\(\int (c e+d e x) (a+b \arctan (c+d x))^2 \, dx\) [17]

Optimal result

Integrand size = 21, antiderivative size = 95 \[ \int (c e+d e x) (a+b \arctan (c+d x))^2 \, dx=-a b e x-\frac {b^2 e (c+d x) \arctan (c+d x)}{d}+\frac {e (a+b \arctan (c+d x))^2}{2 d}+\frac {e (c+d x)^2 (a+b \arctan (c+d x))^2}{2 d}+\frac {b^2 e \log \left (1+(c+d x)^2\right )}{2 d} \] Output:

-a*b*e*x-b^2*e*(d*x+c)*arctan(d*x+c)/d+1/2*e*(a+b*arctan(d*x+c))^2/d+1/2*e 
*(d*x+c)^2*(a+b*arctan(d*x+c))^2/d+1/2*b^2*e*ln(1+(d*x+c)^2)/d
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.13 \[ \int (c e+d e x) (a+b \arctan (c+d x))^2 \, dx=\frac {e \left (a (c+d x) (-2 b+a c+a d x)+2 b \left (-b (c+d x)+a \left (1+c^2+2 c d x+d^2 x^2\right )\right ) \arctan (c+d x)+b^2 \left (1+c^2+2 c d x+d^2 x^2\right ) \arctan (c+d x)^2+b^2 \log \left (1+(c+d x)^2\right )\right )}{2 d} \] Input:

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

Output:

(e*(a*(c + d*x)*(-2*b + a*c + a*d*x) + 2*b*(-(b*(c + d*x)) + a*(1 + c^2 + 
2*c*d*x + d^2*x^2))*ArcTan[c + d*x] + b^2*(1 + c^2 + 2*c*d*x + d^2*x^2)*Ar 
cTan[c + d*x]^2 + b^2*Log[1 + (c + d*x)^2]))/(2*d)
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5566, 27, 5361, 5451, 2009, 5419}

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.

\(\displaystyle \int (c e+d e x) (a+b \arctan (c+d x))^2 \, dx\)

\(\Big \downarrow \) 5566

\(\displaystyle \frac {\int e (c+d x) (a+b \arctan (c+d x))^2d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e \int (c+d x) (a+b \arctan (c+d x))^2d(c+d x)}{d}\)

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5451

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5419

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

Input:

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

Output:

(e*(((c + d*x)^2*(a + b*ArcTan[c + d*x])^2)/2 - b*(a*(c + d*x) + b*(c + d* 
x)*ArcTan[c + d*x] - (a + b*ArcTan[c + d*x])^2/(2*b) - (b*Log[1 + (c + d*x 
)^2])/2)))/d
 

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 5361

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> 
 Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 
1))   Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], 
x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & 
& IntegerQ[m])) && NeQ[m, -1]
 

rule 5419

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo 
l] :> Simp[(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, 
c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
 

rule 5451

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e 
_.)*(x_)^2), x_Symbol] :> Simp[f^2/e   Int[(f*x)^(m - 2)*(a + b*ArcTan[c*x] 
)^p, x], x] - Simp[d*(f^2/e)   Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/(d 
+ e*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && GtQ[m, 1]
 

rule 5566

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

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(\frac {\frac {e \,a^{2} \left (d x +c \right )^{2}}{2}+e \,b^{2} \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )^{2}}{2}+\frac {\arctan \left (d x +c \right )^{2}}{2}-\left (d x +c \right ) \arctan \left (d x +c \right )+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )+2 e a b \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}+\frac {\arctan \left (d x +c \right )}{2}\right )}{d}\)	\(113\)
default	\(\frac {\frac {e \,a^{2} \left (d x +c \right )^{2}}{2}+e \,b^{2} \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )^{2}}{2}+\frac {\arctan \left (d x +c \right )^{2}}{2}-\left (d x +c \right ) \arctan \left (d x +c \right )+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )+2 e a b \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}+\frac {\arctan \left (d x +c \right )}{2}\right )}{d}\)	\(113\)
parts	\(e \,a^{2} \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e \,b^{2} \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )^{2}}{2}+\frac {\arctan \left (d x +c \right )^{2}}{2}-\left (d x +c \right ) \arctan \left (d x +c \right )+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{d}+\frac {2 e a b \left (\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}+\frac {\arctan \left (d x +c \right )}{2}\right )}{d}\)	\(117\)
parallelrisch	\(\frac {d^{3} e \,b^{2} \arctan \left (d x +c \right )^{2} x^{2}+2 x^{2} \arctan \left (d x +c \right ) a b \,d^{3} e +2 c e \,b^{2} \arctan \left (d x +c \right )^{2} x \,d^{2}+x^{2} a^{2} d^{3} e +4 x \arctan \left (d x +c \right ) a b c \,d^{2} e +\arctan \left (d x +c \right )^{2} b^{2} c^{2} d e -2 x \arctan \left (d x +c \right ) b^{2} d^{2} e +2 x \,a^{2} c \,d^{2} e +2 \arctan \left (d x +c \right ) a b \,c^{2} d e -2 x a b \,d^{2} e +e \,b^{2} \arctan \left (d x +c \right )^{2} d -2 \arctan \left (d x +c \right ) b^{2} c d e -5 a^{2} c^{2} d e +e \,b^{2} \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) d +2 \arctan \left (d x +c \right ) a b d e +4 a b c d e -d e \,a^{2}}{2 d^{2}}\)	\(245\)
risch	\(-\frac {e \,b^{2} \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) \ln \left (1+i \left (d x +c \right )\right )^{2}}{8 d}+\frac {e b \left (-2 i a \,d^{2} x^{2}+b \,d^{2} x^{2} \ln \left (1-i \left (d x +c \right )\right )-4 i a c x d +2 b c d x \ln \left (1-i \left (d x +c \right )\right )+2 i b d x +\ln \left (1-i \left (d x +c \right )\right ) b \,c^{2}+b \ln \left (1-i \left (d x +c \right )\right )\right ) \ln \left (1+i \left (d x +c \right )\right )}{4 d}-\frac {e d \,b^{2} x^{2} \ln \left (1-i \left (d x +c \right )\right )^{2}}{8}-\frac {i e \,b^{2} x \ln \left (1-i \left (d x +c \right )\right )}{2}-\frac {e \,b^{2} c x \ln \left (1-i \left (d x +c \right )\right )^{2}}{4}+i e a b c x \ln \left (1-i \left (d x +c \right )\right )-\frac {e \,b^{2} c^{2} \ln \left (1-i \left (d x +c \right )\right )^{2}}{8 d}+\frac {i e d a b \,x^{2} \ln \left (1-i \left (d x +c \right )\right )}{2}+\frac {e \,a^{2} d \,x^{2}}{2}+\frac {e a b \,c^{2} \arctan \left (d x +c \right )}{d}+e \,a^{2} c x -\frac {e \,b^{2} \ln \left (1-i \left (d x +c \right )\right )^{2}}{8 d}-\frac {e \,b^{2} c \arctan \left (d x +c \right )}{d}-a b e x +\frac {e a b \arctan \left (d x +c \right )}{d}+\frac {e \,b^{2} \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right )}{2 d}\)	\(393\)

Input:

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

Output:

1/d*(1/2*e*a^2*(d*x+c)^2+e*b^2*(1/2*(d*x+c)^2*arctan(d*x+c)^2+1/2*arctan(d 
*x+c)^2-(d*x+c)*arctan(d*x+c)+1/2*ln(1+(d*x+c)^2))+2*e*a*b*(1/2*(d*x+c)^2* 
arctan(d*x+c)-1/2*d*x-1/2*c+1/2*arctan(d*x+c)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.58 \[ \int (c e+d e x) (a+b \arctan (c+d x))^2 \, dx=\frac {a^{2} d^{2} e x^{2} + 2 \, {\left (a^{2} c - a b\right )} d e x + b^{2} e \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) + {\left (b^{2} d^{2} e x^{2} + 2 \, b^{2} c d e x + {\left (b^{2} c^{2} + b^{2}\right )} e\right )} \arctan \left (d x + c\right )^{2} + 2 \, {\left (a b d^{2} e x^{2} + {\left (2 \, a b c - b^{2}\right )} d e x + {\left (a b c^{2} - b^{2} c + a b\right )} e\right )} \arctan \left (d x + c\right )}{2 \, d} \] Input:

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

Output:

1/2*(a^2*d^2*e*x^2 + 2*(a^2*c - a*b)*d*e*x + b^2*e*log(d^2*x^2 + 2*c*d*x + 
 c^2 + 1) + (b^2*d^2*e*x^2 + 2*b^2*c*d*e*x + (b^2*c^2 + b^2)*e)*arctan(d*x 
 + c)^2 + 2*(a*b*d^2*e*x^2 + (2*a*b*c - b^2)*d*e*x + (a*b*c^2 - b^2*c + a* 
b)*e)*arctan(d*x + c))/d
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.34 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.53 \[ \int (c e+d e x) (a+b \arctan (c+d x))^2 \, dx=\begin {cases} a^{2} c e x + \frac {a^{2} d e x^{2}}{2} + \frac {a b c^{2} e \operatorname {atan}{\left (c + d x \right )}}{d} + 2 a b c e x \operatorname {atan}{\left (c + d x \right )} + a b d e x^{2} \operatorname {atan}{\left (c + d x \right )} - a b e x + \frac {a b e \operatorname {atan}{\left (c + d x \right )}}{d} + \frac {b^{2} c^{2} e \operatorname {atan}^{2}{\left (c + d x \right )}}{2 d} + b^{2} c e x \operatorname {atan}^{2}{\left (c + d x \right )} - \frac {b^{2} c e \operatorname {atan}{\left (c + d x \right )}}{d} + \frac {b^{2} d e x^{2} \operatorname {atan}^{2}{\left (c + d x \right )}}{2} - b^{2} e x \operatorname {atan}{\left (c + d x \right )} + \frac {b^{2} e \log {\left (\frac {c}{d} + x - \frac {i}{d} \right )}}{d} + \frac {b^{2} e \operatorname {atan}^{2}{\left (c + d x \right )}}{2 d} - \frac {i b^{2} e \operatorname {atan}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\c e x \left (a + b \operatorname {atan}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((a**2*c*e*x + a**2*d*e*x**2/2 + a*b*c**2*e*atan(c + d*x)/d + 2*a 
*b*c*e*x*atan(c + d*x) + a*b*d*e*x**2*atan(c + d*x) - a*b*e*x + a*b*e*atan 
(c + d*x)/d + b**2*c**2*e*atan(c + d*x)**2/(2*d) + b**2*c*e*x*atan(c + d*x 
)**2 - b**2*c*e*atan(c + d*x)/d + b**2*d*e*x**2*atan(c + d*x)**2/2 - b**2* 
e*x*atan(c + d*x) + b**2*e*log(c/d + x - I/d)/d + b**2*e*atan(c + d*x)**2/ 
(2*d) - I*b**2*e*atan(c + d*x)/d, Ne(d, 0)), (c*e*x*(a + b*atan(c))**2, Tr 
ue))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (89) = 178\).

Time = 0.91 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.29 \[ \int (c e+d e x) (a+b \arctan (c+d x))^2 \, dx=\frac {1}{2} \, a^{2} d e x^{2} + {\left (x^{2} \arctan \left (d x + c\right ) - d {\left (\frac {x}{d^{2}} + \frac {{\left (c^{2} - 1\right )} \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{3}} - \frac {c \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right )}{d^{3}}\right )}\right )} a b d e + a^{2} c e x + \frac {{\left (2 \, {\left (d x + c\right )} \arctan \left (d x + c\right ) - \log \left ({\left (d x + c\right )}^{2} + 1\right )\right )} a b c e}{d} + \frac {b^{2} e \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) + {\left (b^{2} d^{2} e x^{2} + 2 \, b^{2} c d e x + {\left (b^{2} c^{2} + b^{2}\right )} e\right )} \arctan \left (d x + c\right )^{2} - 2 \, {\left (b^{2} d e x + b^{2} c e\right )} \arctan \left (d x + c\right )}{2 \, d} \] Input:

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

Output:

1/2*a^2*d*e*x^2 + (x^2*arctan(d*x + c) - d*(x/d^2 + (c^2 - 1)*arctan((d^2* 
x + c*d)/d)/d^3 - c*log(d^2*x^2 + 2*c*d*x + c^2 + 1)/d^3))*a*b*d*e + a^2*c 
*e*x + (2*(d*x + c)*arctan(d*x + c) - log((d*x + c)^2 + 1))*a*b*c*e/d + 1/ 
2*(b^2*e*log(d^2*x^2 + 2*c*d*x + c^2 + 1) + (b^2*d^2*e*x^2 + 2*b^2*c*d*e*x 
 + (b^2*c^2 + b^2)*e)*arctan(d*x + c)^2 - 2*(b^2*d*e*x + b^2*c*e)*arctan(d 
*x + c))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 240 vs. \(2 (89) = 178\).

Time = 0.21 (sec) , antiderivative size = 240, normalized size of antiderivative = 2.53 \[ \int (c e+d e x) (a+b \arctan (c+d x))^2 \, dx=\frac {b^{2} d^{2} e x^{2} \arctan \left (d x + c\right )^{2} + 2 \, a b d^{2} e x^{2} \arctan \left (d x + c\right ) + 2 \, b^{2} c d e x \arctan \left (d x + c\right )^{2} + a^{2} d^{2} e x^{2} + 4 \, a b c d e x \arctan \left (d x + c\right ) + b^{2} c^{2} e \arctan \left (d x + c\right )^{2} + 2 \, a^{2} c d e x + a b c^{2} e \arctan \left (d x + c\right ) - 2 \, b^{2} d e x \arctan \left (d x + c\right ) - a b c^{2} e \arctan \left (-d x - c\right ) - 2 \, a b d e x - b^{2} c e \arctan \left (d x + c\right ) + b^{2} e \arctan \left (d x + c\right )^{2} + b^{2} c e \arctan \left (-d x - c\right ) + a b e \arctan \left (d x + c\right ) - a b e \arctan \left (-d x - c\right ) + b^{2} e \log \left ({\left (d x + c\right )}^{2} + 1\right )}{2 \, d} \] Input:

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

Output:

1/2*(b^2*d^2*e*x^2*arctan(d*x + c)^2 + 2*a*b*d^2*e*x^2*arctan(d*x + c) + 2 
*b^2*c*d*e*x*arctan(d*x + c)^2 + a^2*d^2*e*x^2 + 4*a*b*c*d*e*x*arctan(d*x 
+ c) + b^2*c^2*e*arctan(d*x + c)^2 + 2*a^2*c*d*e*x + a*b*c^2*e*arctan(d*x 
+ c) - 2*b^2*d*e*x*arctan(d*x + c) - a*b*c^2*e*arctan(-d*x - c) - 2*a*b*d* 
e*x - b^2*c*e*arctan(d*x + c) + b^2*e*arctan(d*x + c)^2 + b^2*c*e*arctan(- 
d*x - c) + a*b*e*arctan(d*x + c) - a*b*e*arctan(-d*x - c) + b^2*e*log((d*x 
 + c)^2 + 1))/d
 

Mupad [B] (verification not implemented)

Time = 2.13 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.27 \[ \int (c e+d e x) (a+b \arctan (c+d x))^2 \, dx={\mathrm {atan}\left (c+d\,x\right )}^2\,\left (\frac {e\,b^2\,c^2+e\,b^2}{2\,d}+b^2\,c\,e\,x+\frac {b^2\,d\,e\,x^2}{2}\right )-x\,\left (a\,e\,\left (b-3\,a\,c\right )+2\,a^2\,c\,e\right )-d^2\,\mathrm {atan}\left (c+d\,x\right )\,\left (\frac {x\,\left (b^2\,e-2\,a\,b\,c\,e\right )}{d^2}-\frac {a\,b\,e\,x^2}{d}\right )+\frac {b^2\,e\,\ln \left (c^2+2\,c\,d\,x+d^2\,x^2+1\right )}{2\,d}+\frac {a^2\,d\,e\,x^2}{2}+\frac {b\,e\,\mathrm {atan}\left (\frac {b\,c\,e\,\left (a\,c^2-b\,c+a\right )+b\,d\,e\,x\,\left (a\,c^2-b\,c+a\right )}{-e\,b^2\,c+a\,e\,b\,c^2+a\,e\,b}\right )\,\left (a\,c^2-b\,c+a\right )}{d} \] Input:

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

Output:

atan(c + d*x)^2*((b^2*e + b^2*c^2*e)/(2*d) + b^2*c*e*x + (b^2*d*e*x^2)/2) 
- x*(a*e*(b - 3*a*c) + 2*a^2*c*e) - d^2*atan(c + d*x)*((x*(b^2*e - 2*a*b*c 
*e))/d^2 - (a*b*e*x^2)/d) + (b^2*e*log(c^2 + d^2*x^2 + 2*c*d*x + 1))/(2*d) 
 + (a^2*d*e*x^2)/2 + (b*e*atan((b*c*e*(a - b*c + a*c^2) + b*d*e*x*(a - b*c 
 + a*c^2))/(a*b*e - b^2*c*e + a*b*c^2*e))*(a - b*c + a*c^2))/d
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.01 \[ \int (c e+d e x) (a+b \arctan (c+d x))^2 \, dx=\frac {e \left (\mathit {atan} \left (d x +c \right )^{2} b^{2} c^{2}+2 \mathit {atan} \left (d x +c \right )^{2} b^{2} c d x +\mathit {atan} \left (d x +c \right )^{2} b^{2} d^{2} x^{2}+\mathit {atan} \left (d x +c \right )^{2} b^{2}+2 \mathit {atan} \left (d x +c \right ) a b \,c^{2}+4 \mathit {atan} \left (d x +c \right ) a b c d x +2 \mathit {atan} \left (d x +c \right ) a b \,d^{2} x^{2}+2 \mathit {atan} \left (d x +c \right ) a b -2 \mathit {atan} \left (d x +c \right ) b^{2} c -2 \mathit {atan} \left (d x +c \right ) b^{2} d x +\mathrm {log}\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b^{2}+2 a^{2} c d x +a^{2} d^{2} x^{2}-2 a b d x \right )}{2 d} \] Input:

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

Output:

(e*(atan(c + d*x)**2*b**2*c**2 + 2*atan(c + d*x)**2*b**2*c*d*x + atan(c + 
d*x)**2*b**2*d**2*x**2 + atan(c + d*x)**2*b**2 + 2*atan(c + d*x)*a*b*c**2 
+ 4*atan(c + d*x)*a*b*c*d*x + 2*atan(c + d*x)*a*b*d**2*x**2 + 2*atan(c + d 
*x)*a*b - 2*atan(c + d*x)*b**2*c - 2*atan(c + d*x)*b**2*d*x + log(c**2 + 2 
*c*d*x + d**2*x**2 + 1)*b**2 + 2*a**2*c*d*x + a**2*d**2*x**2 - 2*a*b*d*x)) 
/(2*d)