\(\int \frac {x^3}{(c+a^2 c x^2)^3 \arctan (a x)^3} \, dx\) [634]

Optimal result

Integrand size = 22, antiderivative size = 177 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)^2}-\frac {x}{2 a^3 c^3 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {2}{a^4 c^3 \left (1+a^2 x^2\right )^2 \arctan (a x)}-\frac {3}{2 a^4 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {1-a^2 x^2}{2 a^4 c^3 \left (1+a^2 x^2\right ) \arctan (a x)}-\frac {\text {Si}(2 \arctan (a x))}{2 a^4 c^3}+\frac {\text {Si}(4 \arctan (a x))}{a^4 c^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.41 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=\frac {\frac {a^2 x^2 \left (-a x+\left (-3+a^2 x^2\right ) \arctan (a x)\right )}{\left (1+a^2 x^2\right )^2 \arctan (a x)^2}-\text {Si}(2 \arctan (a x))+2 \text {Si}(4 \arctan (a x))}{2 a^4 c^3} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 2.05 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.54, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {5499, 27, 5467, 5503, 5437, 5499, 5437, 5505, 4906, 27, 2009, 3042, 3780}

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.

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

\(\Big \downarrow \) 5499

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5467

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

\(\Big \downarrow \) 5503

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

\(\Big \downarrow \) 5437

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

\(\Big \downarrow \) 5499

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

\(\Big \downarrow \) 5437

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

\(\Big \downarrow \) 5505

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

Input:

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

Output:

(-1/2*x/(a*(1 + a^2*x^2)*ArcTan[a*x]^2) - (1 - a^2*x^2)/(2*a^2*(1 + a^2*x^ 
2)*ArcTan[a*x]) - SinIntegral[2*ArcTan[a*x]]/a^2)/(a^2*c^3) - (-1/2*x/(a*( 
1 + a^2*x^2)^2*ArcTan[a*x]^2) - (3*a*((-(1/(a*(1 + a^2*x^2)*ArcTan[a*x])) 
- SinIntegral[2*ArcTan[a*x]]/a)/a^2 - (-(1/(a*(1 + a^2*x^2)^2*ArcTan[a*x]) 
) - (4*(SinIntegral[2*ArcTan[a*x]]/4 + SinIntegral[4*ArcTan[a*x]]/8))/a)/a 
^2))/2 + (-(1/(a*(1 + a^2*x^2)^2*ArcTan[a*x])) - (4*(SinIntegral[2*ArcTan[ 
a*x]]/4 + SinIntegral[4*ArcTan[a*x]]/8))/a)/(2*a))/(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 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

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 5437

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

rule 5467

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

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]
 

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 = 8.79 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.51

method	result	size
derivativedivides	\(-\frac {8 \,\operatorname {Si}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-16 \,\operatorname {Si}\left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+4 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-4 \arctan \left (a x \right ) \cos \left (4 \arctan \left (a x \right )\right )+2 \sin \left (2 \arctan \left (a x \right )\right )-\sin \left (4 \arctan \left (a x \right )\right )}{16 a^{4} c^{3} \arctan \left (a x \right )^{2}}\)	\(90\)
default	\(-\frac {8 \,\operatorname {Si}\left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-16 \,\operatorname {Si}\left (4 \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+4 \cos \left (2 \arctan \left (a x \right )\right ) \arctan \left (a x \right )-4 \arctan \left (a x \right ) \cos \left (4 \arctan \left (a x \right )\right )+2 \sin \left (2 \arctan \left (a x \right )\right )-\sin \left (4 \arctan \left (a x \right )\right )}{16 a^{4} c^{3} \arctan \left (a x \right )^{2}}\)	\(90\)

Input:

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

Output:

-1/16/a^4/c^3*(8*Si(2*arctan(a*x))*arctan(a*x)^2-16*Si(4*arctan(a*x))*arct 
an(a*x)^2+4*cos(2*arctan(a*x))*arctan(a*x)-4*arctan(a*x)*cos(4*arctan(a*x) 
)+2*sin(2*arctan(a*x))-sin(4*arctan(a*x)))/arctan(a*x)^2
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.85 \[ \int \frac {x^3}{\left (c+a^2 c x^2\right )^3 \arctan (a x)^3} \, dx=-\frac {2 \, a^{3} x^{3} + 2 \, {\left (-i \, a^{4} x^{4} - 2 i \, a^{2} x^{2} - i\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (\frac {a^{4} x^{4} + 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} - 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) + 2 \, {\left (i \, a^{4} x^{4} + 2 i \, a^{2} x^{2} + i\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (\frac {a^{4} x^{4} - 4 i \, a^{3} x^{3} - 6 \, a^{2} x^{2} + 4 i \, a x + 1}{a^{4} x^{4} + 2 \, a^{2} x^{2} + 1}\right ) - {\left (-i \, a^{4} x^{4} - 2 i \, a^{2} x^{2} - i\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} + 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - {\left (i \, a^{4} x^{4} + 2 i \, a^{2} x^{2} + i\right )} \arctan \left (a x\right )^{2} \operatorname {log\_integral}\left (-\frac {a^{2} x^{2} - 2 i \, a x - 1}{a^{2} x^{2} + 1}\right ) - 2 \, {\left (a^{4} x^{4} - 3 \, a^{2} x^{2}\right )} \arctan \left (a x\right )}{4 \, {\left (a^{8} c^{3} x^{4} + 2 \, a^{6} c^{3} x^{2} + a^{4} c^{3}\right )} \arctan \left (a x\right )^{2}} \] Input:

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

Output:

-1/4*(2*a^3*x^3 + 2*(-I*a^4*x^4 - 2*I*a^2*x^2 - I)*arctan(a*x)^2*log_integ 
ral((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)) + 2*(I*a^4*x^4 + 2*I*a^2*x^2 + I)*arctan(a*x)^2*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)) - ( 
-I*a^4*x^4 - 2*I*a^2*x^2 - I)*arctan(a*x)^2*log_integral(-(a^2*x^2 + 2*I*a 
*x - 1)/(a^2*x^2 + 1)) - (I*a^4*x^4 + 2*I*a^2*x^2 + I)*arctan(a*x)^2*log_i 
ntegral(-(a^2*x^2 - 2*I*a*x - 1)/(a^2*x^2 + 1)) - 2*(a^4*x^4 - 3*a^2*x^2)* 
arctan(a*x))/((a^8*c^3*x^4 + 2*a^6*c^3*x^2 + a^4*c^3)*arctan(a*x)^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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