\(\int \frac {\arctan (a x)^{3/2}}{(c+a^2 c x^2)^{5/2}} \, dx\) [833]

Optimal result

Integrand size = 23, antiderivative size = 252 \[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\sqrt {\arctan (a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{a c^2 \sqrt {c+a^2 c x^2}}+\frac {x \arctan (a x)^{3/2}}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {2 x \arctan (a x)^{3/2}}{3 c^2 \sqrt {c+a^2 c x^2}}-\frac {9 \sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{8 a c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {\frac {\pi }{6}} \sqrt {1+a^2 x^2} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )}{24 a c^2 \sqrt {c+a^2 c x^2}} \] Output:

1/6*arctan(a*x)^(1/2)/a/c/(a^2*c*x^2+c)^(3/2)+arctan(a*x)^(1/2)/a/c^2/(a^2 
*c*x^2+c)^(1/2)+1/3*x*arctan(a*x)^(3/2)/c/(a^2*c*x^2+c)^(3/2)+2/3*x*arctan 
(a*x)^(3/2)/c^2/(a^2*c*x^2+c)^(1/2)-9/16*2^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2 
)*FresnelC(2^(1/2)/Pi^(1/2)*arctan(a*x)^(1/2))/a/c^2/(a^2*c*x^2+c)^(1/2)-1 
/144*6^(1/2)*Pi^(1/2)*(a^2*x^2+1)^(1/2)*FresnelC(6^(1/2)/Pi^(1/2)*arctan(a 
*x)^(1/2))/a/c^2/(a^2*c*x^2+c)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.39 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.37 \[ \int \frac {\arctan (a x)^{3/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {336 \arctan (a x)+288 a^2 x^2 \arctan (a x)+288 a x \arctan (a x)^2+192 a^3 x^3 \arctan (a x)^2+81 i \left (1+a^2 x^2\right )^{3/2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-i \arctan (a x)\right )-81 i \left (1+a^2 x^2\right )^{3/2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},i \arctan (a x)\right )+i \sqrt {3+3 a^2 x^2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-3 i \arctan (a x)\right )+i a^2 x^2 \sqrt {3+3 a^2 x^2} \sqrt {-i \arctan (a x)} \Gamma \left (\frac {1}{2},-3 i \arctan (a x)\right )-i \sqrt {3+3 a^2 x^2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},3 i \arctan (a x)\right )-i a^2 x^2 \sqrt {3+3 a^2 x^2} \sqrt {i \arctan (a x)} \Gamma \left (\frac {1}{2},3 i \arctan (a x)\right )}{288 c^2 \left (a+a^3 x^2\right ) \sqrt {c+a^2 c x^2} \sqrt {\arctan (a x)}} \] Input:

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

Output:

(336*ArcTan[a*x] + 288*a^2*x^2*ArcTan[a*x] + 288*a*x*ArcTan[a*x]^2 + 192*a 
^3*x^3*ArcTan[a*x]^2 + (81*I)*(1 + a^2*x^2)^(3/2)*Sqrt[(-I)*ArcTan[a*x]]*G 
amma[1/2, (-I)*ArcTan[a*x]] - (81*I)*(1 + a^2*x^2)^(3/2)*Sqrt[I*ArcTan[a*x 
]]*Gamma[1/2, I*ArcTan[a*x]] + I*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I)*ArcTan[a*x] 
]*Gamma[1/2, (-3*I)*ArcTan[a*x]] + I*a^2*x^2*Sqrt[3 + 3*a^2*x^2]*Sqrt[(-I) 
*ArcTan[a*x]]*Gamma[1/2, (-3*I)*ArcTan[a*x]] - I*Sqrt[3 + 3*a^2*x^2]*Sqrt[ 
I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]] - I*a^2*x^2*Sqrt[3 + 3*a^2*x^ 
2]*Sqrt[I*ArcTan[a*x]]*Gamma[1/2, (3*I)*ArcTan[a*x]])/(288*c^2*(a + a^3*x^ 
2)*Sqrt[c + a^2*c*x^2]*Sqrt[ArcTan[a*x]])
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.18, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {5435, 5433, 5440, 5439, 3042, 3785, 3793, 2009, 3833}

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.

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

\(\Big \downarrow \) 5435

\(\displaystyle -\frac {1}{12} \int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx+\frac {2 \int \frac {\arctan (a x)^{3/2}}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 c}+\frac {x \arctan (a x)^{3/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5433

\(\displaystyle -\frac {1}{12} \int \frac {1}{\left (a^2 c x^2+c\right )^{5/2} \sqrt {\arctan (a x)}}dx+\frac {2 \left (-\frac {3}{4} \int \frac {1}{\left (a^2 c x^2+c\right )^{3/2} \sqrt {\arctan (a x)}}dx+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {x \arctan (a x)^{3/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5440

\(\displaystyle -\frac {\sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{5/2} \sqrt {\arctan (a x)}}dx}{12 c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \left (-\frac {3 \sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{3/2} \sqrt {\arctan (a x)}}dx}{4 c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {x \arctan (a x)^{3/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 5439

\(\displaystyle -\frac {\sqrt {a^2 x^2+1} \int \frac {1}{\left (a^2 x^2+1\right )^{3/2} \sqrt {\arctan (a x)}}d\arctan (a x)}{12 a c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \left (-\frac {3 \sqrt {a^2 x^2+1} \int \frac {1}{\sqrt {a^2 x^2+1} \sqrt {\arctan (a x)}}d\arctan (a x)}{4 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {x \arctan (a x)^{3/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\sqrt {a^2 x^2+1} \int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^3}{\sqrt {\arctan (a x)}}d\arctan (a x)}{12 a c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \left (-\frac {3 \sqrt {a^2 x^2+1} \int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )}{\sqrt {\arctan (a x)}}d\arctan (a x)}{4 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {x \arctan (a x)^{3/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 3785

\(\displaystyle -\frac {\sqrt {a^2 x^2+1} \int \frac {\sin \left (\arctan (a x)+\frac {\pi }{2}\right )^3}{\sqrt {\arctan (a x)}}d\arctan (a x)}{12 a c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \left (-\frac {3 \sqrt {a^2 x^2+1} \int \frac {1}{\sqrt {a^2 x^2+1}}d\sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {x \arctan (a x)^{3/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 3793

\(\displaystyle -\frac {\sqrt {a^2 x^2+1} \int \left (\frac {\cos (3 \arctan (a x))}{4 \sqrt {\arctan (a x)}}+\frac {3}{4 \sqrt {a^2 x^2+1} \sqrt {\arctan (a x)}}\right )d\arctan (a x)}{12 a c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \left (-\frac {3 \sqrt {a^2 x^2+1} \int \frac {1}{\sqrt {a^2 x^2+1}}d\sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {x \arctan (a x)^{3/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 \left (-\frac {3 \sqrt {a^2 x^2+1} \int \frac {1}{\sqrt {a^2 x^2+1}}d\sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{3 c}-\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{12 a c^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\)

\(\Big \downarrow \) 3833

\(\displaystyle -\frac {\sqrt {a^2 x^2+1} \left (\frac {3}{2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\arctan (a x)}\right )\right )}{12 a c^2 \sqrt {a^2 c x^2+c}}+\frac {2 \left (-\frac {3 \sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \operatorname {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\arctan (a x)}\right )}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^{3/2}}{c \sqrt {a^2 c x^2+c}}+\frac {3 \sqrt {\arctan (a x)}}{2 a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {x \arctan (a x)^{3/2}}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {\sqrt {\arctan (a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}\)

Input:

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

Output:

Sqrt[ArcTan[a*x]]/(6*a*c*(c + a^2*c*x^2)^(3/2)) + (x*ArcTan[a*x]^(3/2))/(3 
*c*(c + a^2*c*x^2)^(3/2)) + (2*((3*Sqrt[ArcTan[a*x]])/(2*a*c*Sqrt[c + a^2* 
c*x^2]) + (x*ArcTan[a*x]^(3/2))/(c*Sqrt[c + a^2*c*x^2]) - (3*Sqrt[Pi/2]*Sq 
rt[1 + a^2*x^2]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcTan[a*x]]])/(2*a*c*Sqrt[c + a^ 
2*c*x^2])))/(3*c) - (Sqrt[1 + a^2*x^2]*((3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]* 
Sqrt[ArcTan[a*x]]])/2 + (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcTan[a*x]]] 
)/2))/(12*a*c^2*Sqrt[c + a^2*c*x^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 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[2/d   Subst[Int[Cos[f*(x^2/d)], x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, 
d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]
 

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 3833

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[ 
d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]
 

rule 5433

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_ 
Symbol] :> Simp[b*p*((a + b*ArcTan[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2])), x] 
 + (Simp[x*((a + b*ArcTan[c*x])^p/(d*Sqrt[d + e*x^2])), x] - Simp[b^2*p*(p 
- 1)   Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[ 
{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]
 

rule 5435

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_), x_S 
ymbol] :> Simp[b*p*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^(p - 1)/(4*c*d* 
(q + 1)^2)), x] + (-Simp[x*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*d* 
(q + 1))), x] + Simp[(2*q + 3)/(2*d*(q + 1))   Int[(d + e*x^2)^(q + 1)*(a + 
 b*ArcTan[c*x])^p, x], x] - Simp[b^2*p*((p - 1)/(4*(q + 1)^2))   Int[(d + e 
*x^2)^q*(a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] & 
& EqQ[e, c^2*d] && LtQ[q, -1] && GtQ[p, 1] && NeQ[q, -3/2]
 

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])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(c)*(8*sqrt(a**2*x**2 + 1)*sqrt(atan(a*x))*atan(a*x)*a**3*x**3 + 12*s 
qrt(a**2*x**2 + 1)*sqrt(atan(a*x))*atan(a*x)*a*x + 12*sqrt(a**2*x**2 + 1)* 
sqrt(atan(a*x))*a**2*x**2 + 14*sqrt(a**2*x**2 + 1)*sqrt(atan(a*x)) - 6*int 
((sqrt(a**2*x**2 + 1)*sqrt(atan(a*x))*x**2)/(atan(a*x)*a**6*x**6 + 3*atan( 
a*x)*a**4*x**4 + 3*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**7*x**4 - 12*int( 
(sqrt(a**2*x**2 + 1)*sqrt(atan(a*x))*x**2)/(atan(a*x)*a**6*x**6 + 3*atan(a 
*x)*a**4*x**4 + 3*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**5*x**2 - 6*int((s 
qrt(a**2*x**2 + 1)*sqrt(atan(a*x))*x**2)/(atan(a*x)*a**6*x**6 + 3*atan(a*x 
)*a**4*x**4 + 3*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**3 - 7*int((sqrt(a** 
2*x**2 + 1)*sqrt(atan(a*x)))/(atan(a*x)*a**6*x**6 + 3*atan(a*x)*a**4*x**4 
+ 3*atan(a*x)*a**2*x**2 + atan(a*x)),x)*a**5*x**4 - 14*int((sqrt(a**2*x**2 
 + 1)*sqrt(atan(a*x)))/(atan(a*x)*a**6*x**6 + 3*atan(a*x)*a**4*x**4 + 3*at 
an(a*x)*a**2*x**2 + atan(a*x)),x)*a**3*x**2 - 7*int((sqrt(a**2*x**2 + 1)*s 
qrt(atan(a*x)))/(atan(a*x)*a**6*x**6 + 3*atan(a*x)*a**4*x**4 + 3*atan(a*x) 
*a**2*x**2 + atan(a*x)),x)*a))/(12*a*c**3*(a**4*x**4 + 2*a**2*x**2 + 1))