\(\int \frac {\arctan (a x)^2}{x^2 (c+a^2 c x^2)^2} \, dx\) [296]

Optimal result

Integrand size = 22, antiderivative size = 177 \[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a^2 x}{4 c^2 \left (1+a^2 x^2\right )}+\frac {a \arctan (a x)}{4 c^2}-\frac {a \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {i a \arctan (a x)^2}{c^2}-\frac {\arctan (a x)^2}{c^2 x}-\frac {a^2 x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}-\frac {a \arctan (a x)^3}{2 c^2}+\frac {2 a \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{c^2}-\frac {i a \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{c^2} \] Output:

1/4*a^2*x/c^2/(a^2*x^2+1)+1/4*a*arctan(a*x)/c^2-1/2*a*arctan(a*x)/c^2/(a^2 
*x^2+1)-I*a*arctan(a*x)^2/c^2-arctan(a*x)^2/c^2/x-1/2*a^2*x*arctan(a*x)^2/ 
c^2/(a^2*x^2+1)-1/2*a*arctan(a*x)^3/c^2+2*a*arctan(a*x)*ln(2-2/(1-I*a*x))/ 
c^2-I*a*polylog(2,-1+2/(1-I*a*x))/c^2
 

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.62 \[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {4 a x \arctan (a x)^3+2 a x \arctan (a x) \left (\cos (2 \arctan (a x))-8 \log \left (1-e^{2 i \arctan (a x)}\right )\right )+8 i a x \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )-a x \sin (2 \arctan (a x))+2 \arctan (a x)^2 (4+4 i a x+a x \sin (2 \arctan (a x)))}{8 c^2 x} \] Input:

Integrate[ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)^2),x]
 

Output:

-1/8*(4*a*x*ArcTan[a*x]^3 + 2*a*x*ArcTan[a*x]*(Cos[2*ArcTan[a*x]] - 8*Log[ 
1 - E^((2*I)*ArcTan[a*x])]) + (8*I)*a*x*PolyLog[2, E^((2*I)*ArcTan[a*x])] 
- a*x*Sin[2*ArcTan[a*x]] + 2*ArcTan[a*x]^2*(4 + (4*I)*a*x + a*x*Sin[2*ArcT 
an[a*x]]))/(c^2*x)
 

Rubi [A] (verified)

Time = 1.35 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5501, 27, 5427, 5453, 5361, 5419, 5459, 5403, 2897, 5465, 215, 216}

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.

\(\displaystyle \int \frac {\arctan (a x)^2}{x^2 \left (a^2 c x^2+c\right )^2} \, dx\)

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5427

\(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x^2 \left (a^2 x^2+1\right )}dx}{c^2}-\frac {a^2 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{c^2}\)

\(\Big \downarrow \) 5453

\(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x^2}dx-a^2 \int \frac {\arctan (a x)^2}{a^2 x^2+1}dx}{c^2}-\frac {a^2 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{c^2}\)

\(\Big \downarrow \) 5361

\(\displaystyle \frac {a^2 \left (-\int \frac {\arctan (a x)^2}{a^2 x^2+1}dx\right )+2 a \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx-\frac {\arctan (a x)^2}{x}}{c^2}-\frac {a^2 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{c^2}\)

\(\Big \downarrow \) 5419

\(\displaystyle \frac {2 a \int \frac {\arctan (a x)}{x \left (a^2 x^2+1\right )}dx-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}}{c^2}-\frac {a^2 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{c^2}\)

\(\Big \downarrow \) 5459

\(\displaystyle -\frac {a^2 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{c^2}+\frac {2 a \left (i \int \frac {\arctan (a x)}{x (a x+i)}dx-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}}{c^2}\)

\(\Big \downarrow \) 5403

\(\displaystyle -\frac {a^2 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{c^2}+\frac {2 a \left (i \left (i a \int \frac {\log \left (2-\frac {2}{1-i a x}\right )}{a^2 x^2+1}dx-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}}{c^2}\)

\(\Big \downarrow \) 2897

\(\displaystyle -\frac {a^2 \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{c^2}+\frac {2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}}{c^2}\)

\(\Big \downarrow \) 5465

\(\displaystyle -\frac {a^2 \left (-a \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{c^2}+\frac {2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}}{c^2}\)

\(\Big \downarrow \) 215

\(\displaystyle -\frac {a^2 \left (-a \left (\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )}{c^2}+\frac {2 a \left (i \left (-i \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {2}{1-i a x}-1\right )\right )-\frac {1}{2} i \arctan (a x)^2\right )-\frac {1}{3} a \arctan (a x)^3-\frac {\arctan (a x)^2}{x}}{c^2}\)

\(\Big \downarrow \) 216

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

Input:

Int[ArcTan[a*x]^2/(x^2*(c + a^2*c*x^2)^2),x]
 

Output:

-((a^2*((x*ArcTan[a*x]^2)/(2*(1 + a^2*x^2)) + ArcTan[a*x]^3/(6*a) - a*(-1/ 
2*ArcTan[a*x]/(a^2*(1 + a^2*x^2)) + (x/(2*(1 + a^2*x^2)) + ArcTan[a*x]/(2* 
a))/(2*a))))/c^2) + (-(ArcTan[a*x]^2/x) - (a*ArcTan[a*x]^3)/3 + 2*a*((-1/2 
*I)*ArcTan[a*x]^2 + I*((-I)*ArcTan[a*x]*Log[2 - 2/(1 - I*a*x)] - PolyLog[2 
, -1 + 2/(1 - I*a*x)]/2)))/c^2
 

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 215

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) 
/(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1))   Int[(a + b*x^2)^(p + 1 
), x], x] /; FreeQ[{a, b}, x] && LtQ[p, -1] && (IntegerQ[4*p] || IntegerQ[6 
*p])
 

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 2897

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[Pq^m*((1 - u)/ 
D[u, x])]}, Simp[C*PolyLog[2, 1 - u], x] /; FreeQ[C, x]] /; IntegerQ[m] && 
PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponents[u, 
 x][[2]], Expon[Pq, x]]
 

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 5403

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_ 
Symbol] :> Simp[(a + b*ArcTan[c*x])^p*(Log[2 - 2/(1 + e*(x/d))]/d), x] - Si 
mp[b*c*(p/d)   Int[(a + b*ArcTan[c*x])^(p - 1)*(Log[2 - 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 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 5427

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

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 5459

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

rule 5465

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

rule 5501

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 
)^(q_), x_Symbol] :> Simp[1/d   Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c 
*x])^p, x], x] - Simp[e/d   Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x]) 
^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2* 
q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.77

method	result	size
derivativedivides	\(a \left (-\frac {a x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {3 \arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\arctan \left (a x \right )^{2}}{c^{2} a x}-\frac {-\arctan \left (a x \right )^{3}+\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}+2}-2 \arctan \left (a x \right ) \ln \left (a x \right )-\frac {a x}{4 \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{4}-i \ln \left (a x \right ) \ln \left (i a x +1\right )+i \ln \left (a x \right ) \ln \left (-i a x +1\right )-i \operatorname {dilog}\left (i a x +1\right )+i \operatorname {dilog}\left (-i a x +1\right )+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{2}}{c^{2}}\right )\)	\(313\)
default	\(a \left (-\frac {a x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {3 \arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\arctan \left (a x \right )^{2}}{c^{2} a x}-\frac {-\arctan \left (a x \right )^{3}+\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}+2}-2 \arctan \left (a x \right ) \ln \left (a x \right )-\frac {a x}{4 \left (a^{2} x^{2}+1\right )}-\frac {\arctan \left (a x \right )}{4}-i \ln \left (a x \right ) \ln \left (i a x +1\right )+i \ln \left (a x \right ) \ln \left (-i a x +1\right )-i \operatorname {dilog}\left (i a x +1\right )+i \operatorname {dilog}\left (-i a x +1\right )+\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{2}-\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{2}}{c^{2}}\right )\)	\(313\)
parts	\(-\frac {a^{2} x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {3 a \arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\arctan \left (a x \right )^{2}}{c^{2} x}-\frac {2 \left (-\frac {a \arctan \left (a x \right )^{3}}{2}-\frac {a \left (-\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+2 \arctan \left (a x \right ) \ln \left (a x \right )+\frac {a x}{4 a^{2} x^{2}+4}+\frac {\arctan \left (a x \right )}{4}+i \ln \left (a x \right ) \ln \left (i a x +1\right )-i \ln \left (a x \right ) \ln \left (-i a x +1\right )+i \operatorname {dilog}\left (i a x +1\right )-i \operatorname {dilog}\left (-i a x +1\right )-\frac {i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )}{2}+\frac {i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )}{2}\right )}{2}\right )}{c^{2}}\)	\(317\)

Input:

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

Output:

a*(-1/2*a*x*arctan(a*x)^2/c^2/(a^2*x^2+1)-3/2*arctan(a*x)^3/c^2-1/c^2*arct 
an(a*x)^2/a/x-1/c^2*(-arctan(a*x)^3+arctan(a*x)*ln(a^2*x^2+1)+1/2*arctan(a 
*x)/(a^2*x^2+1)-2*arctan(a*x)*ln(a*x)-1/4*a*x/(a^2*x^2+1)-1/4*arctan(a*x)- 
I*ln(a*x)*ln(1+I*a*x)+I*ln(a*x)*ln(1-I*a*x)-I*dilog(1+I*a*x)+I*dilog(1-I*a 
*x)+1/2*I*(ln(a*x-I)*ln(a^2*x^2+1)-1/2*ln(a*x-I)^2-dilog(-1/2*I*(a*x+I))-l 
n(a*x-I)*ln(-1/2*I*(a*x+I)))-1/2*I*(ln(a*x+I)*ln(a^2*x^2+1)-1/2*ln(a*x+I)^ 
2-dilog(1/2*I*(a*x-I))-ln(a*x+I)*ln(1/2*I*(a*x-I)))))
 

Fricas [F]

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

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

Output:

integral(arctan(a*x)^2/(a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2), x)
 

Sympy [F]

\[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\frac {\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{a^{4} x^{6} + 2 a^{2} x^{4} + x^{2}}\, dx}{c^{2}} \] Input:

integrate(atan(a*x)**2/x**2/(a**2*c*x**2+c)**2,x)
 

Output:

Integral(atan(a*x)**2/(a**4*x**6 + 2*a**2*x**4 + x**2), x)/c**2
 

Maxima [F]

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

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

Output:

-1/32*(6*a^3*x^3*arctan2(1, a*x) - 6*a^2*x^2 + 8*(a^3*x^3 + a*x)*arctan(a* 
x)^3 + 12*a*x*arctan(a*x) + 4*(3*a^2*x^2 + 2)*arctan(a*x)^2 + 6*a*x*arctan 
2(1, a*x) - (3*a^2*x^2 + 2)*log(a^2*x^2 + 1)^2 + 192*(a^6*c^2*x^3 + a^4*c^ 
2*x)*integrate(1/16*x^2*log(a^2*x^2 + 1)/(a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^ 
2), x) - 128*(a^2*c^2*x^3 + c^2*x)*integrate(1/64*(4*(a^2*x^2 + 1)^(7/2)*a 
^2*arctan(a*x)*log(a^2*x^2 + 1)*sin(6*arctan(a*x)) - 24*(a^2*x^2 + 1)^3*a^ 
2*arctan(a*x)*log(a^2*x^2 + 1)*sin(5*arctan(a*x)) + 52*(a^2*x^2 + 1)^(5/2) 
*a^2*arctan(a*x)*log(a^2*x^2 + 1)*sin(4*arctan(a*x)) - 48*(a^2*x^2 + 1)^2* 
a^2*arctan(a*x)*log(a^2*x^2 + 1)*sin(3*arctan(a*x)) + 16*(a^2*x^2 + 1)^(3/ 
2)*a^2*arctan(a*x)*log(a^2*x^2 + 1)*sin(2*arctan(a*x)) - (4*(a^2*x^2 + 1)^ 
(7/2)*a^2*arctan(a*x)^2 - (a^2*x^2 + 1)^(7/2)*a^2*log(a^2*x^2 + 1)^2)*cos( 
6*arctan(a*x)) + 6*(4*(a^2*x^2 + 1)^3*a^2*arctan(a*x)^2 - (a^2*x^2 + 1)^3* 
a^2*log(a^2*x^2 + 1)^2)*cos(5*arctan(a*x)) - 13*(4*(a^2*x^2 + 1)^(5/2)*a^2 
*arctan(a*x)^2 - (a^2*x^2 + 1)^(5/2)*a^2*log(a^2*x^2 + 1)^2)*cos(4*arctan( 
a*x)) + 12*(4*(a^2*x^2 + 1)^2*a^2*arctan(a*x)^2 - (a^2*x^2 + 1)^2*a^2*log( 
a^2*x^2 + 1)^2)*cos(3*arctan(a*x)) - 4*(4*(a^2*x^2 + 1)^(3/2)*a^2*arctan(a 
*x)^2 - (a^2*x^2 + 1)^(3/2)*a^2*log(a^2*x^2 + 1)^2)*cos(2*arctan(a*x)))*sq 
rt(a^2*x^2 + 1)/((a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^6*cos(6*arctan(a*x))^2 
+ (a^2*c^2*x^2 + c^2)*(a^2*x^2 + 1)^6*sin(6*arctan(a*x))^2 + 36*(a^2*c^2*x 
^2 + c^2)*(a^2*x^2 + 1)^5*cos(5*arctan(a*x))^2 + 36*(a^2*c^2*x^2 + c^2)...
 

Giac [F]

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

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

Output:

integrate(arctan(a*x)^2/((a^2*c*x^2 + c)^2*x^2), x)
 

Mupad [F(-1)]

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

int(atan(a*x)^2/(x^2*(c + a^2*c*x^2)^2),x)
 

Output:

int(atan(a*x)^2/(x^2*(c + a^2*c*x^2)^2), x)
 

Reduce [F]

\[ \int \frac {\arctan (a x)^2}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\frac {-2 \mathit {atan} \left (a x \right )^{3} a^{3} x^{3}-2 \mathit {atan} \left (a x \right )^{3} a x -6 \mathit {atan} \left (a x \right )^{2} a^{2} x^{2}-4 \mathit {atan} \left (a x \right )^{2}+3 \mathit {atan} \left (a x \right ) a^{3} x^{3}-3 \mathit {atan} \left (a x \right ) a x +8 \left (\int \frac {\mathit {atan} \left (a x \right )}{a^{4} x^{5}+2 a^{2} x^{3}+x}d x \right ) a^{3} x^{3}+8 \left (\int \frac {\mathit {atan} \left (a x \right )}{a^{4} x^{5}+2 a^{2} x^{3}+x}d x \right ) a x +3 a^{2} x^{2}}{4 c^{2} x \left (a^{2} x^{2}+1\right )} \] Input:

int(atan(a*x)^2/x^2/(a^2*c*x^2+c)^2,x)
                                                                                    
                                                                                    
 

Output:

( - 2*atan(a*x)**3*a**3*x**3 - 2*atan(a*x)**3*a*x - 6*atan(a*x)**2*a**2*x* 
*2 - 4*atan(a*x)**2 + 3*atan(a*x)*a**3*x**3 - 3*atan(a*x)*a*x + 8*int(atan 
(a*x)/(a**4*x**5 + 2*a**2*x**3 + x),x)*a**3*x**3 + 8*int(atan(a*x)/(a**4*x 
**5 + 2*a**2*x**3 + x),x)*a*x + 3*a**2*x**2)/(4*c**2*x*(a**2*x**2 + 1))