\(\int \frac {\arctan (a x)^2}{(c+a^2 c x^2)^2} \, dx\) [294]

Optimal result

Integrand size = 19, antiderivative size = 100 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {x}{4 c^2 \left (1+a^2 x^2\right )}-\frac {\arctan (a x)}{4 a c^2}+\frac {\arctan (a x)}{2 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^2}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^3}{6 a c^2} \] Output:

-1/4*x/c^2/(a^2*x^2+1)-1/4*arctan(a*x)/a/c^2+1/2*arctan(a*x)/a/c^2/(a^2*x^ 
2+1)+1/2*x*arctan(a*x)^2/c^2/(a^2*x^2+1)+1/6*arctan(a*x)^3/a/c^2
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.65 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {-3 a x+\left (3-3 a^2 x^2\right ) \arctan (a x)+6 a x \arctan (a x)^2+2 \left (1+a^2 x^2\right ) \arctan (a x)^3}{12 c^2 \left (a+a^3 x^2\right )} \] Input:

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

Output:

(-3*a*x + (3 - 3*a^2*x^2)*ArcTan[a*x] + 6*a*x*ArcTan[a*x]^2 + 2*(1 + a^2*x 
^2)*ArcTan[a*x]^3)/(12*c^2*(a + a^3*x^2))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.06, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {5427, 27, 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.

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

\(\Big \downarrow \) 5427

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 215

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

\(\Big \downarrow \) 216

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

Input:

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

Output:

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

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 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]
 

Maple [A] (verified)

Time = 0.69 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.75

method	result	size
parallelrisch	\(\frac {2 \arctan \left (a x \right )^{3} a^{2} x^{2}-3 x^{2} a^{2} \arctan \left (a x \right )+6 a \arctan \left (a x \right )^{2} x +2 \arctan \left (a x \right )^{3}-3 a x +3 \arctan \left (a x \right )}{12 c^{2} \left (a^{2} x^{2}+1\right ) a}\)	\(75\)
derivativedivides	\(\frac {\frac {a x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {a x}{4 a^{2} x^{2}+4}+\frac {\arctan \left (a x \right )}{4}}{c^{2}}}{a}\)	\(93\)
default	\(\frac {\frac {a x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{3}}{2 c^{2}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3}-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {a x}{4 a^{2} x^{2}+4}+\frac {\arctan \left (a x \right )}{4}}{c^{2}}}{a}\)	\(93\)
parts	\(\frac {x \arctan \left (a x \right )^{2}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{3}}{2 a \,c^{2}}-\frac {\frac {\arctan \left (a x \right )^{3}}{3 a}+\frac {-\frac {\arctan \left (a x \right )}{2 \left (a^{2} x^{2}+1\right )}+\frac {a x}{4 a^{2} x^{2}+4}+\frac {\arctan \left (a x \right )}{4}}{a}}{c^{2}}\)	\(99\)
risch	\(\frac {i \ln \left (i a x +1\right )^{3}}{48 a \,c^{2}}-\frac {i \left (a^{2} x^{2} \ln \left (-i a x +1\right )+\ln \left (-i a x +1\right )-2 i a x \right ) \ln \left (i a x +1\right )^{2}}{16 c^{2} \left (a^{2} x^{2}+1\right ) a}+\frac {i \left (a^{2} x^{2} \ln \left (-i a x +1\right )^{2}+\ln \left (-i a x +1\right )^{2}-4 i a x \ln \left (-i a x +1\right )-4\right ) \ln \left (i a x +1\right )}{16 c^{2} \left (a x +i\right ) \left (a x -i\right ) a}-\frac {i \left (a^{2} x^{2} \ln \left (-i a x +1\right )^{3}-6 i a x \ln \left (-i a x +1\right )^{2}+6 a^{2} x^{2} \ln \left (-i a x +1\right )-6 \ln \left (i a x +1\right ) a^{2} x^{2}-12 i a x +\ln \left (-i a x +1\right )^{3}-6 \ln \left (-i a x +1\right )-6 \ln \left (i a x +1\right )\right )}{48 c^{2} \left (a x +i\right ) \left (a x -i\right ) a}\)	\(280\)

Input:

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

Output:

1/12*(2*arctan(a*x)^3*a^2*x^2-3*x^2*a^2*arctan(a*x)+6*a*arctan(a*x)^2*x+2* 
arctan(a*x)^3-3*a*x+3*arctan(a*x))/c^2/(a^2*x^2+1)/a
 

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.67 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {6 \, a x \arctan \left (a x\right )^{2} + 2 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x - 3 \, {\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )}{12 \, {\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \] Input:

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

Output:

1/12*(6*a*x*arctan(a*x)^2 + 2*(a^2*x^2 + 1)*arctan(a*x)^3 - 3*a*x - 3*(a^2 
*x^2 - 1)*arctan(a*x))/(a^3*c^2*x^2 + a*c^2)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.46 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {1}{2} \, {\left (\frac {x}{a^{2} c^{2} x^{2} + c^{2}} + \frac {\arctan \left (a x\right )}{a c^{2}}\right )} \arctan \left (a x\right )^{2} + \frac {{\left (2 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x - 3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a^{2}}{12 \, {\left (a^{5} c^{2} x^{2} + a^{3} c^{2}\right )}} - \frac {{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 1\right )} a \arctan \left (a x\right )}{2 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )}} \] Input:

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

Output:

1/2*(x/(a^2*c^2*x^2 + c^2) + arctan(a*x)/(a*c^2))*arctan(a*x)^2 + 1/12*(2* 
(a^2*x^2 + 1)*arctan(a*x)^3 - 3*a*x - 3*(a^2*x^2 + 1)*arctan(a*x))*a^2/(a^ 
5*c^2*x^2 + a^3*c^2) - 1/2*((a^2*x^2 + 1)*arctan(a*x)^2 - 1)*a*arctan(a*x) 
/(a^4*c^2*x^2 + a^2*c^2)
 

Giac [F]

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

integrate(arctan(a*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)
 

Mupad [B] (verification not implemented)

Time = 0.79 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.01 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\mathrm {atan}\left (a\,x\right )}{2\,\left (a^3\,c^2\,x^2+a\,c^2\right )}-\frac {x}{2\,\left (2\,a^2\,c^2\,x^2+2\,c^2\right )}+\frac {x\,{\mathrm {atan}\left (a\,x\right )}^2}{2\,\left (a^2\,c^2\,x^2+c^2\right )}-\frac {\mathrm {atan}\left (a\,x\right )}{4\,a\,c^2}+\frac {{\mathrm {atan}\left (a\,x\right )}^3}{6\,a\,c^2} \] Input:

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

Output:

atan(a*x)/(2*(a*c^2 + a^3*c^2*x^2)) - x/(2*(2*c^2 + 2*a^2*c^2*x^2)) + (x*a 
tan(a*x)^2)/(2*(c^2 + a^2*c^2*x^2)) - atan(a*x)/(4*a*c^2) + atan(a*x)^3/(6 
*a*c^2)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.74 \[ \int \frac {\arctan (a x)^2}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {2 \mathit {atan} \left (a x \right )^{3} a^{2} x^{2}+2 \mathit {atan} \left (a x \right )^{3}+6 \mathit {atan} \left (a x \right )^{2} a x -3 \mathit {atan} \left (a x \right ) a^{2} x^{2}+3 \mathit {atan} \left (a x \right )-3 a x}{12 a \,c^{2} \left (a^{2} x^{2}+1\right )} \] Input:

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

Output:

(2*atan(a*x)**3*a**2*x**2 + 2*atan(a*x)**3 + 6*atan(a*x)**2*a*x - 3*atan(a 
*x)*a**2*x**2 + 3*atan(a*x) - 3*a*x)/(12*a*c**2*(a**2*x**2 + 1))