\(\int \frac {\arctan (a x)^3}{(c+a^2 c x^2)^2} \, dx\) [399]

Optimal result

Integrand size = 19, antiderivative size = 129 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=-\frac {3}{8 a c^2 \left (1+a^2 x^2\right )}-\frac {3 x \arctan (a x)}{4 c^2 \left (1+a^2 x^2\right )}-\frac {3 \arctan (a x)^2}{8 a c^2}+\frac {3 \arctan (a x)^2}{4 a c^2 \left (1+a^2 x^2\right )}+\frac {x \arctan (a x)^3}{2 c^2 \left (1+a^2 x^2\right )}+\frac {\arctan (a x)^4}{8 a c^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.55 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {-3-6 a x \arctan (a x)+\left (3-3 a^2 x^2\right ) \arctan (a x)^2+4 a x \arctan (a x)^3+\left (1+a^2 x^2\right ) \arctan (a x)^4}{8 c^2 \left (a+a^3 x^2\right )} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.02, 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, 5427, 241}

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.

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

\(\Big \downarrow \) 5427

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 5427

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

\(\Big \downarrow \) 241

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

Input:

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

Output:

(x*ArcTan[a*x]^3)/(2*c^2*(1 + a^2*x^2)) + ArcTan[a*x]^4/(8*a*c^2) - (3*a*( 
-1/2*ArcTan[a*x]^2/(a^2*(1 + a^2*x^2)) + (1/(4*a*(1 + a^2*x^2)) + (x*ArcTa 
n[a*x])/(2*(1 + a^2*x^2)) + ArcTan[a*x]^2/(4*a))/a))/(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 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 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 = 1.68 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.68

method	result	size
parallelrisch	\(\frac {\arctan \left (a x \right )^{4} a^{2} x^{2}-3 \arctan \left (a x \right )^{2} x^{2} a^{2}+4 x \arctan \left (a x \right )^{3} a +3 a^{2} x^{2}+\arctan \left (a x \right )^{4}-6 \arctan \left (a x \right ) a x +3 \arctan \left (a x \right )^{2}}{8 c^{2} \left (a^{2} x^{2}+1\right ) a}\)	\(88\)
derivativedivides	\(\frac {\frac {\arctan \left (a x \right )^{3} a x}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{4}}{2 c^{2}}-\frac {3 \left (\frac {\arctan \left (a x \right )^{4}}{4}-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right ) a x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )^{2}}{4}+\frac {1}{4 a^{2} x^{2}+4}\right )}{2 c^{2}}}{a}\)	\(114\)
default	\(\frac {\frac {\arctan \left (a x \right )^{3} a x}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{4}}{2 c^{2}}-\frac {3 \left (\frac {\arctan \left (a x \right )^{4}}{4}-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right ) a x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )^{2}}{4}+\frac {1}{4 a^{2} x^{2}+4}\right )}{2 c^{2}}}{a}\)	\(114\)
parts	\(\frac {x \arctan \left (a x \right )^{3}}{2 c^{2} \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right )^{4}}{2 a \,c^{2}}-\frac {3 \left (\frac {\arctan \left (a x \right )^{4}}{4 a}+\frac {-\frac {\arctan \left (a x \right )^{2}}{2 \left (a^{2} x^{2}+1\right )}+\frac {\arctan \left (a x \right ) a x}{2 a^{2} x^{2}+2}+\frac {\arctan \left (a x \right )^{2}}{4}+\frac {1}{4 a^{2} x^{2}+4}}{a}\right )}{2 c^{2}}\)	\(120\)
risch	\(\frac {\ln \left (i a x +1\right )^{4}}{128 a \,c^{2}}-\frac {\left (a^{2} x^{2} \ln \left (-i a x +1\right )+\ln \left (-i a x +1\right )-2 i a x \right ) \ln \left (i a x +1\right )^{3}}{32 c^{2} \left (a^{2} x^{2}+1\right ) a}+\frac {3 \left (a^{2} x^{2} \ln \left (-i a x +1\right )^{2}+2 a^{2} x^{2}+\ln \left (-i a x +1\right )^{2}-4 i a x \ln \left (-i a x +1\right )-2\right ) \ln \left (i a x +1\right )^{2}}{64 c^{2} \left (a x +i\right ) \left (a x -i\right ) a}-\frac {\left (a^{2} x^{2} \ln \left (-i a x +1\right )^{3}+6 a^{2} x^{2} \ln \left (-i a x +1\right )+\ln \left (-i a x +1\right )^{3}-6 i a x \ln \left (-i a x +1\right )^{2}-6 \ln \left (-i a x +1\right )-12 i a x \right ) \ln \left (i a x +1\right )}{32 c^{2} \left (a x +i\right ) \left (a x -i\right ) a}+\frac {a^{2} x^{2} \ln \left (-i a x +1\right )^{4}+12 a^{2} x^{2} \ln \left (-i a x +1\right )^{2}+\ln \left (-i a x +1\right )^{4}-8 i a x \ln \left (-i a x +1\right )^{3}-12 \ln \left (-i a x +1\right )^{2}-48 i a x \ln \left (-i a x +1\right )-48}{128 c^{2} \left (a x +i\right ) \left (a x -i\right ) a}\)	\(379\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.57 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {4 \, a x \arctan \left (a x\right )^{3} + {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} - 6 \, a x \arctan \left (a x\right ) - 3 \, {\left (a^{2} x^{2} - 1\right )} \arctan \left (a x\right )^{2} - 3}{8 \, {\left (a^{3} c^{2} x^{2} + a c^{2}\right )}} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.65 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {1}{2} \, {\left (\frac {x}{a^{2} c^{2} x^{2} + c^{2}} + \frac {\arctan \left (a x\right )}{a c^{2}}\right )} \arctan \left (a x\right )^{3} - \frac {3 \, {\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} - 1\right )} a \arctan \left (a x\right )^{2}}{4 \, {\left (a^{4} c^{2} x^{2} + a^{2} c^{2}\right )}} - \frac {1}{8} \, {\left (\frac {{\left ({\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{4} - 3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{2} + 3\right )} a^{2}}{a^{6} c^{2} x^{2} + a^{4} c^{2}} - \frac {2 \, {\left (2 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )^{3} - 3 \, a x - 3 \, {\left (a^{2} x^{2} + 1\right )} \arctan \left (a x\right )\right )} a \arctan \left (a x\right )}{a^{5} c^{2} x^{2} + a^{3} c^{2}}\right )} a \] Input:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.92 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {{\mathrm {atan}\left (a\,x\right )}^4}{8\,a\,c^2}-{\mathrm {atan}\left (a\,x\right )}^2\,\left (\frac {3}{8\,a\,c^2}-\frac {3}{4\,a^3\,c^2\,\left (\frac {1}{a^2}+x^2\right )}\right )-\frac {3}{2\,a\,\left (4\,a^2\,c^2\,x^2+4\,c^2\right )}-\frac {3\,x\,\mathrm {atan}\left (a\,x\right )}{4\,a^2\,c^2\,\left (\frac {1}{a^2}+x^2\right )}+\frac {x\,{\mathrm {atan}\left (a\,x\right )}^3}{2\,a^2\,c^2\,\left (\frac {1}{a^2}+x^2\right )} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.67 \[ \int \frac {\arctan (a x)^3}{\left (c+a^2 c x^2\right )^2} \, dx=\frac {\mathit {atan} \left (a x \right )^{4} a^{2} x^{2}+\mathit {atan} \left (a x \right )^{4}+4 \mathit {atan} \left (a x \right )^{3} a x -3 \mathit {atan} \left (a x \right )^{2} a^{2} x^{2}+3 \mathit {atan} \left (a x \right )^{2}-6 \mathit {atan} \left (a x \right ) a x +3 a^{2} x^{2}}{8 a \,c^{2} \left (a^{2} x^{2}+1\right )} \] Input:

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

Output:

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