\(\int \frac {1}{(c+a^2 c x^2)^{5/2} \arctan (a x)^3} \, dx\) [670]

Optimal result

Integrand size = 21, antiderivative size = 145 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=-\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {3 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{8 a c^2 \sqrt {c+a^2 c x^2}}-\frac {9 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{8 a c^2 \sqrt {c+a^2 c x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\frac {-4+12 a x \arctan (a x)-3 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2 \operatorname {CosIntegral}(\arctan (a x))-9 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2 \operatorname {CosIntegral}(3 \arctan (a x))}{8 c^2 \left (a+a^3 x^2\right ) \sqrt {c+a^2 c x^2} \arctan (a x)^2} \] Input:

Integrate[1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]
 

Output:

(-4 + 12*a*x*ArcTan[a*x] - 3*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*CosIntegral 
[ArcTan[a*x]] - 9*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^2*CosIntegral[3*ArcTan[a 
*x]])/(8*c^2*(a + a^3*x^2)*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)
 

Rubi [A] (verified)

Time = 1.83 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.22, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {5437, 5503, 5440, 5439, 3042, 3793, 2009, 5506, 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.

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

\(\Big \downarrow \) 5437

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

\(\Big \downarrow \) 5503

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

\(\Big \downarrow \) 5440

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

\(\Big \downarrow \) 5439

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {3}{2} a \left (\frac {\sqrt {a^2 x^2+1} \int \left (\frac {\cos (3 \arctan (a x))}{4 \arctan (a x)}+\frac {3}{4 \sqrt {a^2 x^2+1} \arctan (a x)}\right )d\arctan (a x)}{a^2 c^2 \sqrt {a^2 c x^2+c}}-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {3}{2} a \left (-2 a \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2} \arctan (a x)}dx+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5506

\(\displaystyle -\frac {3}{2} a \left (-\frac {2 a \sqrt {a^2 x^2+1} \int \frac {x^2}{\left (a^2 x^2+1\right )^{5/2} \arctan (a x)}dx}{c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{4} \operatorname {CosIntegral}(\arctan (a x))+\frac {1}{4} \operatorname {CosIntegral}(3 \arctan (a x))\right )}{a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {x}{a c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}\right )-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5505

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

\(\Big \downarrow \) 4906

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

\(\Big \downarrow \) 2009

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

Input:

Int[1/((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3),x]
 

Output:

-1/2*1/(a*c*(c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2) - (3*a*(-(x/(a*c*(c + a^2 
*c*x^2)^(3/2)*ArcTan[a*x])) - (2*Sqrt[1 + a^2*x^2]*(CosIntegral[ArcTan[a*x 
]]/4 - CosIntegral[3*ArcTan[a*x]]/4))/(a^2*c^2*Sqrt[c + a^2*c*x^2]) + (Sqr 
t[1 + a^2*x^2]*((3*CosIntegral[ArcTan[a*x]])/4 + CosIntegral[3*ArcTan[a*x] 
]/4))/(a^2*c^2*Sqrt[c + a^2*c*x^2])))/2
 

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 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 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 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 5440

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^2)^(q_), x_ 
Symbol] :> Simp[d^(q + 1/2)*(Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2])   Int[(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] && 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])
 

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 = 5.79 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.39

method	result	size
default	\(\frac {\left (3 \arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (i \arctan \left (a x \right )\right ) a^{4} x^{4}+3 \arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (-i \arctan \left (a x \right )\right ) a^{4} x^{4}+9 \arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{4} x^{4}+9 \arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{4} x^{4}+6 \arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (i \arctan \left (a x \right )\right ) a^{2} x^{2}+6 \arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (-i \arctan \left (a x \right )\right ) a^{2} x^{2}+18 \arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{2} x^{2}+18 \arctan \left (a x \right )^{2} \operatorname {expIntegral}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{2} x^{2}+24 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +3 \,\operatorname {expIntegral}_{1}\left (i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+3 \,\operatorname {expIntegral}_{1}\left (-i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+9 \,\operatorname {expIntegral}_{1}\left (3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+9 \,\operatorname {expIntegral}_{1}\left (-3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-8 \sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{16 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )^{2} a \,c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\)	\(346\)

Input:

int(1/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x,method=_RETURNVERBOSE)
                                                                                    
                                                                                    
 

Output:

1/16*(3*arctan(a*x)^2*Ei(1,I*arctan(a*x))*a^4*x^4+3*arctan(a*x)^2*Ei(1,-I* 
arctan(a*x))*a^4*x^4+9*arctan(a*x)^2*Ei(1,3*I*arctan(a*x))*a^4*x^4+9*arcta 
n(a*x)^2*Ei(1,-3*I*arctan(a*x))*a^4*x^4+6*arctan(a*x)^2*Ei(1,I*arctan(a*x) 
)*a^2*x^2+6*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+24*arctan(a*x)*(a^2*x^2+1)^(1/2)*a*x+3*Ei(1,I*arctan(a*x))*arctan(a*x)^2 
+3*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*(a^2*x^2+1)^(1/2))*(c*(a*x-I)*( 
a*x+I))^(1/2)/(a^2*x^2+1)^(1/2)/arctan(a*x)^2/a/c^3/(a^4*x^4+2*a^2*x^2+1)
 

Fricas [F]

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

integrate(1/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="fricas")
 

Output:

integral(sqrt(a^2*c*x^2 + c)/((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]

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

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

Output:

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

Maxima [F]

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

integrate(1/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(1/(a^2*c*x^2+c)^(5/2)/arctan(a*x)^3,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(1/(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)