\(\int \frac {\arctan (a x)}{x^2 (c+a^2 c x^2)^2} \, dx\) [189]

Optimal result

Integrand size = 20, antiderivative size = 97 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {a}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{c^2 x}-\frac {a^2 x \arctan (a x)}{2 c^2 \left (1+a^2 x^2\right )}-\frac {3 a \arctan (a x)^2}{4 c^2}+\frac {a \log (x)}{c^2}-\frac {a \log \left (1+a^2 x^2\right )}{2 c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.97 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {a}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\left (2+3 a^2 x^2\right ) \arctan (a x)}{2 c^2 x \left (1+a^2 x^2\right )}-\frac {3 a \arctan (a x)^2}{4 c^2}+\frac {a \log (x)}{c^2}-\frac {a \log \left (1+a^2 x^2\right )}{2 c^2} \] Input:

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

Output:

-1/4*a/(c^2*(1 + a^2*x^2)) - ((2 + 3*a^2*x^2)*ArcTan[a*x])/(2*c^2*x*(1 + a 
^2*x^2)) - (3*a*ArcTan[a*x]^2)/(4*c^2) + (a*Log[x])/c^2 - (a*Log[1 + a^2*x 
^2])/(2*c^2)
 

Rubi [A] (verified)

Time = 0.64 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {5501, 27, 5427, 241, 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.

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

\(\Big \downarrow \) 5501

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5427

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 5453

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

\(\Big \downarrow \) 5361

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 47

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 5419

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

Input:

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

Output:

-((a^2*(1/(4*a*(1 + a^2*x^2)) + (x*ArcTan[a*x])/(2*(1 + a^2*x^2)) + ArcTan 
[a*x]^2/(4*a)))/c^2) + (-(ArcTan[a*x]/x) - (a*ArcTan[a*x]^2)/2 + (a*(Log[x 
^2] - Log[1 + a^2*x^2]))/2)/c^2
 

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 241

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

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 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 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 = 0.49 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(a \left (-\frac {a x \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {3 \arctan \left (a x \right )^{2}}{2 c^{2}}-\frac {\arctan \left (a x \right )}{c^{2} a x}-\frac {-\frac {3 \arctan \left (a x \right )^{2}}{2}+\frac {1}{2 a^{2} x^{2}+2}+\ln \left (a^{2} x^{2}+1\right )-2 \ln \left (a x \right )}{2 c^{2}}\right )\)	\(95\)
default	\(a \left (-\frac {a x \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {3 \arctan \left (a x \right )^{2}}{2 c^{2}}-\frac {\arctan \left (a x \right )}{c^{2} a x}-\frac {-\frac {3 \arctan \left (a x \right )^{2}}{2}+\frac {1}{2 a^{2} x^{2}+2}+\ln \left (a^{2} x^{2}+1\right )-2 \ln \left (a x \right )}{2 c^{2}}\right )\)	\(95\)
parts	\(-\frac {a^{2} x \arctan \left (a x \right )}{2 c^{2} \left (a^{2} x^{2}+1\right )}-\frac {3 a \arctan \left (a x \right )^{2}}{2 c^{2}}-\frac {\arctan \left (a x \right )}{c^{2} x}-\frac {-\frac {3 a \arctan \left (a x \right )^{2}}{4}-\frac {a \left (-\frac {1}{2 \left (a^{2} x^{2}+1\right )}-\ln \left (a^{2} x^{2}+1\right )+2 \ln \left (a x \right )\right )}{2}}{c^{2}}\)	\(100\)
parallelrisch	\(\frac {-3 a^{3} \arctan \left (a x \right )^{2} x^{3}+4 \ln \left (x \right ) a^{3} x^{3}-2 a^{3} \ln \left (a^{2} x^{2}+1\right ) x^{3}+a^{3} x^{3}-6 x^{2} a^{2} \arctan \left (a x \right )-3 a \arctan \left (a x \right )^{2} x +4 a x \ln \left (x \right )-2 a \ln \left (a^{2} x^{2}+1\right ) x -4 \arctan \left (a x \right )}{4 x \,c^{2} \left (a^{2} x^{2}+1\right )}\)	\(118\)
risch	\(-\frac {a \ln \left (-i a x +1\right )}{16 c^{2} \left (-i a x -1\right )}-\frac {a \ln \left (-i a x +1\right )}{8 c^{2} \left (-i a x +1\right )}-\frac {3 a \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (-i a x +1\right )}{8 c^{2}}+\frac {3 a \ln \left (\frac {1}{2}+\frac {i a x}{2}\right ) \ln \left (\frac {1}{2}-\frac {i a x}{2}\right )}{4 c^{2}}+\frac {i \ln \left (i a x +1\right )}{2 c^{2} x}-\frac {a}{8 c^{2} \left (-i a x +1\right )}+\frac {3 a \operatorname {dilog}\left (\frac {1}{2}-\frac {i a x}{2}\right )}{8 c^{2}}+\frac {a \ln \left (-i a x \right )}{2 c^{2}}-\frac {a \ln \left (-i a x +1\right )}{2 c^{2}}+\frac {3 a \ln \left (-i a x +1\right )^{2}}{16 c^{2}}-\frac {a \ln \left (i a x +1\right )}{8 c^{2} \left (i a x +1\right )}-\frac {3 a \ln \left (\frac {1}{2}-\frac {i a x}{2}\right ) \ln \left (i a x +1\right )}{8 c^{2}}+\frac {i a^{2} \ln \left (-i a x +1\right ) x}{16 c^{2} \left (-i a x -1\right )}-\frac {i a^{2} \ln \left (i a x +1\right ) x}{16 c^{2} \left (i a x -1\right )}+\frac {a \ln \left (i a x \right )}{2 c^{2}}-\frac {a \ln \left (i a x +1\right )}{2 c^{2}}+\frac {3 a \ln \left (i a x +1\right )^{2}}{16 c^{2}}-\frac {i \ln \left (-i a x +1\right )}{2 c^{2} x}-\frac {a}{8 c^{2} \left (i a x +1\right )}+\frac {3 a \operatorname {dilog}\left (\frac {1}{2}+\frac {i a x}{2}\right )}{8 c^{2}}-\frac {a \ln \left (i a x +1\right )}{16 c^{2} \left (i a x -1\right )}+\frac {a \ln \left (a^{2} x^{2}+1\right )}{16 c^{2}}\)	\(406\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {3 \, {\left (a^{3} x^{3} + a x\right )} \arctan \left (a x\right )^{2} + a x + 2 \, {\left (3 \, a^{2} x^{2} + 2\right )} \arctan \left (a x\right ) + 2 \, {\left (a^{3} x^{3} + a x\right )} \log \left (a^{2} x^{2} + 1\right ) - 4 \, {\left (a^{3} x^{3} + a x\right )} \log \left (x\right )}{4 \, {\left (a^{2} c^{2} x^{3} + c^{2} x\right )}} \] Input:

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

Output:

-1/4*(3*(a^3*x^3 + a*x)*arctan(a*x)^2 + a*x + 2*(3*a^2*x^2 + 2)*arctan(a*x 
) + 2*(a^3*x^3 + a*x)*log(a^2*x^2 + 1) - 4*(a^3*x^3 + a*x)*log(x))/(a^2*c^ 
2*x^3 + c^2*x)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (88) = 176\).

Time = 0.66 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.82 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\begin {cases} \frac {4 a^{3} x^{3} \log {\left (x \right )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} - \frac {2 a^{3} x^{3} \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} - \frac {3 a^{3} x^{3} \operatorname {atan}^{2}{\left (a x \right )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} - \frac {6 a^{2} x^{2} \operatorname {atan}{\left (a x \right )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} + \frac {4 a x \log {\left (x \right )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} - \frac {2 a x \log {\left (x^{2} + \frac {1}{a^{2}} \right )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} - \frac {3 a x \operatorname {atan}^{2}{\left (a x \right )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} - \frac {a x}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} - \frac {4 \operatorname {atan}{\left (a x \right )}}{4 a^{2} c^{2} x^{3} + 4 c^{2} x} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((4*a**3*x**3*log(x)/(4*a**2*c**2*x**3 + 4*c**2*x) - 2*a**3*x**3* 
log(x**2 + a**(-2))/(4*a**2*c**2*x**3 + 4*c**2*x) - 3*a**3*x**3*atan(a*x)* 
*2/(4*a**2*c**2*x**3 + 4*c**2*x) - 6*a**2*x**2*atan(a*x)/(4*a**2*c**2*x**3 
 + 4*c**2*x) + 4*a*x*log(x)/(4*a**2*c**2*x**3 + 4*c**2*x) - 2*a*x*log(x**2 
 + a**(-2))/(4*a**2*c**2*x**3 + 4*c**2*x) - 3*a*x*atan(a*x)**2/(4*a**2*c** 
2*x**3 + 4*c**2*x) - a*x/(4*a**2*c**2*x**3 + 4*c**2*x) - 4*atan(a*x)/(4*a* 
*2*c**2*x**3 + 4*c**2*x), Ne(a, 0)), (0, True))
 

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.23 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=-\frac {1}{2} \, {\left (\frac {3 \, a^{2} x^{2} + 2}{a^{2} c^{2} x^{3} + c^{2} x} + \frac {3 \, a \arctan \left (a x\right )}{c^{2}}\right )} \arctan \left (a x\right ) + \frac {{\left (3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 2 \, {\left (a^{2} x^{2} + 1\right )} \log \left (a^{2} x^{2} + 1\right ) + 4 \, {\left (a^{2} x^{2} + 1\right )} \log \left (x\right ) - 1\right )} a}{4 \, {\left (a^{2} c^{2} x^{2} + c^{2}\right )}} \] Input:

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

Output:

-1/2*((3*a^2*x^2 + 2)/(a^2*c^2*x^3 + c^2*x) + 3*a*arctan(a*x)/c^2)*arctan( 
a*x) + 1/4*(3*(a^2*x^2 + 1)*arctan(a*x)^2 - 2*(a^2*x^2 + 1)*log(a^2*x^2 + 
1) + 4*(a^2*x^2 + 1)*log(x) - 1)*a/(a^2*c^2*x^2 + c^2)
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.74 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.94 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\frac {a\,\ln \left (x\right )}{c^2}-\frac {a\,\ln \left (a^2\,x^2+1\right )}{2\,c^2}-\frac {\mathrm {atan}\left (a\,x\right )\,\left (\frac {1}{a^2\,c^2}+\frac {3\,x^2}{2\,c^2}\right )}{\frac {x}{a^2}+x^3}-\frac {a}{2\,\left (2\,a^2\,c^2\,x^2+2\,c^2\right )}-\frac {3\,a\,{\mathrm {atan}\left (a\,x\right )}^2}{4\,c^2} \] Input:

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

Output:

(a*log(x))/c^2 - (a*log(a^2*x^2 + 1))/(2*c^2) - (atan(a*x)*(1/(a^2*c^2) + 
(3*x^2)/(2*c^2)))/(x/a^2 + x^3) - a/(2*(2*c^2 + 2*a^2*c^2*x^2)) - (3*a*ata 
n(a*x)^2)/(4*c^2)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.21 \[ \int \frac {\arctan (a x)}{x^2 \left (c+a^2 c x^2\right )^2} \, dx=\frac {-3 \mathit {atan} \left (a x \right )^{2} a^{3} x^{3}-3 \mathit {atan} \left (a x \right )^{2} a x -6 \mathit {atan} \left (a x \right ) a^{2} x^{2}-4 \mathit {atan} \left (a x \right )-2 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a^{3} x^{3}-2 \,\mathrm {log}\left (a^{2} x^{2}+1\right ) a x +4 \,\mathrm {log}\left (x \right ) a^{3} x^{3}+4 \,\mathrm {log}\left (x \right ) a x +a^{3} x^{3}}{4 c^{2} x \left (a^{2} x^{2}+1\right )} \] Input:

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

Output:

( - 3*atan(a*x)**2*a**3*x**3 - 3*atan(a*x)**2*a*x - 6*atan(a*x)*a**2*x**2 
- 4*atan(a*x) - 2*log(a**2*x**2 + 1)*a**3*x**3 - 2*log(a**2*x**2 + 1)*a*x 
+ 4*log(x)*a**3*x**3 + 4*log(x)*a*x + a**3*x**3)/(4*c**2*x*(a**2*x**2 + 1) 
)