\(\int \frac {\arctan (a x)^2}{x^4 (c+a^2 c x^2)} \, dx\) [290]

Optimal result

Integrand size = 22, antiderivative size = 166 \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx=-\frac {a^2}{3 c x}-\frac {a^3 \arctan (a x)}{3 c}-\frac {a \arctan (a x)}{3 c x^2}+\frac {4 i a^3 \arctan (a x)^2}{3 c}-\frac {\arctan (a x)^2}{3 c x^3}+\frac {a^2 \arctan (a x)^2}{c x}+\frac {a^3 \arctan (a x)^3}{3 c}-\frac {8 a^3 \arctan (a x) \log \left (2-\frac {2}{1-i a x}\right )}{3 c}+\frac {4 i a^3 \operatorname {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )}{3 c} \] Output:

-1/3*a^2/c/x-1/3*a^3*arctan(a*x)/c-1/3*a*arctan(a*x)/c/x^2+4/3*I*a^3*arcta 
n(a*x)^2/c-1/3*arctan(a*x)^2/c/x^3+a^2*arctan(a*x)^2/c/x+1/3*a^3*arctan(a* 
x)^3/c-8/3*a^3*arctan(a*x)*ln(2-2/(1-I*a*x))/c+4/3*I*a^3*polylog(2,-1+2/(1 
-I*a*x))/c
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.72 \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx=\frac {a^3 \left (-\frac {1-4 \arctan (a x)^2+\frac {\left (1+a^2 x^2\right ) \arctan (a x)^2}{a^2 x^2}}{a x}+\arctan (a x) \left (-\frac {1+a^2 x^2}{a^2 x^2}+\arctan (a x) (4 i+\arctan (a x))-8 \log \left (1-e^{2 i \arctan (a x)}\right )\right )+4 i \operatorname {PolyLog}\left (2,e^{2 i \arctan (a x)}\right )\right )}{3 c} \] Input:

Integrate[ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)),x]
 

Output:

(a^3*(-((1 - 4*ArcTan[a*x]^2 + ((1 + a^2*x^2)*ArcTan[a*x]^2)/(a^2*x^2))/(a 
*x)) + ArcTan[a*x]*(-((1 + a^2*x^2)/(a^2*x^2)) + ArcTan[a*x]*(4*I + ArcTan 
[a*x]) - 8*Log[1 - E^((2*I)*ArcTan[a*x])]) + (4*I)*PolyLog[2, E^((2*I)*Arc 
Tan[a*x])]))/(3*c)
 

Rubi [A] (verified)

Time = 1.40 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.30, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5453, 27, 5361, 5453, 5361, 264, 216, 5419, 5459, 5403, 2897}

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.

\(\displaystyle \int \frac {\arctan (a x)^2}{x^4 \left (a^2 c x^2+c\right )} \, dx\)

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 264

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 5419

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

\(\Big \downarrow \) 5459

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

\(\Big \downarrow \) 5403

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

\(\Big \downarrow \) 2897

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

Input:

Int[ArcTan[a*x]^2/(x^4*(c + a^2*c*x^2)),x]
 

Output:

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

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 264

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( 
m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c 
^2*(m + 1)))   Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p 
}, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
 

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 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]
 

Maple [A] (verified)

Time = 1.32 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.77

method	result	size
derivativedivides	\(a^{3} \left (-\frac {\arctan \left (a x \right )^{2}}{3 c \,a^{3} x^{3}}+\frac {\arctan \left (a x \right )^{2}}{c a x}+\frac {\arctan \left (a x \right )^{3}}{c}-\frac {2 \left (-2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}}+4 \arctan \left (a x \right ) \ln \left (a x \right )+\frac {1}{2 a x}+\frac {\arctan \left (a x \right )}{2}-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )+2 i \ln \left (a x \right ) \ln \left (i a x +1\right )-2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+2 i \operatorname {dilog}\left (i a x +1\right )-2 i \operatorname {dilog}\left (-i a x +1\right )+\arctan \left (a x \right )^{3}\right )}{3 c}\right )\)	\(293\)
default	\(a^{3} \left (-\frac {\arctan \left (a x \right )^{2}}{3 c \,a^{3} x^{3}}+\frac {\arctan \left (a x \right )^{2}}{c a x}+\frac {\arctan \left (a x \right )^{3}}{c}-\frac {2 \left (-2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}}+4 \arctan \left (a x \right ) \ln \left (a x \right )+\frac {1}{2 a x}+\frac {\arctan \left (a x \right )}{2}-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )+2 i \ln \left (a x \right ) \ln \left (i a x +1\right )-2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+2 i \operatorname {dilog}\left (i a x +1\right )-2 i \operatorname {dilog}\left (-i a x +1\right )+\arctan \left (a x \right )^{3}\right )}{3 c}\right )\)	\(293\)
parts	\(\frac {a^{3} \arctan \left (a x \right )^{3}}{c}-\frac {\arctan \left (a x \right )^{2}}{3 c \,x^{3}}+\frac {a^{2} \arctan \left (a x \right )^{2}}{c x}-\frac {2 \left (a^{3} \left (-2 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )+\frac {\arctan \left (a x \right )}{2 a^{2} x^{2}}+4 \arctan \left (a x \right ) \ln \left (a x \right )+\frac {1}{2 a x}+\frac {\arctan \left (a x \right )}{2}-i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x -i\right )^{2}}{2}-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )\right )+i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\frac {\ln \left (a x +i\right )^{2}}{2}-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )\right )+2 i \ln \left (a x \right ) \ln \left (i a x +1\right )-2 i \ln \left (a x \right ) \ln \left (-i a x +1\right )+2 i \operatorname {dilog}\left (i a x +1\right )-2 i \operatorname {dilog}\left (-i a x +1\right )\right )+a^{3} \arctan \left (a x \right )^{3}\right )}{3 c}\)	\(298\)

Input:

int(arctan(a*x)^2/x^4/(a^2*c*x^2+c),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [F]

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

integrate(arctan(a*x)^2/x^4/(a^2*c*x^2+c),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx=\text {Timed out} \] Input:

integrate(arctan(a*x)^2/x^4/(a^2*c*x^2+c),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

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

integrate(arctan(a*x)^2/x^4/(a^2*c*x^2+c),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^4\,\left (c\,a^2\,x^2+c\right )} \,d x \] Input:

int(atan(a*x)^2/(x^4*(c + a^2*c*x^2)),x)
 

Output:

int(atan(a*x)^2/(x^4*(c + a^2*c*x^2)), x)
 

Reduce [F]

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

int(atan(a*x)^2/x^4/(a^2*c*x^2+c),x)
 

Output:

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