\(\int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^3} \, dx\) [20]

Optimal result

Integrand size = 23, antiderivative size = 117 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^3} \, dx=-\frac {b (a+b \arctan (c+d x))}{d e^3 (c+d x)}-\frac {(a+b \arctan (c+d x))^2}{2 d e^3}-\frac {(a+b \arctan (c+d x))^2}{2 d e^3 (c+d x)^2}+\frac {b^2 \log (c+d x)}{d e^3}-\frac {b^2 \log \left (1+(c+d x)^2\right )}{2 d e^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.66 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^3} \, dx=-\frac {a^2+2 a b c+2 a b d x+2 b \left (b (c+d x)+a \left (1+c^2+2 c d x+d^2 x^2\right )\right ) \arctan (c+d x)+b^2 \left (1+c^2+2 c d x+d^2 x^2\right ) \arctan (c+d x)^2-2 b^2 (c+d x)^2 \log (c+d x)+b^2 c^2 \log \left (1+c^2+2 c d x+d^2 x^2\right )+2 b^2 c d x \log \left (1+c^2+2 c d x+d^2 x^2\right )+b^2 d^2 x^2 \log \left (1+c^2+2 c d x+d^2 x^2\right )}{2 d e^3 (c+d x)^2} \] Input:

Integrate[(a + b*ArcTan[c + d*x])^2/(c*e + d*e*x)^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.84, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {5566, 27, 5361, 5453, 5361, 243, 47, 14, 16, 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.

\(\displaystyle \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^3} \, dx\)

\(\Big \downarrow \) 5566

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 5419

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

Input:

Int[(a + b*ArcTan[c + d*x])^2/(c*e + d*e*x)^3,x]
 

Output:

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

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

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 47

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c 
 - a*d)   Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d)   Int[1/(c + d*x), x 
], x] /; FreeQ[{a, b, c, d}, x]
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 5453

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e 
_.)*(x_)^2), x_Symbol] :> Simp[1/d   Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], 
 x] - Simp[e/(d*f^2)   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] && LtQ[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 = 0.45 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.10

method	result	size
derivativedivides	\(\frac {-\frac {a^{2}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{2} \left (-\frac {\arctan \left (d x +c \right )^{2}}{2 \left (d x +c \right )^{2}}-\frac {\arctan \left (d x +c \right )}{d x +c}-\frac {\arctan \left (d x +c \right )^{2}}{2}+\ln \left (d x +c \right )-\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{e^{3}}+\frac {2 a b \left (-\frac {\arctan \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {1}{2 \left (d x +c \right )}-\frac {\arctan \left (d x +c \right )}{2}\right )}{e^{3}}}{d}\)	\(129\)
default	\(\frac {-\frac {a^{2}}{2 e^{3} \left (d x +c \right )^{2}}+\frac {b^{2} \left (-\frac {\arctan \left (d x +c \right )^{2}}{2 \left (d x +c \right )^{2}}-\frac {\arctan \left (d x +c \right )}{d x +c}-\frac {\arctan \left (d x +c \right )^{2}}{2}+\ln \left (d x +c \right )-\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{e^{3}}+\frac {2 a b \left (-\frac {\arctan \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {1}{2 \left (d x +c \right )}-\frac {\arctan \left (d x +c \right )}{2}\right )}{e^{3}}}{d}\)	\(129\)
parts	\(-\frac {a^{2}}{2 e^{3} \left (d x +c \right )^{2} d}+\frac {b^{2} \left (-\frac {\arctan \left (d x +c \right )^{2}}{2 \left (d x +c \right )^{2}}-\frac {\arctan \left (d x +c \right )}{d x +c}-\frac {\arctan \left (d x +c \right )^{2}}{2}+\ln \left (d x +c \right )-\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{2}\right )}{e^{3} d}+\frac {2 a b \left (-\frac {\arctan \left (d x +c \right )}{2 \left (d x +c \right )^{2}}-\frac {1}{2 \left (d x +c \right )}-\frac {\arctan \left (d x +c \right )}{2}\right )}{e^{3} d}\)	\(134\)
parallelrisch	\(\frac {a b \,d^{4} x^{2}-3 a b \,c^{2} d^{2}+4 \ln \left (d x +c \right ) b^{2} c^{3} d^{2}-2 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b^{2} c^{3} d^{2}-4 \arctan \left (d x +c \right ) b^{2} c^{2} d^{2}-2 \arctan \left (d x +c \right )^{2} b^{2} c^{3} d^{2}-2 b^{2} \arctan \left (d x +c \right )^{2} c \,d^{2}-8 x \arctan \left (d x +c \right ) a b \,c^{2} d^{3}-4 x^{2} \arctan \left (d x +c \right ) a b c \,d^{4}-2 a^{2} c \,d^{2}+4 \ln \left (d x +c \right ) x^{2} b^{2} c \,d^{4}-2 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) x^{2} b^{2} c \,d^{4}+8 \ln \left (d x +c \right ) x \,b^{2} c^{2} d^{3}-4 b^{2} c^{2} \arctan \left (d x +c \right )^{2} x \,d^{3}-2 b^{2} d^{4} \arctan \left (d x +c \right )^{2} x^{2} c -4 x \arctan \left (d x +c \right ) b^{2} c \,d^{3}-4 \arctan \left (d x +c \right ) a b \,c^{3} d^{2}-4 \arctan \left (d x +c \right ) a b c \,d^{2}-2 x a b c \,d^{3}-4 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) x \,b^{2} c^{2} d^{3}}{4 \left (d x +c \right )^{2} c \,d^{3} e^{3}}\)	\(372\)
risch	\(\frac {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 e^{3} \left (d x +c \right )^{2} d}-\frac {b \left (b \,d^{2} x^{2} \ln \left (1-i \left (d x +c \right )\right )+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}-2 i b c -2 i a +b \ln \left (1-i \left (d x +c \right )\right )\right ) \ln \left (1+i \left (d x +c \right )\right )}{4 e^{3} \left (d x +c \right )^{2} d}-\frac {-b^{2} \ln \left (1-i \left (d x +c \right )\right )^{2}-b^{2} c^{2} \ln \left (1-i \left (d x +c \right )\right )^{2}+4 \ln \left (\left (-3 i b d -a d \right ) x -a c -3 b -3 i b c +i a \right ) b^{2} c^{2}+4 \ln \left (\left (3 i b d -a d \right ) x -a c -3 b +3 i b c -i a \right ) b^{2} c^{2}+4 a^{2}+8 a b x d -8 i \ln \left (\left (-3 i b d -a d \right ) x -a c -3 b -3 i b c +i a \right ) a b c d x +8 i \ln \left (\left (3 i b d -a d \right ) x -a c -3 b +3 i b c -i a \right ) a b c d x -8 \ln \left (-d x -c \right ) b^{2} c^{2}+8 a b c +4 i \ln \left (\left (3 i b d -a d \right ) x -a c -3 b +3 i b c -i a \right ) a b \,d^{2} x^{2}-4 i \ln \left (\left (-3 i b d -a d \right ) x -a c -3 b -3 i b c +i a \right ) a b \,d^{2} x^{2}+8 \ln \left (\left (-3 i b d -a d \right ) x -a c -3 b -3 i b c +i a \right ) b^{2} c d x +8 \ln \left (\left (3 i b d -a d \right ) x -a c -3 b +3 i b c -i a \right ) b^{2} c d x -2 b^{2} c d x \ln \left (1-i \left (d x +c \right )\right )^{2}-4 i \ln \left (\left (-3 i b d -a d \right ) x -a c -3 b -3 i b c +i a \right ) a b \,c^{2}+4 i \ln \left (\left (3 i b d -a d \right ) x -a c -3 b +3 i b c -i a \right ) a b \,c^{2}+4 i b^{2} x \ln \left (1-i \left (d x +c \right )\right ) d -16 \ln \left (-d x -c \right ) b^{2} c d x +4 i b^{2} c \ln \left (1-i \left (d x +c \right )\right )+4 i a b \ln \left (1-i \left (d x +c \right )\right )-b^{2} d^{2} x^{2} \ln \left (1-i \left (d x +c \right )\right )^{2}+4 \ln \left (\left (-3 i b d -a d \right ) x -a c -3 b -3 i b c +i a \right ) b^{2} d^{2} x^{2}+4 \ln \left (\left (3 i b d -a d \right ) x -a c -3 b +3 i b c -i a \right ) b^{2} d^{2} x^{2}-8 \ln \left (-d x -c \right ) b^{2} d^{2} x^{2}}{8 e^{3} \left (d x +c \right )^{2} d}\)	\(849\)

Input:

int((a+b*arctan(d*x+c))^2/(d*e*x+c*e)^3,x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.79 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^3} \, dx=-\frac {2 \, a b d x + 2 \, a b c + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + b^{2}\right )} \arctan \left (d x + c\right )^{2} + a^{2} + 2 \, {\left (a b d^{2} x^{2} + a b c^{2} + b^{2} c + {\left (2 \, a b c + b^{2}\right )} d x + a b\right )} \arctan \left (d x + c\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (d x + c\right )}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \] Input:

integrate((a+b*arctan(d*x+c))^2/(d*e*x+c*e)^3,x, algorithm="fricas")
 

Output:

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 9.22 (sec) , antiderivative size = 1107, normalized size of antiderivative = 9.46 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^3} \, dx =\text {Too large to display} \] Input:

integrate((a+b*atan(d*x+c))**2/(d*e*x+c*e)**3,x)
 

Output:

Piecewise((-a**2/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2) - 2* 
a*b*c**2*atan(c + d*x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2 
) - 4*a*b*c*d*x*atan(c + d*x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e* 
*3*x**2) - 2*a*b*c/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2) - 
2*a*b*d**2*x**2*atan(c + d*x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e* 
*3*x**2) - 2*a*b*d*x/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2) 
- 2*a*b*atan(c + d*x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2) 
 + 2*b**2*c**2*log(c/d + x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3 
*x**2) - 2*b**2*c**2*log(c/d + x - I/d)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 
 2*d**3*e**3*x**2) - b**2*c**2*atan(c + d*x)**2/(2*c**2*d*e**3 + 4*c*d**2* 
e**3*x + 2*d**3*e**3*x**2) + 2*I*b**2*c**2*atan(c + d*x)/(2*c**2*d*e**3 + 
4*c*d**2*e**3*x + 2*d**3*e**3*x**2) + 4*b**2*c*d*x*log(c/d + x)/(2*c**2*d* 
e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2) - 4*b**2*c*d*x*log(c/d + x - I/ 
d)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2) - 2*b**2*c*d*x*ata 
n(c + d*x)**2/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x**2) + 4*I*b 
**2*c*d*x*atan(c + d*x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3*x** 
2) - 2*b**2*c*atan(c + d*x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 2*d**3*e**3 
*x**2) + 2*b**2*d**2*x**2*log(c/d + x)/(2*c**2*d*e**3 + 4*c*d**2*e**3*x + 
2*d**3*e**3*x**2) - 2*b**2*d**2*x**2*log(c/d + x - I/d)/(2*c**2*d*e**3 + 4 
*c*d**2*e**3*x + 2*d**3*e**3*x**2) - b**2*d**2*x**2*atan(c + d*x)**2/(2...
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (111) = 222\).

Time = 0.21 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.29 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^3} \, dx=-{\left (d {\left (\frac {1}{d^{3} e^{3} x + c d^{2} e^{3}} + \frac {\arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{2} e^{3}}\right )} + \frac {\arctan \left (d x + c\right )}{d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}}\right )} a b - \frac {1}{2} \, {\left (2 \, d {\left (\frac {1}{d^{3} e^{3} x + c d^{2} e^{3}} + \frac {\arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{2} e^{3}}\right )} \arctan \left (d x + c\right ) - \frac {\arctan \left (d x + c\right )^{2} - \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) + 2 \, \log \left (d x + c\right )}{d e^{3}}\right )} b^{2} - \frac {b^{2} \arctan \left (d x + c\right )^{2}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} - \frac {a^{2}}{2 \, {\left (d^{3} e^{3} x^{2} + 2 \, c d^{2} e^{3} x + c^{2} d e^{3}\right )}} \] Input:

integrate((a+b*arctan(d*x+c))^2/(d*e*x+c*e)^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

\[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^3} \, dx=\int { \frac {{\left (b \arctan \left (d x + c\right ) + a\right )}^{2}}{{\left (d e x + c e\right )}^{3}} \,d x } \] Input:

integrate((a+b*arctan(d*x+c))^2/(d*e*x+c*e)^3,x, algorithm="giac")
 

Output:

integrate((b*arctan(d*x + c) + a)^2/(d*e*x + c*e)^3, x)
 

Mupad [B] (verification not implemented)

Time = 3.49 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.98 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^3} \, dx=\frac {b^2\,\ln \left (c+d\,x\right )}{d\,e^3}-\frac {\frac {a^2+2\,b\,c\,a}{2\,d}+a\,b\,x}{c^2\,e^3+2\,c\,d\,e^3\,x+d^2\,e^3\,x^2}-\frac {\mathrm {atan}\left (c+d\,x\right )\,\left (\frac {b^2\,c}{d^3\,e^3}+\frac {b^2\,x}{d^2\,e^3}+\frac {a\,b}{d^3\,e^3}\right )}{x^2+\frac {c^2}{d^2}+\frac {2\,c\,x}{d}}-{\mathrm {atan}\left (c+d\,x\right )}^2\,\left (\frac {b^2}{2\,d\,e^3}+\frac {b^2}{2\,d^3\,e^3\,\left (x^2+\frac {c^2}{d^2}+\frac {2\,c\,x}{d}\right )}\right )+\frac {\ln \left (c+d\,x-\mathrm {i}\right )\,\left (-\frac {b^2}{2}+\frac {a\,b\,1{}\mathrm {i}}{2}\right )}{d\,e^3}-\frac {\ln \left (c+d\,x+1{}\mathrm {i}\right )\,\left (\frac {b^2}{2}+\frac {1{}\mathrm {i}\,a\,b}{2}\right )}{d\,e^3} \] Input:

int((a + b*atan(c + d*x))^2/(c*e + d*e*x)^3,x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 336, normalized size of antiderivative = 2.87 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^3} \, dx=\frac {-\mathit {atan} \left (d x +c \right )^{2} b^{2} c^{3}-2 \mathit {atan} \left (d x +c \right )^{2} b^{2} c^{2} d x -\mathit {atan} \left (d x +c \right )^{2} b^{2} c \,d^{2} x^{2}-\mathit {atan} \left (d x +c \right )^{2} b^{2} c -2 \mathit {atan} \left (d x +c \right ) a b \,c^{3}-4 \mathit {atan} \left (d x +c \right ) a b \,c^{2} d x -2 \mathit {atan} \left (d x +c \right ) a b c \,d^{2} x^{2}-2 \mathit {atan} \left (d x +c \right ) a b c -2 \mathit {atan} \left (d x +c \right ) b^{2} c^{2}-2 \mathit {atan} \left (d x +c \right ) b^{2} c d x -\mathrm {log}\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b^{2} c^{3}-2 \,\mathrm {log}\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b^{2} c^{2} d x -\mathrm {log}\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b^{2} c \,d^{2} x^{2}+2 \,\mathrm {log}\left (d x +c \right ) b^{2} c^{3}+4 \,\mathrm {log}\left (d x +c \right ) b^{2} c^{2} d x +2 \,\mathrm {log}\left (d x +c \right ) b^{2} c \,d^{2} x^{2}-a^{2} c -a b \,c^{2}+a b \,d^{2} x^{2}}{2 c d \,e^{3} \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:

int((a+b*atan(d*x+c))^2/(d*e*x+c*e)^3,x)
 

Output:

( - atan(c + d*x)**2*b**2*c**3 - 2*atan(c + d*x)**2*b**2*c**2*d*x - atan(c 
 + d*x)**2*b**2*c*d**2*x**2 - atan(c + d*x)**2*b**2*c - 2*atan(c + d*x)*a* 
b*c**3 - 4*atan(c + d*x)*a*b*c**2*d*x - 2*atan(c + d*x)*a*b*c*d**2*x**2 - 
2*atan(c + d*x)*a*b*c - 2*atan(c + d*x)*b**2*c**2 - 2*atan(c + d*x)*b**2*c 
*d*x - log(c**2 + 2*c*d*x + d**2*x**2 + 1)*b**2*c**3 - 2*log(c**2 + 2*c*d* 
x + d**2*x**2 + 1)*b**2*c**2*d*x - log(c**2 + 2*c*d*x + d**2*x**2 + 1)*b** 
2*c*d**2*x**2 + 2*log(c + d*x)*b**2*c**3 + 4*log(c + d*x)*b**2*c**2*d*x + 
2*log(c + d*x)*b**2*c*d**2*x**2 - a**2*c - a*b*c**2 + a*b*d**2*x**2)/(2*c* 
d*e**3*(c**2 + 2*c*d*x + d**2*x**2))