\(\int \frac {x^2 \arctan (a x)^2}{(c+a^2 c x^2)^3} \, dx\) [300]

Optimal result

Integrand size = 22, antiderivative size = 181 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {x}{32 a^2 c^3 \left (1+a^2 x^2\right )^2}-\frac {x}{64 a^2 c^3 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{64 a^3 c^3}-\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )^2}+\frac {\arctan (a x)}{8 a^3 c^3 \left (1+a^2 x^2\right )}-\frac {x \arctan (a x)^2}{4 a^2 c^3 \left (1+a^2 x^2\right )^2}+\frac {x \arctan (a x)^2}{8 a^2 c^3 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{24 a^3 c^3} \] Output:

1/32*x/a^2/c^3/(a^2*x^2+1)^2-1/64*x/a^2/c^3/(a^2*x^2+1)-1/64*arctan(a*x)/a 
^3/c^3-1/8*arctan(a*x)/a^3/c^3/(a^2*x^2+1)^2+1/8*arctan(a*x)/a^3/c^3/(a^2* 
x^2+1)-1/4*x*arctan(a*x)^2/a^2/c^3/(a^2*x^2+1)^2+1/8*x*arctan(a*x)^2/a^2/c 
^3/(a^2*x^2+1)+1/24*arctan(a*x)^3/a^3/c^3
 

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.52 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {3 a x-3 a^3 x^3-3 \left (1-6 a^2 x^2+a^4 x^4\right ) \arctan (a x)+24 a x \left (-1+a^2 x^2\right ) \arctan (a x)^2+8 \left (1+a^2 x^2\right )^2 \arctan (a x)^3}{192 a^3 c^3 \left (1+a^2 x^2\right )^2} \] Input:

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

Output:

(3*a*x - 3*a^3*x^3 - 3*(1 - 6*a^2*x^2 + a^4*x^4)*ArcTan[a*x] + 24*a*x*(-1 
+ a^2*x^2)*ArcTan[a*x]^2 + 8*(1 + a^2*x^2)^2*ArcTan[a*x]^3)/(192*a^3*c^3*( 
1 + a^2*x^2)^2)
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.72, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5499, 27, 5427, 5435, 215, 215, 216, 5427, 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.

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

\(\Big \downarrow \) 5499

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5427

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

\(\Big \downarrow \) 5435

\(\displaystyle \frac {-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}}{a^2 c^3}-\frac {\frac {3}{4} \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx-\frac {1}{8} \int \frac {1}{\left (a^2 x^2+1\right )^3}dx+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}}{a^2 c^3}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}}{a^2 c^3}-\frac {\frac {3}{4} \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {1}{8} \left (-\frac {3}{4} \int \frac {1}{\left (a^2 x^2+1\right )^2}dx-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}}{a^2 c^3}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}}{a^2 c^3}-\frac {\frac {3}{4} \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}}{a^2 c^3}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}}{a^2 c^3}-\frac {\frac {3}{4} \int \frac {\arctan (a x)^2}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )}{a^2 c^3}\)

\(\Big \downarrow \) 5427

\(\displaystyle \frac {-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}}{a^2 c^3}-\frac {\frac {3}{4} \left (-a \int \frac {x \arctan (a x)}{\left (a^2 x^2+1\right )^2}dx+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )}{a^2 c^3}\)

\(\Big \downarrow \) 5465

\(\displaystyle \frac {-a \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}}{a^2 c^3}-\frac {\frac {3}{4} \left (-a \left (\frac {\int \frac {1}{\left (a^2 x^2+1\right )^2}dx}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )}{a^2 c^3}\)

\(\Big \downarrow \) 215

\(\displaystyle \frac {-a \left (\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}}{a^2 c^3}-\frac {\frac {3}{4} \left (-a \left (\frac {\frac {1}{2} \int \frac {1}{a^2 x^2+1}dx+\frac {x}{2 \left (a^2 x^2+1\right )}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)^3}{6 a}\right )+\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )}{a^2 c^3}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}}{a^2 c^3}-\frac {\frac {x \arctan (a x)^2}{4 \left (a^2 x^2+1\right )^2}+\frac {\arctan (a x)}{8 a \left (a^2 x^2+1\right )^2}+\frac {1}{8} \left (-\frac {3}{4} \left (\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}\right )-\frac {x}{4 \left (a^2 x^2+1\right )^2}\right )+\frac {3}{4} \left (\frac {x \arctan (a x)^2}{2 \left (a^2 x^2+1\right )}-a \left (\frac {\frac {x}{2 \left (a^2 x^2+1\right )}+\frac {\arctan (a x)}{2 a}}{2 a}-\frac {\arctan (a x)}{2 a^2 \left (a^2 x^2+1\right )}\right )+\frac {\arctan (a x)^3}{6 a}\right )}{a^2 c^3}\)

Input:

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

Output:

((x*ArcTan[a*x]^2)/(2*(1 + a^2*x^2)) + ArcTan[a*x]^3/(6*a) - a*(-1/2*ArcTa 
n[a*x]/(a^2*(1 + a^2*x^2)) + (x/(2*(1 + a^2*x^2)) + ArcTan[a*x]/(2*a))/(2* 
a)))/(a^2*c^3) - (ArcTan[a*x]/(8*a*(1 + a^2*x^2)^2) + (x*ArcTan[a*x]^2)/(4 
*(1 + a^2*x^2)^2) + (-1/4*x/(1 + a^2*x^2)^2 - (3*(x/(2*(1 + a^2*x^2)) + Ar 
cTan[a*x]/(2*a)))/4)/8 + (3*((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))))/4)/(a^2*c^3)
 

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 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 5435

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_S 
ymbol] :> Simp[b*p*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p - 1)/(4*c*d* 
(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*d* 
(q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1))   Int[(d + e*x^2)^(q + 1)*(a + 
 b*ArcTan[c*x])^p, x], x] - Simp[b^2*p*((p - 1)/(4*(q + 1)^2))   Int[(d + e 
*x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] & 
& EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
 

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 5499

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 
)^(q_), x_Symbol] :> Simp[1/e   Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*Ar 
cTan[c*x])^p, x], x] - Simp[d/e   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] && IGtQ[m, 1] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.68

method	result	size
parallelrisch	\(\frac {8 \arctan \left (a x \right )^{3} a^{4} x^{4}-3 x^{4} \arctan \left (a x \right ) a^{4}+24 a^{3} \arctan \left (a x \right )^{2} x^{3}+16 \arctan \left (a x \right )^{3} a^{2} x^{2}-3 a^{3} x^{3}+18 x^{2} a^{2} \arctan \left (a x \right )-24 a \arctan \left (a x \right )^{2} x +8 \arctan \left (a x \right )^{3}+3 a x -3 \arctan \left (a x \right )}{192 c^{3} \left (a^{2} x^{2}+1\right )^{2} a^{3}}\)	\(123\)
derivativedivides	\(\frac {\frac {\arctan \left (a x \right )^{2} a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {a x \arctan \left (a x \right )^{2}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{3}}{8 c^{3}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\frac {1}{8} a^{3} x^{3}-\frac {1}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )}{16}}{4 c^{3}}}{a^{3}}\)	\(149\)
default	\(\frac {\frac {\arctan \left (a x \right )^{2} a^{3} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {a x \arctan \left (a x \right )^{2}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{3}}{8 c^{3}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\frac {1}{8} a^{3} x^{3}-\frac {1}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )}{16}}{4 c^{3}}}{a^{3}}\)	\(149\)
parts	\(\frac {\arctan \left (a x \right )^{2} x^{3}}{8 c^{3} \left (a^{2} x^{2}+1\right )^{2}}-\frac {x \arctan \left (a x \right )^{2}}{8 a^{2} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )^{3}}{8 a^{3} c^{3}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3 a^{3}}+\frac {-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\frac {1}{8} a^{3} x^{3}-\frac {1}{8} a x}{2 \left (a^{2} x^{2}+1\right )^{2}}+\frac {\arctan \left (a x \right )}{16}}{a^{3}}}{4 c^{3}}\)	\(155\)
risch	\(\frac {i \ln \left (i a x +1\right )^{3}}{192 a^{3} c^{3}}-\frac {i \left (x^{4} \ln \left (-i a x +1\right ) a^{4}+2 a^{2} x^{2} \ln \left (-i a x +1\right )-2 i a^{3} x^{3}+\ln \left (-i a x +1\right )+2 i a x \right ) \ln \left (i a x +1\right )^{2}}{64 a^{3} c^{3} \left (a^{2} x^{2}+1\right )^{2}}+\frac {i \left (a^{4} x^{4} \ln \left (-i a x +1\right )^{2}+2 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}-4 i x^{3} \ln \left (-i a x +1\right ) a^{3}-4 a^{2} x^{2}+\ln \left (-i a x +1\right )^{2}+4 i a x \ln \left (-i a x +1\right )\right ) \ln \left (i a x +1\right )}{64 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}-\frac {i \left (2 a^{4} x^{4} \ln \left (-i a x +1\right )^{3}+3 \ln \left (i a x -1\right ) a^{4} x^{4}-3 \ln \left (-i a x -1\right ) a^{4} x^{4}-12 i a^{3} x^{3} \ln \left (-i a x +1\right )^{2}-6 i a^{3} x^{3}+4 a^{2} x^{2} \ln \left (-i a x +1\right )^{3}+6 \ln \left (i a x -1\right ) a^{2} x^{2}-6 \ln \left (-i a x -1\right ) a^{2} x^{2}+12 i a x \ln \left (-i a x +1\right )^{2}-24 a^{2} x^{2} \ln \left (-i a x +1\right )+6 i a x +2 \ln \left (-i a x +1\right )^{3}+3 \ln \left (i a x -1\right )-3 \ln \left (-i a x -1\right )\right )}{384 \left (a x +i\right )^{2} c^{3} \left (a x -i\right )^{2} a^{3}}\)	\(444\)

Input:

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

Output:

1/192*(8*arctan(a*x)^3*a^4*x^4-3*x^4*arctan(a*x)*a^4+24*a^3*arctan(a*x)^2* 
x^3+16*arctan(a*x)^3*a^2*x^2-3*a^3*x^3+18*x^2*a^2*arctan(a*x)-24*a*arctan( 
a*x)^2*x+8*arctan(a*x)^3+3*a*x-3*arctan(a*x))/c^3/(a^2*x^2+1)^2/a^3
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.63 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=-\frac {3 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 24 \, {\left (a^{3} x^{3} - a x\right )} \arctan \left (a x\right )^{2} - 3 \, a x + 3 \, {\left (a^{4} x^{4} - 6 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )}{192 \, {\left (a^{7} c^{3} x^{4} + 2 \, a^{5} c^{3} x^{2} + a^{3} c^{3}\right )}} \] Input:

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

Output:

-1/192*(3*a^3*x^3 - 8*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x)^3 - 24*(a^3*x^ 
3 - a*x)*arctan(a*x)^2 - 3*a*x + 3*(a^4*x^4 - 6*a^2*x^2 + 1)*arctan(a*x))/ 
(a^7*c^3*x^4 + 2*a^5*c^3*x^2 + a^3*c^3)
 

Sympy [F]

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

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

Output:

Integral(x**2*atan(a*x)**2/(a**6*x**6 + 3*a**4*x**4 + 3*a**2*x**2 + 1), x) 
/c**3
 

Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.28 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {1}{8} \, {\left (\frac {a^{2} x^{3} - x}{a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}} + \frac {\arctan \left (a x\right )}{a^{3} c^{3}}\right )} \arctan \left (a x\right )^{2} - \frac {{\left (3 \, a^{3} x^{3} - 8 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x + 3 \, {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{192 \, {\left (a^{9} c^{3} x^{4} + 2 \, a^{7} c^{3} x^{2} + a^{5} c^{3}\right )}} + \frac {{\left (a^{2} x^{2} - {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2}\right )} a \arctan \left (a x\right )}{8 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )}} \] Input:

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

Output:

1/8*((a^2*x^3 - x)/(a^6*c^3*x^4 + 2*a^4*c^3*x^2 + a^2*c^3) + arctan(a*x)/( 
a^3*c^3))*arctan(a*x)^2 - 1/192*(3*a^3*x^3 - 8*(a^4*x^4 + 2*a^2*x^2 + 1)*a 
rctan(a*x)^3 - 3*a*x + 3*(a^4*x^4 + 2*a^2*x^2 + 1)*arctan(a*x))*a^2/(a^9*c 
^3*x^4 + 2*a^7*c^3*x^2 + a^5*c^3) + 1/8*(a^2*x^2 - (a^4*x^4 + 2*a^2*x^2 + 
1)*arctan(a*x)^2)*a*arctan(a*x)/(a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3)
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.75 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.83 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {\frac {x}{8\,a^2}-\frac {x^3}{8}}{8\,a^4\,c^3\,x^4+16\,a^2\,c^3\,x^2+8\,c^3}-\frac {{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {x}{8\,a^4\,c^3}-\frac {x^3}{8\,a^2\,c^3}\right )}{\frac {1}{a^2}+2\,x^2+a^2\,x^4}-\frac {\mathrm {atan}\left (a\,x\right )}{64\,a^3\,c^3}+\frac {{\mathrm {atan}\left (a\,x\right )}^3}{24\,a^3\,c^3}+\frac {x^2\,\mathrm {atan}\left (a\,x\right )}{8\,a^3\,c^3\,\left (\frac {1}{a^2}+2\,x^2+a^2\,x^4\right )} \] Input:

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

Output:

(x/(8*a^2) - x^3/8)/(8*c^3 + 16*a^2*c^3*x^2 + 8*a^4*c^3*x^4) - (atan(a*x)^ 
2*(x/(8*a^4*c^3) - x^3/(8*a^2*c^3)))/(1/a^2 + 2*x^2 + a^2*x^4) - atan(a*x) 
/(64*a^3*c^3) + atan(a*x)^3/(24*a^3*c^3) + (x^2*atan(a*x))/(8*a^3*c^3*(1/a 
^2 + 2*x^2 + a^2*x^4))
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.72 \[ \int \frac {x^2 \arctan (a x)^2}{\left (c+a^2 c x^2\right )^3} \, dx=\frac {8 \mathit {atan} \left (a x \right )^{3} a^{4} x^{4}+16 \mathit {atan} \left (a x \right )^{3} a^{2} x^{2}+8 \mathit {atan} \left (a x \right )^{3}+24 \mathit {atan} \left (a x \right )^{2} a^{3} x^{3}-24 \mathit {atan} \left (a x \right )^{2} a x -3 \mathit {atan} \left (a x \right ) a^{4} x^{4}+18 \mathit {atan} \left (a x \right ) a^{2} x^{2}-3 \mathit {atan} \left (a x \right )-3 a^{3} x^{3}+3 a x}{192 a^{3} c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )} \] Input:

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

Output:

(8*atan(a*x)**3*a**4*x**4 + 16*atan(a*x)**3*a**2*x**2 + 8*atan(a*x)**3 + 2 
4*atan(a*x)**2*a**3*x**3 - 24*atan(a*x)**2*a*x - 3*atan(a*x)*a**4*x**4 + 1 
8*atan(a*x)*a**2*x**2 - 3*atan(a*x) - 3*a**3*x**3 + 3*a*x)/(192*a**3*c**3* 
(a**4*x**4 + 2*a**2*x**2 + 1))