*f^2 
*arccot(d*x+c)^2*(d*x+c)^2*c+1/d*f*arccot(d*x+c)^2*(d*x+c)^2*e+1/d^2*f^2*a 
rccot(d*x+c)^2*(d*x+c)*c^2-2/d*f*arccot(d*x+c)^2*(d*x+c)*c*e+arccot(d*x+c) 
^2*(d*x+c)*e^2-1/3/d^2*f^2*arccot(d*x+c)^2*c^3+1/d*f*arccot(d*x+c)^2*c^2*e 
-arccot(d*x+c)^2*c*e^2+1/3*d/f*arccot(d*x+c)^2*e^3+2/3/d^2/f*(1/2*arccot(d 
*x+c)*f^3*(d*x+c)^2-3*arccot(d*x+c)*c*f^3*(d*x+c)+3*arccot(d*x+c)*d*e*f^2* 
(d*x+c)+3/2*arccot(d*x+c)*ln(1+(d*x+c)^2)*c^2*f^3-3*arccot(d*x+c)*ln(1+(d* 
x+c)^2)*c*d*e*f^2+3/2*arccot(d*x+c)*ln(1+(d*x+c)^2)*d^2*e^2*f-1/2*arccot(d 
*x+c)*ln(1+(d*x+c)^2)*f^3-arccot(d*x+c)*arctan(d*x+c)*c^3*f^3+3*arccot(d*x 
+c)*arctan(d*x+c)*c^2*d*e*f^2-3*arccot(d*x+c)*arctan(d*x+c)*c*d^2*e^2*f+ar 
ccot(d*x+c)*arctan(d*x+c)*d^3*e^3+3*arccot(d*x+c)*arctan(d*x+c)*c*f^3-3*ar 
ccot(d*x+c)*arctan(d*x+c)*d*e*f^2+1/2*f^2*(f*(d*x+c)+1/2*(-6*c*f+6*d*e)*ln 
(1+(d*x+c)^2)-f*arctan(d*x+c))+1/2*f*(3*c^2*f^2-6*c*d*e*f+3*d^2*e^2-f^2)*( 
-1/2*I*(ln(d*x+c-I)*ln(1+(d*x+c)^2)-1/2*ln(d*x+c-I)^2-dilog(-1/2*I*(d*x+c+ 
I))-ln(d*x+c-I)*ln(-1/2*I*(d*x+c+I)))+1/2*I*(ln(d*x+c+I)*ln(1+(d*x+c)^2)-1 
/2*ln(d*x+c+I)^2-dilog(1/2*I*(d*x+c-I))-ln(d*x+c+I)*ln(1/2*I*(d*x+c-I))))+ 
1/4*(-2*c^3*f^3+6*c^2*d*e*f^2-6*c*d^2*e^2*f+2*d^3*e^3+6*c*f^3-6*d*e*f^2)*a 
rctan(d*x+c)^2))+2/d^2*c*f*e*b*a-1/3*a*b/d^3*f^2*ln(1+(d*x+c)^2)-2*a*b/d^2 
*f*ln(1+(d*x+c)^2)*c*e+2*b/d^2*arctan(d*x+c)*a*c^2*e*f-2*b/d*arctan(d*x+c) 
*a*c*e^2+1/3/d*f^2*b*a*x^2-5/3/d^3*c^2*f^2*b*a+2*a*b*f*arccot(d*x+c)*e*...
 

Fricas [F]

\[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:

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

Output:

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)*arc 
cot(d*x + c), x)
 

Sympy [F]

\[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {acot}{\left (c + d x \right )}\right )^{2} \left (e + f x\right )^{2}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:

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

Output:

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*arc 
tan2(1, d*x + c))*d*f^2)*x^3 + 36*(b^2*c^2*arctan2(1, d*x + c)^2 + b^2*arc 
tan2(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*a 
rctan2(1, d*x + c)^2 + b^2*arctan2(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*arctan2(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*...
 

Giac [F]

\[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\int { {\left (f x + e\right )}^{2} {\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx=\int {\left (e+f\,x\right )}^2\,{\left (a+b\,\mathrm {acot}\left (c+d\,x\right )\right )}^2 \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int (e+f x)^2 \left (a+b \cot ^{-1}(c+d x)\right )^2 \, dx =\text {Too large to display} \] Input:

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

Output:

( - 2*acot(c + d*x)**2*b**2*c**3*f**2 + 3*acot(c + d*x)**2*b**2*c**2*d*e*f 
 - 2*acot(c + d*x)**2*b**2*c*f**2 + 3*acot(c + d*x)**2*b**2*d**3*e**2*x + 
3*acot(c + d*x)**2*b**2*d**3*e*f*x**2 + acot(c + d*x)**2*b**2*d**3*f**2*x* 
*3 + 3*acot(c + d*x)**2*b**2*d*e*f + 2*acot(c + d*x)*a*b*c**3*f**2 - 6*aco 
t(c + d*x)*a*b*c**2*d*e*f + 6*acot(c + d*x)*a*b*c*d**2*e**2 - 6*acot(c + d 
*x)*a*b*c*f**2 + 6*acot(c + d*x)*a*b*d**3*e**2*x + 6*acot(c + d*x)*a*b*d** 
3*e*f*x**2 + 2*acot(c + d*x)*a*b*d**3*f**2*x**3 + 6*acot(c + d*x)*a*b*d*e* 
f - 5*acot(c + d*x)*b**2*c**2*f**2 + 6*acot(c + d*x)*b**2*c*d*e*f - 4*acot 
(c + d*x)*b**2*c*d*f**2*x + 6*acot(c + d*x)*b**2*d**2*e*f*x + acot(c + d*x 
)*b**2*d**2*f**2*x**2 + acot(c + d*x)*b**2*f**2 + 6*int((acot(c + d*x)*x)/ 
(c**2 + 2*c*d*x + d**2*x**2 + 1),x)*b**2*c**2*d**2*f**2 - 12*int((acot(c + 
 d*x)*x)/(c**2 + 2*c*d*x + d**2*x**2 + 1),x)*b**2*c*d**3*e*f + 6*int((acot 
(c + d*x)*x)/(c**2 + 2*c*d*x + d**2*x**2 + 1),x)*b**2*d**4*e**2 - 2*int((a 
cot(c + d*x)*x)/(c**2 + 2*c*d*x + d**2*x**2 + 1),x)*b**2*d**2*f**2 + 3*log 
(c**2 + 2*c*d*x + d**2*x**2 + 1)*a*b*c**2*f**2 - 6*log(c**2 + 2*c*d*x + d* 
*2*x**2 + 1)*a*b*c*d*e*f + 3*log(c**2 + 2*c*d*x + d**2*x**2 + 1)*a*b*d**2* 
e**2 - log(c**2 + 2*c*d*x + d**2*x**2 + 1)*a*b*f**2 - 3*log(c**2 + 2*c*d*x 
 + d**2*x**2 + 1)*b**2*c*f**2 + 3*log(c**2 + 2*c*d*x + d**2*x**2 + 1)*b**2 
*d*e*f + 3*a**2*d**3*e**2*x + 3*a**2*d**3*e*f*x**2 + a**2*d**3*f**2*x**3 - 
 4*a*b*c*d*f**2*x + 6*a*b*d**2*e*f*x + a*b*d**2*f**2*x**2 + b**2*d*f**2...