\(\int (c e+d e x)^2 (a+b \arctan (c+d x))^2 \, dx\) [16]

Optimal result

Integrand size = 23, antiderivative size = 183 \[ \int (c e+d e x)^2 (a+b \arctan (c+d x))^2 \, dx=\frac {1}{3} b^2 e^2 x-\frac {b^2 e^2 \arctan (c+d x)}{3 d}-\frac {b e^2 (c+d x)^2 (a+b \arctan (c+d x))}{3 d}-\frac {i e^2 (a+b \arctan (c+d x))^2}{3 d}+\frac {e^2 (c+d x)^3 (a+b \arctan (c+d x))^2}{3 d}-\frac {2 b e^2 (a+b \arctan (c+d x)) \log \left (\frac {2}{1+i (c+d x)}\right )}{3 d}-\frac {i b^2 e^2 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i (c+d x)}\right )}{3 d} \] Output:

1/3*b^2*e^2*x-1/3*b^2*e^2*arctan(d*x+c)/d-1/3*b*e^2*(d*x+c)^2*(a+b*arctan( 
d*x+c))/d-1/3*I*e^2*(a+b*arctan(d*x+c))^2/d+1/3*e^2*(d*x+c)^3*(a+b*arctan( 
d*x+c))^2/d-2/3*b*e^2*(a+b*arctan(d*x+c))*ln(2/(1+I*(d*x+c)))/d-1/3*I*b^2* 
e^2*polylog(2,1-2/(1+I*(d*x+c)))/d
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.89 \[ \int (c e+d e x)^2 (a+b \arctan (c+d x))^2 \, dx=\frac {e^2 \left (a^2 (c+d x)^3+a b \left (-(c+d x)^2+2 (c+d x)^3 \arctan (c+d x)+\log \left (1+(c+d x)^2\right )\right )+b^2 \left (c+d x-\arctan (c+d x)-(c+d x)^2 \arctan (c+d x)+i \arctan (c+d x)^2+(c+d x)^3 \arctan (c+d x)^2-2 \arctan (c+d x) \log \left (1+e^{2 i \arctan (c+d x)}\right )+i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (c+d x)}\right )\right )\right )}{3 d} \] Input:

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

Output:

(e^2*(a^2*(c + d*x)^3 + a*b*(-(c + d*x)^2 + 2*(c + d*x)^3*ArcTan[c + d*x] 
+ Log[1 + (c + d*x)^2]) + b^2*(c + d*x - ArcTan[c + d*x] - (c + d*x)^2*Arc 
Tan[c + d*x] + I*ArcTan[c + d*x]^2 + (c + d*x)^3*ArcTan[c + d*x]^2 - 2*Arc 
Tan[c + d*x]*Log[1 + E^((2*I)*ArcTan[c + d*x])] + I*PolyLog[2, -E^((2*I)*A 
rcTan[c + d*x])])))/(3*d)
 

Rubi [A] (verified)

Time = 0.79 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.82, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {5566, 27, 5361, 5451, 5361, 262, 216, 5455, 5379, 2849, 2752}

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.

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

\(\Big \downarrow \) 5566

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5451

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 5455

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

\(\Big \downarrow \) 5379

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

\(\Big \downarrow \) 2849

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

\(\Big \downarrow \) 2752

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

Input:

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

Output:

(e^2*(((c + d*x)^3*(a + b*ArcTan[c + d*x])^2)/3 - (2*b*(-1/2*(b*(c + d*x - 
 ArcTan[c + d*x])) + ((c + d*x)^2*(a + b*ArcTan[c + d*x]))/2 + ((I/2)*(a + 
 b*ArcTan[c + d*x])^2)/b + (a + b*ArcTan[c + d*x])*Log[2/(1 + I*(c + d*x)) 
] + (I/2)*b*PolyLog[2, 1 - 2/(1 + I*(c + d*x))]))/3))/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 216

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

rule 262

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

rule 2752

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLo 
g[2, 1 - c*x], x] /; FreeQ[{c, d, e}, x] && EqQ[e + c*d, 0]
 

rule 2849

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> Simp 
[-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 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 5379

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] 
 :> Simp[(-(a + b*ArcTan[c*x])^p)*(Log[2/(1 + e*(x/d))]/e), x] + Simp[b*c*( 
p/e)   Int[(a + b*ArcTan[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 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 5455

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

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 = 0.83 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.51

method	result	size
derivativedivides	\(\frac {\frac {e^{2} a^{2} \left (d x +c \right )^{3}}{3}+b^{2} e^{2} \left (\frac {\left (d x +c \right )^{3} \arctan \left (d x +c \right )^{2}}{3}-\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{3}+\frac {\arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{3}+\frac {d x}{3}+\frac {c}{3}-\frac {\arctan \left (d x +c \right )}{3}+\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{6}\right )+2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \arctan \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2}}{6}+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{6}\right )}{d}\)	\(276\)
default	\(\frac {\frac {e^{2} a^{2} \left (d x +c \right )^{3}}{3}+b^{2} e^{2} \left (\frac {\left (d x +c \right )^{3} \arctan \left (d x +c \right )^{2}}{3}-\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{3}+\frac {\arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{3}+\frac {d x}{3}+\frac {c}{3}-\frac {\arctan \left (d x +c \right )}{3}+\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{6}\right )+2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \arctan \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2}}{6}+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{6}\right )}{d}\)	\(276\)
parts	\(\frac {e^{2} a^{2} \left (d x +c \right )^{3}}{3 d}+\frac {b^{2} e^{2} \left (\frac {\left (d x +c \right )^{3} \arctan \left (d x +c \right )^{2}}{3}-\frac {\left (d x +c \right )^{2} \arctan \left (d x +c \right )}{3}+\frac {\arctan \left (d x +c \right ) \ln \left (1+\left (d x +c \right )^{2}\right )}{3}+\frac {d x}{3}+\frac {c}{3}-\frac {\arctan \left (d x +c \right )}{3}+\frac {i \left (\ln \left (d x +c -i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (d x +c +i\right )}{2}\right )-\ln \left (d x +c -i\right ) \ln \left (-\frac {i \left (d x +c +i\right )}{2}\right )\right )}{6}-\frac {i \left (\ln \left (d x +c +i\right ) \ln \left (1+\left (d x +c \right )^{2}\right )-\frac {\ln \left (d x +c +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (d x +c -i\right )}{2}\right )-\ln \left (d x +c +i\right ) \ln \left (\frac {i \left (d x +c -i\right )}{2}\right )\right )}{6}\right )}{d}+\frac {2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \arctan \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2}}{6}+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{6}\right )}{d}\)	\(281\)
risch	\(\text {Expression too large to display}\)	\(1383\)

Input:

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

Output:

1/d*(1/3*e^2*a^2*(d*x+c)^3+b^2*e^2*(1/3*(d*x+c)^3*arctan(d*x+c)^2-1/3*(d*x 
+c)^2*arctan(d*x+c)+1/3*arctan(d*x+c)*ln(1+(d*x+c)^2)+1/3*d*x+1/3*c-1/3*ar 
ctan(d*x+c)+1/6*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)))-1/6*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))))+2*e^2*a*b*(1/3*(d*x+c)^3*arctan(d*x+c)-1/6*(d*x+c)^2+1/6*ln(1+( 
d*x+c)^2)))
 

Fricas [F]

\[ \int (c e+d e x)^2 (a+b \arctan (c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:

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

Output:

integral(a^2*d^2*e^2*x^2 + 2*a^2*c*d*e^2*x + a^2*c^2*e^2 + (b^2*d^2*e^2*x^ 
2 + 2*b^2*c*d*e^2*x + b^2*c^2*e^2)*arctan(d*x + c)^2 + 2*(a*b*d^2*e^2*x^2 
+ 2*a*b*c*d*e^2*x + a*b*c^2*e^2)*arctan(d*x + c), x)
 

Sympy [F]

\[ \int (c e+d e x)^2 (a+b \arctan (c+d x))^2 \, dx=e^{2} \left (\int a^{2} c^{2}\, dx + \int a^{2} d^{2} x^{2}\, dx + \int b^{2} c^{2} \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c^{2} \operatorname {atan}{\left (c + d x \right )}\, dx + \int 2 a^{2} c d x\, dx + \int b^{2} d^{2} x^{2} \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d^{2} x^{2} \operatorname {atan}{\left (c + d x \right )}\, dx + \int 2 b^{2} c d x \operatorname {atan}^{2}{\left (c + d x \right )}\, dx + \int 4 a b c d x \operatorname {atan}{\left (c + d x \right )}\, dx\right ) \] Input:

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

Output:

e**2*(Integral(a**2*c**2, x) + Integral(a**2*d**2*x**2, x) + Integral(b**2 
*c**2*atan(c + d*x)**2, x) + Integral(2*a*b*c**2*atan(c + d*x), x) + Integ 
ral(2*a**2*c*d*x, x) + Integral(b**2*d**2*x**2*atan(c + d*x)**2, x) + Inte 
gral(2*a*b*d**2*x**2*atan(c + d*x), x) + Integral(2*b**2*c*d*x*atan(c + d* 
x)**2, x) + Integral(4*a*b*c*d*x*atan(c + d*x), x))
 

Maxima [F]

\[ \int (c e+d e x)^2 (a+b \arctan (c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:

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

Output:

3/4*b^2*c^4*e^2*arctan(d*x + c)^2*arctan((d^2*x + c*d)/d)/d - 1/4*(3*arcta 
n(d*x + c)*arctan((d^2*x + c*d)/d)^2/d - arctan((d^2*x + c*d)/d)^3/d)*b^2* 
c^4*e^2 + 1/3*a^2*d^2*e^2*x^3 + 36*b^2*d^4*e^2*integrate(1/48*x^4*arctan(d 
*x + c)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^2*d^4*e^2*integrate(1/48 
*x^4*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) 
+ 144*b^2*c*d^3*e^2*integrate(1/48*x^3*arctan(d*x + c)^2/(d^2*x^2 + 2*c*d* 
x + c^2 + 1), x) + 4*b^2*d^4*e^2*integrate(1/48*x^4*log(d^2*x^2 + 2*c*d*x 
+ c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 12*b^2*c*d^3*e^2*integrate( 
1/48*x^3*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1), 
 x) + 216*b^2*c^2*d^2*e^2*integrate(1/48*x^2*arctan(d*x + c)^2/(d^2*x^2 + 
2*c*d*x + c^2 + 1), x) + 16*b^2*c*d^3*e^2*integrate(1/48*x^3*log(d^2*x^2 + 
 2*c*d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 18*b^2*c^2*d^2*e^2 
*integrate(1/48*x^2*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x 
+ c^2 + 1), x) + 144*b^2*c^3*d*e^2*integrate(1/48*x*arctan(d*x + c)^2/(d^2 
*x^2 + 2*c*d*x + c^2 + 1), x) + 24*b^2*c^2*d^2*e^2*integrate(1/48*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) + 12*b^2*c^ 
3*d*e^2*integrate(1/48*x*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c 
*d*x + c^2 + 1), x) + 12*b^2*c^3*d*e^2*integrate(1/48*x*log(d^2*x^2 + 2*c* 
d*x + c^2 + 1)/(d^2*x^2 + 2*c*d*x + c^2 + 1), x) + 3*b^2*c^4*e^2*integrate 
(1/48*log(d^2*x^2 + 2*c*d*x + c^2 + 1)^2/(d^2*x^2 + 2*c*d*x + c^2 + 1),...
 

Giac [F]

\[ \int (c e+d e x)^2 (a+b \arctan (c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \arctan \left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:

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

Output:

sage0*x
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c e+d e x)^2 (a+b \arctan (c+d x))^2 \, dx=\frac {e^{2} \left (\mathit {atan} \left (d x +c \right )^{2} b^{2} c^{3}+3 \mathit {atan} \left (d x +c \right )^{2} b^{2} c^{2} d x +3 \mathit {atan} \left (d x +c \right )^{2} b^{2} c \,d^{2} x^{2}+\mathit {atan} \left (d x +c \right )^{2} b^{2} c +\mathit {atan} \left (d x +c \right )^{2} b^{2} d^{3} x^{3}+2 \mathit {atan} \left (d x +c \right ) a b \,c^{3}+6 \mathit {atan} \left (d x +c \right ) a b \,c^{2} d x +6 \mathit {atan} \left (d x +c \right ) a b c \,d^{2} x^{2}+2 \mathit {atan} \left (d x +c \right ) a b \,d^{3} x^{3}-\mathit {atan} \left (d x +c \right ) b^{2} c^{2}-2 \mathit {atan} \left (d x +c \right ) b^{2} c d x -\mathit {atan} \left (d x +c \right ) b^{2} d^{2} x^{2}-\mathit {atan} \left (d x +c \right ) b^{2}+2 \left (\int \frac {\mathit {atan} \left (d x +c \right ) x}{d^{2} x^{2}+2 c d x +c^{2}+1}d x \right ) b^{2} d^{2}+\mathrm {log}\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) a b +3 a^{2} c^{2} d x +3 a^{2} c \,d^{2} x^{2}+a^{2} d^{3} x^{3}-2 a b c d x -a b \,d^{2} x^{2}+b^{2} d x \right )}{3 d} \] Input:

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

Output:

(e**2*(atan(c + d*x)**2*b**2*c**3 + 3*atan(c + d*x)**2*b**2*c**2*d*x + 3*a 
tan(c + d*x)**2*b**2*c*d**2*x**2 + atan(c + d*x)**2*b**2*c + atan(c + d*x) 
**2*b**2*d**3*x**3 + 2*atan(c + d*x)*a*b*c**3 + 6*atan(c + d*x)*a*b*c**2*d 
*x + 6*atan(c + d*x)*a*b*c*d**2*x**2 + 2*atan(c + d*x)*a*b*d**3*x**3 - ata 
n(c + d*x)*b**2*c**2 - 2*atan(c + d*x)*b**2*c*d*x - atan(c + d*x)*b**2*d** 
2*x**2 - atan(c + d*x)*b**2 + 2*int((atan(c + d*x)*x)/(c**2 + 2*c*d*x + d* 
*2*x**2 + 1),x)*b**2*d**2 + log(c**2 + 2*c*d*x + d**2*x**2 + 1)*a*b + 3*a* 
*2*c**2*d*x + 3*a**2*c*d**2*x**2 + a**2*d**3*x**3 - 2*a*b*c*d*x - a*b*d**2 
*x**2 + b**2*d*x))/(3*d)