\(\int \frac {x}{(c+a^2 c x^2)^3 \arctan (a x)^2} \, dx\) [560]

Optimal result

Integrand size = 20, antiderivative size = 61 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=-\frac {x}{a c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}+\frac {\operatorname {CosIntegral}(2 \arctan (a x))}{2 a^2 c^3}+\frac {\operatorname {CosIntegral}(4 \arctan (a x))}{2 a^2 c^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.23 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\frac {-2 a x+\left (1+a^2 x^2\right )^2 \arctan (a x) \operatorname {CosIntegral}(2 \arctan (a x))+\left (1+a^2 x^2\right )^2 \arctan (a x) \operatorname {CosIntegral}(4 \arctan (a x))}{2 c^3 \left (a+a^3 x^2\right )^2 \arctan (a x)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.87 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.56, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.450, Rules used = {5503, 27, 5439, 3042, 3793, 2009, 5505, 4906, 2009}

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.

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

\(\Big \downarrow \) 5503

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5439

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 5505

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

\(\Big \downarrow \) 4906

\(\displaystyle -\frac {3 \int \left (\frac {1}{8 \arctan (a x)}-\frac {\cos (4 \arctan (a x))}{8 \arctan (a x)}\right )d\arctan (a x)}{a^2 c^3}+\frac {\frac {1}{2} \operatorname {CosIntegral}(2 \arctan (a x))+\frac {1}{8} \operatorname {CosIntegral}(4 \arctan (a x))+\frac {3}{8} \log (\arctan (a x))}{a^2 c^3}-\frac {x}{a c^3 \left (a^2 x^2+1\right )^2 \arctan (a x)}\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-(x/(a*c^3*(1 + a^2*x^2)^2*ArcTan[a*x])) - (3*(-1/8*CosIntegral[4*ArcTan[a 
*x]] + Log[ArcTan[a*x]]/8))/(a^2*c^3) + (CosIntegral[2*ArcTan[a*x]]/2 + Co 
sIntegral[4*ArcTan[a*x]]/8 + (3*Log[ArcTan[a*x]])/8)/(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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f 
, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
 

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

rule 5439

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ 
Symbol] :> Simp[d^q/c   Subst[Int[(a + b*x)^p/Cos[x]^(2*(q + 1)), x], x, Ar 
cTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && ILtQ[2*( 
q + 1), 0] && (IntegerQ[q] || GtQ[d, 0])
 

rule 5503

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[x^m*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p 
+ 1)/(b*c*d*(p + 1))), x] + (-Simp[c*((m + 2*q + 2)/(b*(p + 1)))   Int[x^(m 
 + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x] - Simp[m/(b*c*(p + 
1))   Int[x^(m - 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p + 1), x], x]) /; F 
reeQ[{a, b, c, d, e, m}, x] && EqQ[e, c^2*d] && IntegerQ[m] && LtQ[q, -1] & 
& LtQ[p, -1] && NeQ[m + 2*q + 2, 0]
 

rule 5505

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1)   Subst[Int[(a + b*x)^p*(Sin[x]^m/ 
Cos[x]^(m + 2*(q + 1))), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p 
}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q 
] || GtQ[d, 0])
 

Maple [A] (verified)

Time = 9.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.98

method	result	size
derivativedivides	\(\frac {4 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+4 \,\operatorname {Ci}\left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-2 \sin \left (2 \arctan \left (a x \right )\right )-\sin \left (4 \arctan \left (a x \right )\right )}{8 a^{2} c^{3} \arctan \left (a x \right )}\)	\(60\)
default	\(\frac {4 \,\operatorname {Ci}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )+4 \,\operatorname {Ci}\left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-2 \sin \left (2 \arctan \left (a x \right )\right )-\sin \left (4 \arctan \left (a x \right )\right )}{8 a^{2} c^{3} \arctan \left (a x \right )}\)	\(60\)

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 286, normalized size of antiderivative = 4.69 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^2} \, dx=\frac {{\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (\frac {a^{4} x^{4} + 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) + {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (\frac {a^{4} x^{4} - 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) + {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) + {\left (a^{4} x^{4} + 2 \, a^{2} x^{2} + 1\right )} \arctan \left (a x\right ) \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 4 \, a x}{4 \, {\left (a^{6} c^{3} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{2} c^{3}\right )} \arctan \left (a x\right )} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

-((a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c^3)*arctan(a*x)*integrate((3*a^2*x^2 - 
 1)/((a^7*c^3*x^6 + 3*a^5*c^3*x^4 + 3*a^3*c^3*x^2 + a*c^3)*arctan(a*x)), x 
) + x)/((a^5*c^3*x^4 + 2*a^3*c^3*x^2 + a*c^3)*arctan(a*x))
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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