\(\int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^5} \, dx\) [22]

Optimal result

Integrand size = 23, antiderivative size = 170 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^5} \, dx=-\frac {b^2}{12 d e^5 (c+d x)^2}-\frac {b (a+b \arctan (c+d x))}{6 d e^5 (c+d x)^3}+\frac {b (a+b \arctan (c+d x))}{2 d e^5 (c+d x)}+\frac {(a+b \arctan (c+d x))^2}{4 d e^5}-\frac {(a+b \arctan (c+d x))^2}{4 d e^5 (c+d x)^4}-\frac {2 b^2 \log (c+d x)}{3 d e^5}+\frac {b^2 \log \left (1+(c+d x)^2\right )}{3 d e^5} \] Output:

-1/12*b^2/d/e^5/(d*x+c)^2-1/6*b*(a+b*arctan(d*x+c))/d/e^5/(d*x+c)^3+1/2*b* 
(a+b*arctan(d*x+c))/d/e^5/(d*x+c)+1/4*(a+b*arctan(d*x+c))^2/d/e^5-1/4*(a+b 
*arctan(d*x+c))^2/d/e^5/(d*x+c)^4-2/3*b^2*ln(d*x+c)/d/e^5+1/3*b^2*ln(1+(d* 
x+c)^2)/d/e^5
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.44 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^5} \, dx=-\frac {3 a^2+2 a b (c+d x)+b^2 (c+d x)^2-6 a b (c+d x)^3-2 b \left (b \left (-c+3 c^3-d x+9 c^2 d x+9 c d^2 x^2+3 d^3 x^3\right )+3 a \left (-1+c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4\right )\right ) \arctan (c+d x)-3 b^2 \left (-1+c^4+4 c^3 d x+6 c^2 d^2 x^2+4 c d^3 x^3+d^4 x^4\right ) \arctan (c+d x)^2+8 b^2 (c+d x)^4 \log (c+d x)-4 b^2 (c+d x)^4 \log \left (1+c^2+2 c d x+d^2 x^2\right )}{12 d e^5 (c+d x)^4} \] Input:

Integrate[(a + b*ArcTan[c + d*x])^2/(c*e + d*e*x)^5,x]
 

Output:

-1/12*(3*a^2 + 2*a*b*(c + d*x) + b^2*(c + d*x)^2 - 6*a*b*(c + d*x)^3 - 2*b 
*(b*(-c + 3*c^3 - d*x + 9*c^2*d*x + 9*c*d^2*x^2 + 3*d^3*x^3) + 3*a*(-1 + c 
^4 + 4*c^3*d*x + 6*c^2*d^2*x^2 + 4*c*d^3*x^3 + d^4*x^4))*ArcTan[c + d*x] - 
 3*b^2*(-1 + c^4 + 4*c^3*d*x + 6*c^2*d^2*x^2 + 4*c*d^3*x^3 + d^4*x^4)*ArcT 
an[c + d*x]^2 + 8*b^2*(c + d*x)^4*Log[c + d*x] - 4*b^2*(c + d*x)^4*Log[1 + 
 c^2 + 2*c*d*x + d^2*x^2])/(d*e^5*(c + d*x)^4)
 

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.92, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.652, Rules used = {5566, 27, 5361, 5453, 5361, 243, 54, 2009, 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.

\(\displaystyle \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^5} \, dx\)

\(\Big \downarrow \) 5566

\(\displaystyle \frac {\int \frac {(a+b \arctan (c+d x))^2}{e^5 (c+d x)^5}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(a+b \arctan (c+d x))^2}{(c+d x)^5}d(c+d x)}{d e^5}\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {\frac {1}{2} b \int \frac {a+b \arctan (c+d x)}{(c+d x)^4 \left ((c+d x)^2+1\right )}d(c+d x)-\frac {(a+b \arctan (c+d x))^2}{4 (c+d x)^4}}{d e^5}\)

\(\Big \downarrow \) 5453

\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \arctan (c+d x)}{(c+d x)^4}d(c+d x)-\int \frac {a+b \arctan (c+d x)}{(c+d x)^2 \left ((c+d x)^2+1\right )}d(c+d x)\right )-\frac {(a+b \arctan (c+d x))^2}{4 (c+d x)^4}}{d e^5}\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {\frac {1}{2} b \left (-\int \frac {a+b \arctan (c+d x)}{(c+d x)^2 \left ((c+d x)^2+1\right )}d(c+d x)+\frac {1}{3} b \int \frac {1}{(c+d x)^3 \left ((c+d x)^2+1\right )}d(c+d x)-\frac {a+b \arctan (c+d x)}{3 (c+d x)^3}\right )-\frac {(a+b \arctan (c+d x))^2}{4 (c+d x)^4}}{d e^5}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {1}{2} b \left (-\int \frac {a+b \arctan (c+d x)}{(c+d x)^2 \left ((c+d x)^2+1\right )}d(c+d x)+\frac {1}{6} b \int \frac {1}{(c+d x)^4 \left ((c+d x)^2+1\right )}d(c+d x)^2-\frac {a+b \arctan (c+d x)}{3 (c+d x)^3}\right )-\frac {(a+b \arctan (c+d x))^2}{4 (c+d x)^4}}{d e^5}\)

\(\Big \downarrow \) 54

\(\displaystyle \frac {\frac {1}{2} b \left (-\int \frac {a+b \arctan (c+d x)}{(c+d x)^2 \left ((c+d x)^2+1\right )}d(c+d x)+\frac {1}{6} b \int \left (-\frac {1}{(c+d x)^2}+\frac {1}{(c+d x)^4}+\frac {1}{(c+d x)^2+1}\right )d(c+d x)^2-\frac {a+b \arctan (c+d x)}{3 (c+d x)^3}\right )-\frac {(a+b \arctan (c+d x))^2}{4 (c+d x)^4}}{d e^5}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {1}{2} b \left (-\int \frac {a+b \arctan (c+d x)}{(c+d x)^2 \left ((c+d x)^2+1\right )}d(c+d x)-\frac {a+b \arctan (c+d x)}{3 (c+d x)^3}+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log \left ((c+d x)^2\right )+\log \left ((c+d x)^2+1\right )\right )\right )-\frac {(a+b \arctan (c+d x))^2}{4 (c+d x)^4}}{d e^5}\)

\(\Big \downarrow \) 5453

\(\displaystyle \frac {\frac {1}{2} b \left (-\int \frac {a+b \arctan (c+d x)}{(c+d x)^2}d(c+d x)+\int \frac {a+b \arctan (c+d x)}{(c+d x)^2+1}d(c+d x)-\frac {a+b \arctan (c+d x)}{3 (c+d x)^3}+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log \left ((c+d x)^2\right )+\log \left ((c+d x)^2+1\right )\right )\right )-\frac {(a+b \arctan (c+d x))^2}{4 (c+d x)^4}}{d e^5}\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \arctan (c+d x)}{(c+d x)^2+1}d(c+d x)-b \int \frac {1}{(c+d x) \left ((c+d x)^2+1\right )}d(c+d x)+\frac {a+b \arctan (c+d x)}{c+d x}-\frac {a+b \arctan (c+d x)}{3 (c+d x)^3}+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log \left ((c+d x)^2\right )+\log \left ((c+d x)^2+1\right )\right )\right )-\frac {(a+b \arctan (c+d x))^2}{4 (c+d x)^4}}{d e^5}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \arctan (c+d x)}{(c+d x)^2+1}d(c+d x)-\frac {1}{2} b \int \frac {1}{(c+d x)^2 \left ((c+d x)^2+1\right )}d(c+d x)^2+\frac {a+b \arctan (c+d x)}{c+d x}-\frac {a+b \arctan (c+d x)}{3 (c+d x)^3}+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log \left ((c+d x)^2\right )+\log \left ((c+d x)^2+1\right )\right )\right )-\frac {(a+b \arctan (c+d x))^2}{4 (c+d x)^4}}{d e^5}\)

\(\Big \downarrow \) 47

\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \arctan (c+d x)}{(c+d x)^2+1}d(c+d x)-\frac {1}{2} b \left (\int \frac {1}{(c+d x)^2}d(c+d x)^2-\int \frac {1}{(c+d x)^2+1}d(c+d x)^2\right )+\frac {a+b \arctan (c+d x)}{c+d x}-\frac {a+b \arctan (c+d x)}{3 (c+d x)^3}+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log \left ((c+d x)^2\right )+\log \left ((c+d x)^2+1\right )\right )\right )-\frac {(a+b \arctan (c+d x))^2}{4 (c+d x)^4}}{d e^5}\)

\(\Big \downarrow \) 14

\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \arctan (c+d x)}{(c+d x)^2+1}d(c+d x)-\frac {1}{2} b \left (\log \left ((c+d x)^2\right )-\int \frac {1}{(c+d x)^2+1}d(c+d x)^2\right )+\frac {a+b \arctan (c+d x)}{c+d x}-\frac {a+b \arctan (c+d x)}{3 (c+d x)^3}+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log \left ((c+d x)^2\right )+\log \left ((c+d x)^2+1\right )\right )\right )-\frac {(a+b \arctan (c+d x))^2}{4 (c+d x)^4}}{d e^5}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\frac {1}{2} b \left (\int \frac {a+b \arctan (c+d x)}{(c+d x)^2+1}d(c+d x)+\frac {a+b \arctan (c+d x)}{c+d x}-\frac {a+b \arctan (c+d x)}{3 (c+d x)^3}-\frac {1}{2} b \left (\log \left ((c+d x)^2\right )-\log \left ((c+d x)^2+1\right )\right )+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log \left ((c+d x)^2\right )+\log \left ((c+d x)^2+1\right )\right )\right )-\frac {(a+b \arctan (c+d x))^2}{4 (c+d x)^4}}{d e^5}\)

\(\Big \downarrow \) 5419

\(\displaystyle \frac {\frac {1}{2} b \left (\frac {(a+b \arctan (c+d x))^2}{2 b}+\frac {a+b \arctan (c+d x)}{c+d x}-\frac {a+b \arctan (c+d x)}{3 (c+d x)^3}-\frac {1}{2} b \left (\log \left ((c+d x)^2\right )-\log \left ((c+d x)^2+1\right )\right )+\frac {1}{6} b \left (-\frac {1}{(c+d x)^2}-\log \left ((c+d x)^2\right )+\log \left ((c+d x)^2+1\right )\right )\right )-\frac {(a+b \arctan (c+d x))^2}{4 (c+d x)^4}}{d e^5}\)

Input:

Int[(a + b*ArcTan[c + d*x])^2/(c*e + d*e*x)^5,x]
 

Output:

(-1/4*(a + b*ArcTan[c + d*x])^2/(c + d*x)^4 + (b*(-1/3*(a + b*ArcTan[c + d 
*x])/(c + d*x)^3 + (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 + (b*(-(c + 
 d*x)^(-2) - Log[(c + d*x)^2] + Log[1 + (c + d*x)^2]))/6))/2)/(d*e^5)
 

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 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

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 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 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 = 0.98 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.96

method	result	size
derivativedivides	\(\frac {-\frac {a^{2}}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b^{2} \left (-\frac {\arctan \left (d x +c \right )^{2}}{4 \left (d x +c \right )^{4}}-\frac {\arctan \left (d x +c \right )}{6 \left (d x +c \right )^{3}}+\frac {\arctan \left (d x +c \right )}{2 d x +2 c}+\frac {\arctan \left (d x +c \right )^{2}}{4}-\frac {1}{12 \left (d x +c \right )^{2}}-\frac {2 \ln \left (d x +c \right )}{3}+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{3}\right )}{e^{5}}+\frac {2 a b \left (-\frac {\arctan \left (d x +c \right )}{4 \left (d x +c \right )^{4}}-\frac {1}{12 \left (d x +c \right )^{3}}+\frac {1}{4 d x +4 c}+\frac {\arctan \left (d x +c \right )}{4}\right )}{e^{5}}}{d}\)	\(164\)
default	\(\frac {-\frac {a^{2}}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b^{2} \left (-\frac {\arctan \left (d x +c \right )^{2}}{4 \left (d x +c \right )^{4}}-\frac {\arctan \left (d x +c \right )}{6 \left (d x +c \right )^{3}}+\frac {\arctan \left (d x +c \right )}{2 d x +2 c}+\frac {\arctan \left (d x +c \right )^{2}}{4}-\frac {1}{12 \left (d x +c \right )^{2}}-\frac {2 \ln \left (d x +c \right )}{3}+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{3}\right )}{e^{5}}+\frac {2 a b \left (-\frac {\arctan \left (d x +c \right )}{4 \left (d x +c \right )^{4}}-\frac {1}{12 \left (d x +c \right )^{3}}+\frac {1}{4 d x +4 c}+\frac {\arctan \left (d x +c \right )}{4}\right )}{e^{5}}}{d}\)	\(164\)
parts	\(-\frac {a^{2}}{4 e^{5} \left (d x +c \right )^{4} d}+\frac {b^{2} \left (-\frac {\arctan \left (d x +c \right )^{2}}{4 \left (d x +c \right )^{4}}-\frac {\arctan \left (d x +c \right )}{6 \left (d x +c \right )^{3}}+\frac {\arctan \left (d x +c \right )}{2 d x +2 c}+\frac {\arctan \left (d x +c \right )^{2}}{4}-\frac {1}{12 \left (d x +c \right )^{2}}-\frac {2 \ln \left (d x +c \right )}{3}+\frac {\ln \left (1+\left (d x +c \right )^{2}\right )}{3}\right )}{e^{5} d}+\frac {2 a b \left (-\frac {\arctan \left (d x +c \right )}{4 \left (d x +c \right )^{4}}-\frac {1}{12 \left (d x +c \right )^{3}}+\frac {1}{4 d x +4 c}+\frac {\arctan \left (d x +c \right )}{4}\right )}{e^{5} d}\)	\(169\)
parallelrisch	\(\frac {18 x^{2} \arctan \left (d x +c \right ) b^{2} c \,d^{7}+12 c \,d^{8} b^{2} \arctan \left (d x +c \right )^{2} x^{3}+18 x^{2} a b c \,d^{7}+18 x \arctan \left (d x +c \right ) b^{2} c^{2} d^{6}+18 x a b \,c^{2} d^{6}+12 x \arctan \left (d x +c \right )^{2} b^{2} c^{3} d^{6}-b^{2} c^{2} d^{5}+6 \arctan \left (d x +c \right ) a b \,c^{4} d^{5}+6 x^{4} \arctan \left (d x +c \right ) a b \,d^{9}+18 x^{2} \arctan \left (d x +c \right )^{2} b^{2} c^{2} d^{7}-32 \ln \left (d x +c \right ) x^{3} b^{2} c \,d^{8}+24 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) x^{2} b^{2} c^{2} d^{7}-48 \ln \left (d x +c \right ) x^{2} b^{2} c^{2} d^{7}+16 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) x^{3} b^{2} c \,d^{8}+16 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) x \,b^{2} c^{3} d^{6}-32 \ln \left (d x +c \right ) x \,b^{2} c^{3} d^{6}+24 x \arctan \left (d x +c \right ) a b \,c^{3} d^{6}+36 x^{2} \arctan \left (d x +c \right ) a b \,c^{2} d^{7}-2 x \,b^{2} c \,d^{6}-2 x a b \,d^{6}+3 d^{9} b^{2} \arctan \left (d x +c \right )^{2} x^{4}-2 x \arctan \left (d x +c \right ) b^{2} d^{6}+6 x^{3} \arctan \left (d x +c \right ) b^{2} d^{8}+6 \arctan \left (d x +c \right ) b^{2} c^{3} d^{5}-2 \arctan \left (d x +c \right ) b^{2} c \,d^{5}-6 \arctan \left (d x +c \right ) a b \,d^{5}+3 \arctan \left (d x +c \right )^{2} b^{2} c^{4} d^{5}-8 \ln \left (d x +c \right ) x^{4} b^{2} d^{9}+4 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) x^{4} b^{2} d^{9}-8 \ln \left (d x +c \right ) b^{2} c^{4} d^{5}+4 \ln \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b^{2} c^{4} d^{5}+6 x^{3} a b \,d^{8}+24 x^{3} \arctan \left (d x +c \right ) a b c \,d^{8}-x^{2} b^{2} d^{7}-3 b^{2} \arctan \left (d x +c \right )^{2} d^{5}-2 a b c \,d^{5}-3 a^{2} d^{5}+6 a b \,c^{3} d^{5}}{12 \left (d x +c \right )^{4} d^{6} e^{5}}\)	\(669\)
risch	\(\text {Expression too large to display}\)	\(1562\)

Input:

int((a+b*arctan(d*x+c))^2/(d*e*x+c*e)^5,x,method=_RETURNVERBOSE)
 

Output:

1/d*(-1/4*a^2/e^5/(d*x+c)^4+b^2/e^5*(-1/4/(d*x+c)^4*arctan(d*x+c)^2-1/6/(d 
*x+c)^3*arctan(d*x+c)+1/2/(d*x+c)*arctan(d*x+c)+1/4*arctan(d*x+c)^2-1/12/( 
d*x+c)^2-2/3*ln(d*x+c)+1/3*ln(1+(d*x+c)^2))+2*a*b/e^5*(-1/4/(d*x+c)^4*arct 
an(d*x+c)-1/12/(d*x+c)^3+1/4/(d*x+c)+1/4*arctan(d*x+c)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (156) = 312\).

Time = 0.15 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.64 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^5} \, dx=\frac {6 \, a b d^{3} x^{3} + 6 \, a b c^{3} + {\left (18 \, a b c - b^{2}\right )} d^{2} x^{2} - b^{2} c^{2} - 2 \, a b c + 2 \, {\left (9 \, a b c^{2} - b^{2} c - a b\right )} d x + 3 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4} - b^{2}\right )} \arctan \left (d x + c\right )^{2} - 3 \, a^{2} + 2 \, {\left (3 \, a b d^{4} x^{4} + 3 \, {\left (4 \, a b c + b^{2}\right )} d^{3} x^{3} + 3 \, a b c^{4} + 3 \, b^{2} c^{3} + 9 \, {\left (2 \, a b c^{2} + b^{2} c\right )} d^{2} x^{2} - b^{2} c + {\left (12 \, a b c^{3} + 9 \, b^{2} c^{2} - b^{2}\right )} d x - 3 \, a b\right )} \arctan \left (d x + c\right ) + 4 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) - 8 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (d x + c\right )}{12 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} \] Input:

integrate((a+b*arctan(d*x+c))^2/(d*e*x+c*e)^5,x, algorithm="fricas")
 

Output:

1/12*(6*a*b*d^3*x^3 + 6*a*b*c^3 + (18*a*b*c - b^2)*d^2*x^2 - b^2*c^2 - 2*a 
*b*c + 2*(9*a*b*c^2 - b^2*c - a*b)*d*x + 3*(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 
+ 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4 - b^2)*arctan(d*x + c)^2 - 3 
*a^2 + 2*(3*a*b*d^4*x^4 + 3*(4*a*b*c + b^2)*d^3*x^3 + 3*a*b*c^4 + 3*b^2*c^ 
3 + 9*(2*a*b*c^2 + b^2*c)*d^2*x^2 - b^2*c + (12*a*b*c^3 + 9*b^2*c^2 - b^2) 
*d*x - 3*a*b)*arctan(d*x + c) + 4*(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c 
^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*log(d^2*x^2 + 2*c*d*x + c^2 + 1) - 8 
*(b^2*d^4*x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2* 
c^4)*log(d*x + c))/(d^5*e^5*x^4 + 4*c*d^4*e^5*x^3 + 6*c^2*d^3*e^5*x^2 + 4* 
c^3*d^2*e^5*x + c^4*d*e^5)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^5} \, dx=\text {Timed out} \] Input:

integrate((a+b*atan(d*x+c))**2/(d*e*x+c*e)**5,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 534 vs. \(2 (156) = 312\).

Time = 0.20 (sec) , antiderivative size = 534, normalized size of antiderivative = 3.14 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^5} \, dx=\frac {1}{6} \, {\left (d {\left (\frac {3 \, d^{2} x^{2} + 6 \, c d x + 3 \, c^{2} - 1}{d^{5} e^{5} x^{3} + 3 \, c d^{4} e^{5} x^{2} + 3 \, c^{2} d^{3} e^{5} x + c^{3} d^{2} e^{5}} + \frac {3 \, \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{2} e^{5}}\right )} - \frac {3 \, \arctan \left (d x + c\right )}{d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}}\right )} a b + \frac {1}{12} \, {\left (2 \, d {\left (\frac {3 \, d^{2} x^{2} + 6 \, c d x + 3 \, c^{2} - 1}{d^{5} e^{5} x^{3} + 3 \, c d^{4} e^{5} x^{2} + 3 \, c^{2} d^{3} e^{5} x + c^{3} d^{2} e^{5}} + \frac {3 \, \arctan \left (\frac {d^{2} x + c d}{d}\right )}{d^{2} e^{5}}\right )} \arctan \left (d x + c\right ) - \frac {{\left (3 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \arctan \left (d x + c\right )^{2} - 4 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d^{2} x^{2} + 2 \, c d x + c^{2} + 1\right ) + 8 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \log \left (d x + c\right ) + 1\right )} d^{2}}{d^{5} e^{5} x^{2} + 2 \, c d^{4} e^{5} x + c^{2} d^{3} e^{5}}\right )} b^{2} - \frac {b^{2} \arctan \left (d x + c\right )^{2}}{4 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} - \frac {a^{2}}{4 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} \] Input:

integrate((a+b*arctan(d*x+c))^2/(d*e*x+c*e)^5,x, algorithm="maxima")
 

Output:

1/6*(d*((3*d^2*x^2 + 6*c*d*x + 3*c^2 - 1)/(d^5*e^5*x^3 + 3*c*d^4*e^5*x^2 + 
 3*c^2*d^3*e^5*x + c^3*d^2*e^5) + 3*arctan((d^2*x + c*d)/d)/(d^2*e^5)) - 3 
*arctan(d*x + c)/(d^5*e^5*x^4 + 4*c*d^4*e^5*x^3 + 6*c^2*d^3*e^5*x^2 + 4*c^ 
3*d^2*e^5*x + c^4*d*e^5))*a*b + 1/12*(2*d*((3*d^2*x^2 + 6*c*d*x + 3*c^2 - 
1)/(d^5*e^5*x^3 + 3*c*d^4*e^5*x^2 + 3*c^2*d^3*e^5*x + c^3*d^2*e^5) + 3*arc 
tan((d^2*x + c*d)/d)/(d^2*e^5))*arctan(d*x + c) - (3*(d^2*x^2 + 2*c*d*x + 
c^2)*arctan(d*x + c)^2 - 4*(d^2*x^2 + 2*c*d*x + c^2)*log(d^2*x^2 + 2*c*d*x 
 + c^2 + 1) + 8*(d^2*x^2 + 2*c*d*x + c^2)*log(d*x + c) + 1)*d^2/(d^5*e^5*x 
^2 + 2*c*d^4*e^5*x + c^2*d^3*e^5))*b^2 - 1/4*b^2*arctan(d*x + c)^2/(d^5*e^ 
5*x^4 + 4*c*d^4*e^5*x^3 + 6*c^2*d^3*e^5*x^2 + 4*c^3*d^2*e^5*x + c^4*d*e^5) 
 - 1/4*a^2/(d^5*e^5*x^4 + 4*c*d^4*e^5*x^3 + 6*c^2*d^3*e^5*x^2 + 4*c^3*d^2* 
e^5*x + c^4*d*e^5)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 836 vs. \(2 (156) = 312\).

Time = 0.34 (sec) , antiderivative size = 836, normalized size of antiderivative = 4.92 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^5} \, dx=\text {Too large to display} \] Input:

integrate((a+b*arctan(d*x+c))^2/(d*e*x+c*e)^5,x, algorithm="giac")
 

Output:

-1/192*(3*b^2*arctan((d*e*x + c*e)/e)^2*tan(1/2*arctan((d*e*x + c*e)/e))^8 
 + 6*a*b*arctan((d*e*x + c*e)/e)*tan(1/2*arctan((d*e*x + c*e)/e))^8 - 12*b 
^2*arctan((d*e*x + c*e)/e)^2*tan(1/2*arctan((d*e*x + c*e)/e))^6 - 4*b^2*ar 
ctan((d*e*x + c*e)/e)*tan(1/2*arctan((d*e*x + c*e)/e))^7 + 3*a^2*tan(1/2*a 
rctan((d*e*x + c*e)/e))^8 - 24*a*b*arctan((d*e*x + c*e)/e)*tan(1/2*arctan( 
(d*e*x + c*e)/e))^6 - 4*a*b*tan(1/2*arctan((d*e*x + c*e)/e))^7 - 30*b^2*ar 
ctan((d*e*x + c*e)/e)^2*tan(1/2*arctan((d*e*x + c*e)/e))^4 + 60*b^2*arctan 
((d*e*x + c*e)/e)*tan(1/2*arctan((d*e*x + c*e)/e))^5 - 12*a^2*tan(1/2*arct 
an((d*e*x + c*e)/e))^6 + 4*b^2*tan(1/2*arctan((d*e*x + c*e)/e))^6 - 60*a*b 
*arctan((d*e*x + c*e)/e)*tan(1/2*arctan((d*e*x + c*e)/e))^4 + 64*b^2*log(1 
6*tan(1/2*arctan((d*e*x + c*e)/e))^2/(tan(1/2*arctan((d*e*x + c*e)/e))^4 + 
 2*tan(1/2*arctan((d*e*x + c*e)/e))^2 + 1))*tan(1/2*arctan((d*e*x + c*e)/e 
))^4 + 60*a*b*tan(1/2*arctan((d*e*x + c*e)/e))^5 - 12*b^2*arctan((d*e*x + 
c*e)/e)^2*tan(1/2*arctan((d*e*x + c*e)/e))^2 - 60*b^2*arctan((d*e*x + c*e) 
/e)*tan(1/2*arctan((d*e*x + c*e)/e))^3 - 30*a^2*tan(1/2*arctan((d*e*x + c* 
e)/e))^4 + 8*b^2*tan(1/2*arctan((d*e*x + c*e)/e))^4 - 24*a*b*arctan((d*e*x 
 + c*e)/e)*tan(1/2*arctan((d*e*x + c*e)/e))^2 - 60*a*b*tan(1/2*arctan((d*e 
*x + c*e)/e))^3 + 3*b^2*arctan((d*e*x + c*e)/e)^2 + 4*b^2*arctan((d*e*x + 
c*e)/e)*tan(1/2*arctan((d*e*x + c*e)/e)) - 12*a^2*tan(1/2*arctan((d*e*x + 
c*e)/e))^2 + 4*b^2*tan(1/2*arctan((d*e*x + c*e)/e))^2 + 6*a*b*arctan((d...
 

Mupad [B] (verification not implemented)

Time = 4.37 (sec) , antiderivative size = 438, normalized size of antiderivative = 2.58 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^5} \, dx={\mathrm {atan}\left (c+d\,x\right )}^2\,\left (\frac {b^2}{4\,d\,e^5}-\frac {b^2}{4\,d^3\,e^5\,\left (\frac {c^4}{d^2}+6\,c^2\,x^2+d^2\,x^4+\frac {4\,c^3\,x}{d}+4\,c\,d\,x^3\right )}\right )-\frac {x^2\,\left (\frac {b^2\,d}{2}-9\,a\,b\,c\,d\right )+x\,\left (b^2\,c-9\,a\,b\,c^2+a\,b\right )+\frac {3\,a^2-6\,a\,b\,c^3+2\,a\,b\,c+b^2\,c^2}{2\,d}-3\,a\,b\,d^2\,x^3}{6\,c^4\,e^5+24\,c^3\,d\,e^5\,x+36\,c^2\,d^2\,e^5\,x^2+24\,c\,d^3\,e^5\,x^3+6\,d^4\,e^5\,x^4}+\frac {\mathrm {atan}\left (c+d\,x\right )\,\left (\frac {b^2\,x^3}{2\,e^5}-\frac {a\,b}{2\,d^3\,e^5}+\frac {b^2\,c\,\left (\frac {c^2-1}{3\,d^2}+\frac {2\,c^2}{3\,d^2}\right )}{2\,d\,e^5}+\frac {b^2\,x\,\left (d\,\left (\frac {c^2-1}{3\,d^2}+\frac {2\,c^2}{3\,d^2}\right )+\frac {2\,c^2}{d}\right )}{2\,d\,e^5}+\frac {3\,b^2\,c\,x^2}{2\,d\,e^5}\right )}{\frac {c^4}{d^2}+6\,c^2\,x^2+d^2\,x^4+\frac {4\,c^3\,x}{d}+4\,c\,d\,x^3}-\frac {2\,b^2\,\ln \left (c+d\,x\right )}{3\,d\,e^5}-\frac {\ln \left (c+d\,x-\mathrm {i}\right )\,\left (-\frac {b^2}{3}+\frac {a\,b\,1{}\mathrm {i}}{4}\right )}{d\,e^5}+\frac {\ln \left (c+d\,x+1{}\mathrm {i}\right )\,\left (\frac {b^2}{3}+\frac {1{}\mathrm {i}\,a\,b}{4}\right )}{d\,e^5} \] Input:

int((a + b*atan(c + d*x))^2/(c*e + d*e*x)^5,x)
 

Output:

atan(c + d*x)^2*(b^2/(4*d*e^5) - b^2/(4*d^3*e^5*(c^4/d^2 + 6*c^2*x^2 + d^2 
*x^4 + (4*c^3*x)/d + 4*c*d*x^3))) - (x^2*((b^2*d)/2 - 9*a*b*c*d) + x*(a*b 
+ b^2*c - 9*a*b*c^2) + (3*a^2 + b^2*c^2 + 2*a*b*c - 6*a*b*c^3)/(2*d) - 3*a 
*b*d^2*x^3)/(6*c^4*e^5 + 6*d^4*e^5*x^4 + 24*c*d^3*e^5*x^3 + 36*c^2*d^2*e^5 
*x^2 + 24*c^3*d*e^5*x) + (atan(c + d*x)*((b^2*x^3)/(2*e^5) - (a*b)/(2*d^3* 
e^5) + (b^2*c*((c^2 - 1)/(3*d^2) + (2*c^2)/(3*d^2)))/(2*d*e^5) + (b^2*x*(d 
*((c^2 - 1)/(3*d^2) + (2*c^2)/(3*d^2)) + (2*c^2)/d))/(2*d*e^5) + (3*b^2*c* 
x^2)/(2*d*e^5)))/(c^4/d^2 + 6*c^2*x^2 + d^2*x^4 + (4*c^3*x)/d + 4*c*d*x^3) 
 - (2*b^2*log(c + d*x))/(3*d*e^5) - (log(c + d*x - 1i)*((a*b*1i)/4 - b^2/3 
))/(d*e^5) + (log(c + d*x + 1i)*((a*b*1i)/4 + b^2/3))/(d*e^5)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 679, normalized size of antiderivative = 3.99 \[ \int \frac {(a+b \arctan (c+d x))^2}{(c e+d e x)^5} \, dx=\frac {-16 \,\mathrm {log}\left (d x +c \right ) b^{2} c^{5}+12 \mathit {atan} \left (d x +c \right ) b^{2} c \,d^{3} x^{3}+8 \,\mathrm {log}\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b^{2} c \,d^{4} x^{4}-16 \,\mathrm {log}\left (d x +c \right ) b^{2} c \,d^{4} x^{4}+18 a b \,c^{2} d^{2} x^{2}-4 \mathit {atan} \left (d x +c \right ) b^{2} c^{2}-6 \mathit {atan} \left (d x +c \right )^{2} b^{2} c +6 \mathit {atan} \left (d x +c \right )^{2} b^{2} c \,d^{4} x^{4}+24 \mathit {atan} \left (d x +c \right )^{2} b^{2} c^{4} d x +36 \mathit {atan} \left (d x +c \right )^{2} b^{2} c^{3} d^{2} x^{2}+24 \mathit {atan} \left (d x +c \right )^{2} b^{2} c^{2} d^{3} x^{3}+36 \mathit {atan} \left (d x +c \right ) b^{2} c^{3} d x +36 \mathit {atan} \left (d x +c \right ) b^{2} c^{2} d^{2} x^{2}+32 \,\mathrm {log}\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b^{2} c^{4} d x +48 \,\mathrm {log}\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b^{2} c^{3} d^{2} x^{2}+32 \,\mathrm {log}\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b^{2} c^{2} d^{3} x^{3}-64 \,\mathrm {log}\left (d x +c \right ) b^{2} c^{4} d x -96 \,\mathrm {log}\left (d x +c \right ) b^{2} c^{3} d^{2} x^{2}-64 \,\mathrm {log}\left (d x +c \right ) b^{2} c^{2} d^{3} x^{3}-2 b^{2} c^{3}-4 \mathit {atan} \left (d x +c \right ) b^{2} c d x -4 a b \,c^{2}+6 \mathit {atan} \left (d x +c \right )^{2} b^{2} c^{5}+12 \mathit {atan} \left (d x +c \right ) b^{2} c^{4}+8 \,\mathrm {log}\left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) b^{2} c^{5}-12 \mathit {atan} \left (d x +c \right ) a b c -4 a b c d x -6 a^{2} c +9 a b \,c^{4}+48 \mathit {atan} \left (d x +c \right ) a b \,c^{4} d x +72 \mathit {atan} \left (d x +c \right ) a b \,c^{3} d^{2} x^{2}+48 \mathit {atan} \left (d x +c \right ) a b \,c^{2} d^{3} x^{3}+12 \mathit {atan} \left (d x +c \right ) a b c \,d^{4} x^{4}+12 \mathit {atan} \left (d x +c \right ) a b \,c^{5}-3 a b \,d^{4} x^{4}-2 b^{2} c \,d^{2} x^{2}-4 b^{2} c^{2} d x +24 a b \,c^{3} d x}{24 c d \,e^{5} \left (d^{4} x^{4}+4 c \,d^{3} x^{3}+6 c^{2} d^{2} x^{2}+4 c^{3} d x +c^{4}\right )} \] Input:

int((a+b*atan(d*x+c))^2/(d*e*x+c*e)^5,x)
 

Output:

(6*atan(c + d*x)**2*b**2*c**5 + 24*atan(c + d*x)**2*b**2*c**4*d*x + 36*ata 
n(c + d*x)**2*b**2*c**3*d**2*x**2 + 24*atan(c + d*x)**2*b**2*c**2*d**3*x** 
3 + 6*atan(c + d*x)**2*b**2*c*d**4*x**4 - 6*atan(c + d*x)**2*b**2*c + 12*a 
tan(c + d*x)*a*b*c**5 + 48*atan(c + d*x)*a*b*c**4*d*x + 72*atan(c + d*x)*a 
*b*c**3*d**2*x**2 + 48*atan(c + d*x)*a*b*c**2*d**3*x**3 + 12*atan(c + d*x) 
*a*b*c*d**4*x**4 - 12*atan(c + d*x)*a*b*c + 12*atan(c + d*x)*b**2*c**4 + 3 
6*atan(c + d*x)*b**2*c**3*d*x + 36*atan(c + d*x)*b**2*c**2*d**2*x**2 - 4*a 
tan(c + d*x)*b**2*c**2 + 12*atan(c + d*x)*b**2*c*d**3*x**3 - 4*atan(c + d* 
x)*b**2*c*d*x + 8*log(c**2 + 2*c*d*x + d**2*x**2 + 1)*b**2*c**5 + 32*log(c 
**2 + 2*c*d*x + d**2*x**2 + 1)*b**2*c**4*d*x + 48*log(c**2 + 2*c*d*x + d** 
2*x**2 + 1)*b**2*c**3*d**2*x**2 + 32*log(c**2 + 2*c*d*x + d**2*x**2 + 1)*b 
**2*c**2*d**3*x**3 + 8*log(c**2 + 2*c*d*x + d**2*x**2 + 1)*b**2*c*d**4*x** 
4 - 16*log(c + d*x)*b**2*c**5 - 64*log(c + d*x)*b**2*c**4*d*x - 96*log(c + 
 d*x)*b**2*c**3*d**2*x**2 - 64*log(c + d*x)*b**2*c**2*d**3*x**3 - 16*log(c 
 + d*x)*b**2*c*d**4*x**4 - 6*a**2*c + 9*a*b*c**4 + 24*a*b*c**3*d*x + 18*a* 
b*c**2*d**2*x**2 - 4*a*b*c**2 - 4*a*b*c*d*x - 3*a*b*d**4*x**4 - 2*b**2*c** 
3 - 4*b**2*c**2*d*x - 2*b**2*c*d**2*x**2)/(24*c*d*e**5*(c**4 + 4*c**3*d*x 
+ 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4))