\(\int \frac {x}{(c+a^2 c x^2)^{5/2} \arctan (a x)^3} \, dx\) [669]

Optimal result

Integrand size = 22, antiderivative size = 175 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=-\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}-\frac {3}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}+\frac {1}{a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \text {Si}(\arctan (a x))}{8 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {9 \sqrt {1+a^2 x^2} \text {Si}(3 \arctan (a x))}{8 a^2 c^2 \sqrt {c+a^2 c x^2}} \] Output:

-1/2*x/a/c/(a^2*c*x^2+c)^(3/2)/arctan(a*x)^2-3/2/a^2/c/(a^2*c*x^2+c)^(3/2) 
/arctan(a*x)+1/a^2/c^2/(a^2*c*x^2+c)^(1/2)/arctan(a*x)-1/8*(a^2*x^2+1)^(1/ 
2)*Si(arctan(a*x))/a^2/c^2/(a^2*c*x^2+c)^(1/2)-9/8*(a^2*x^2+1)^(1/2)*Si(3* 
arctan(a*x))/a^2/c^2/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.67 \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\frac {-4 a x-4 \arctan (a x)+8 a^2 x^2 \arctan (a x)-\left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2 \text {Si}(\arctan (a x))-9 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2 \text {Si}(3 \arctan (a x))}{8 a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \arctan (a x)^2} \] Input:

Integrate[x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]
 

Output:

(-4*a*x - 4*ArcTan[a*x] + 8*a^2*x^2*ArcTan[a*x] - (1 + a^2*x^2)^(3/2)*ArcT 
an[a*x]^2*SinIntegral[ArcTan[a*x]] - 9*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*S 
inIntegral[3*ArcTan[a*x]])/(8*a^2*c^2*(1 + a^2*x^2)*Sqrt[c + a^2*c*x^2]*Ar 
cTan[a*x]^2)
 

Rubi [A] (verified)

Time = 2.26 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.68, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {5503, 5437, 5499, 5437, 5506, 5505, 3042, 3780, 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.

\(\displaystyle \int \frac {x}{\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5503

\(\displaystyle \frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx}{2 a}-a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5437

\(\displaystyle -a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx+\frac {-3 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5499

\(\displaystyle -a \left (\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}dx}{a^2 c}-\frac {\int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)^2}dx}{a^2}\right )+\frac {-3 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{2 a}-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5437

\(\displaystyle \frac {-3 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{2 a}-a \left (\frac {-a \int \frac {x}{\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-3 a \int \frac {x}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5506

\(\displaystyle \frac {-\frac {3 a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{5/2} \arctan (a x)}dx}{c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{2 a}-a \left (\frac {-\frac {a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{3/2} \arctan (a x)}dx}{c \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \sqrt {a^2 c x^2+c}}}{a^2 c}-\frac {-\frac {3 a \sqrt {a^2 x^2+1} \int \frac {x}{\left (a^2 x^2+1\right )^{5/2} \arctan (a x)}dx}{c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}}{a^2}\right )-\frac {x}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5505

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3780

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

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

Input:

Int[x/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]
 

Output:

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

Defintions of rubi rules used

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 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])
 

rule 5506

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

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 6.75 (sec) , antiderivative size = 371, normalized size of antiderivative = 2.12

method	result	size
default	\(\frac {i \left (\arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (i \arctan \left (a x \right )\right ) a^{4} x^{4}-\arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (-i \arctan \left (a x \right )\right ) a^{4} x^{4}+9 \arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{4} x^{4}-9 \arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{4} x^{4}+2 \arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (i \arctan \left (a x \right )\right ) a^{2} x^{2}-2 \arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (-i \arctan \left (a x \right )\right ) a^{2} x^{2}+18 \arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{2} x^{2}-18 \arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{2} x^{2}-16 i \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+\operatorname {expIntegral}_{1}\left (i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-\operatorname {expIntegral}_{1}\left (-i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+9 \,\operatorname {expIntegral}_{1}\left (3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-9 \,\operatorname {expIntegral}_{1}\left (-3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+8 i \sqrt {a^{2} x^{2}+1}\, a x +8 i \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{16 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )^{2} a^{2} c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\)	\(371\)

Input:

int(x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x,method=_RETURNVERBOSE)
 

Output:

1/16*I*(arctan(a*x)^2*Ei(1,I*arctan(a*x))*a^4*x^4-arctan(a*x)^2*Ei(1,-I*ar 
ctan(a*x))*a^4*x^4+9*arctan(a*x)^2*Ei(1,3*I*arctan(a*x))*a^4*x^4-9*arctan( 
a*x)^2*Ei(1,-3*I*arctan(a*x))*a^4*x^4+2*arctan(a*x)^2*Ei(1,I*arctan(a*x))* 
a^2*x^2-2*arctan(a*x)^2*Ei(1,-I*arctan(a*x))*a^2*x^2+18*arctan(a*x)^2*Ei(1 
,3*I*arctan(a*x))*a^2*x^2-18*arctan(a*x)^2*Ei(1,-3*I*arctan(a*x))*a^2*x^2- 
16*I*arctan(a*x)*(a^2*x^2+1)^(1/2)*a^2*x^2+Ei(1,I*arctan(a*x))*arctan(a*x) 
^2-Ei(1,-I*arctan(a*x))*arctan(a*x)^2+9*Ei(1,3*I*arctan(a*x))*arctan(a*x)^ 
2-9*Ei(1,-3*I*arctan(a*x))*arctan(a*x)^2+8*I*(a^2*x^2+1)^(1/2)*a*x+8*I*arc 
tan(a*x)*(a^2*x^2+1)^(1/2))*(c*(a*x-I)*(a*x+I))^(1/2)/(a^2*x^2+1)^(1/2)/ar 
ctan(a*x)^2/a^2/c^3/(a^4*x^4+2*a^2*x^2+1)
 

Fricas [F]

\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \] Input:

integrate(x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="fricas")
 

Output:

integral(sqrt(a^2*c*x^2 + c)*x/((a^6*c^3*x^6 + 3*a^4*c^3*x^4 + 3*a^2*c^3*x 
^2 + c^3)*arctan(a*x)^3), x)
 

Sympy [F]

\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {x}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \] Input:

integrate(x/(a**2*c*x**2+c)**(5/2)/atan(a*x)**3,x)
 

Output:

Integral(x/((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**3), x)
 

Maxima [F]

\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \] Input:

integrate(x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="maxima")
 

Output:

integrate(x/((a^2*c*x^2 + c)^(5/2)*arctan(a*x)^3), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \] Input:

int(x/(atan(a*x)^3*(c + a^2*c*x^2)^(5/2)),x)
 

Output:

int(x/(atan(a*x)^3*(c + a^2*c*x^2)^(5/2)), x)
 

Reduce [F]

\[ \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\frac {\int \frac {x}{\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} a^{4} x^{4}+2 \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} a^{2} x^{2}+\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3}}d x}{\sqrt {c}\, c^{2}} \] Input:

int(x/(a^2*c*x^2+c)^(5/2)/atan(a*x)^3,x)
 

Output:

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