\(\int x^2 (c+a^2 c x^2)^{5/2} \arctan (a x)^2 \, dx\) [324]

Optimal result

Integrand size = 24, antiderivative size = 638 \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\frac {43 c^2 x \sqrt {c+a^2 c x^2}}{4032 a^2}+\frac {29 c^2 x^3 \sqrt {c+a^2 c x^2}}{1680}+\frac {1}{168} a^2 c^2 x^5 \sqrt {c+a^2 c x^2}+\frac {1373 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{20160 a^3}-\frac {737 c^2 x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{10080 a}-\frac {83}{840} a c^2 x^4 \sqrt {c+a^2 c x^2} \arctan (a x)-\frac {1}{28} a^3 c^2 x^6 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {5 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{128 a^2}+\frac {59}{192} c^2 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {17}{48} a^2 c^2 x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {1}{8} a^4 c^2 x^7 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {397 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{5040 a^3}-\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}+\frac {5 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}}-\frac {5 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{64 a^3 \sqrt {c+a^2 c x^2}} \] Output:

43/4032*c^2*x*(a^2*c*x^2+c)^(1/2)/a^2+29/1680*c^2*x^3*(a^2*c*x^2+c)^(1/2)+ 
1/168*a^2*c^2*x^5*(a^2*c*x^2+c)^(1/2)+1373/20160*c^2*(a^2*c*x^2+c)^(1/2)*a 
rctan(a*x)/a^3-737/10080*c^2*x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a-83/840* 
a*c^2*x^4*(a^2*c*x^2+c)^(1/2)*arctan(a*x)-1/28*a^3*c^2*x^6*(a^2*c*x^2+c)^( 
1/2)*arctan(a*x)+5/128*c^2*x*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/a^2+59/192* 
c^2*x^3*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2+17/48*a^2*c^2*x^5*(a^2*c*x^2+c)^ 
(1/2)*arctan(a*x)^2+1/8*a^4*c^2*x^7*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2+5/64 
*I*c^3*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/ 
2))/a^3/(a^2*c*x^2+c)^(1/2)-397/5040*c^(5/2)*arctanh(a*c^(1/2)*x/(a^2*c*x^ 
2+c)^(1/2))/a^3+5/64*I*c^3*(a^2*x^2+1)^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^ 
(1/2))*arctan(a*x)^2/a^3/(a^2*c*x^2+c)^(1/2)-5/64*I*c^3*(a^2*x^2+1)^(1/2)* 
arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^3/(a^2*c*x^2+c)^(1 
/2)+5/64*c^3*(a^2*x^2+1)^(1/2)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a 
^3/(a^2*c*x^2+c)^(1/2)-5/64*c^3*(a^2*x^2+1)^(1/2)*polylog(3,I*(1+I*a*x)/(a 
^2*x^2+1)^(1/2))/a^3/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 3.40 (sec) , antiderivative size = 759, normalized size of antiderivative = 1.19 \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx =\text {Too large to display} \] Input:

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

Output:

(c^2*Sqrt[c + a^2*c*x^2]*(53760*a*x*(1 + a^2*x^2)^(3/2) - 25088*a*x*(1 + a 
^2*x^2)^(5/2) + 7006*a*x*(1 + a^2*x^2)^(7/2) - 203264*ArcCoth[(a*x)/Sqrt[1 
 + a^2*x^2]] + 53760*(1 + a^2*x^2)^(3/2)*ArcTan[a*x] + 5376*(1 + a^2*x^2)^ 
(5/2)*ArcTan[a*x] - 38134*(1 + a^2*x^2)^(7/2)*ArcTan[a*x] + 564480*a*x*(1 
+ a^2*x^2)^(3/2)*ArcTan[a*x]^2 + 524160*a*x*(1 + a^2*x^2)^(5/2)*ArcTan[a*x 
]^2 + 185325*a*x*(1 + a^2*x^2)^(7/2)*ArcTan[a*x]^2 + (201600*I)*ArcTan[E^( 
I*ArcTan[a*x])]*ArcTan[a*x]^2 + 161280*(1 + a^2*x^2)^2*ArcTan[a*x]*Cos[3*A 
rcTan[a*x]] + 49280*(1 + a^2*x^2)^3*ArcTan[a*x]*Cos[3*ArcTan[a*x]] - 7658* 
(1 + a^2*x^2)^4*ArcTan[a*x]*Cos[3*ArcTan[a*x]] - 40320*(1 + a^2*x^2)^3*Arc 
Tan[a*x]*Cos[5*ArcTan[a*x]] - 10990*(1 + a^2*x^2)^4*ArcTan[a*x]*Cos[5*ArcT 
an[a*x]] + 3150*(1 + a^2*x^2)^4*ArcTan[a*x]*Cos[7*ArcTan[a*x]] - (201600*I 
)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (201600*I)*ArcTan[a*x]* 
PolyLog[2, I*E^(I*ArcTan[a*x])] + 201600*PolyLog[3, (-I)*E^(I*ArcTan[a*x]) 
] - 201600*PolyLog[3, I*E^(I*ArcTan[a*x])] + 53760*(1 + a^2*x^2)^2*Sin[3*A 
rcTan[a*x]] - 48384*(1 + a^2*x^2)^3*Sin[3*ArcTan[a*x]] + 12246*(1 + a^2*x^ 
2)^4*Sin[3*ArcTan[a*x]] - 80640*(1 + a^2*x^2)^2*ArcTan[a*x]^2*Sin[3*ArcTan 
[a*x]] - 315840*(1 + a^2*x^2)^3*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 93975*( 
1 + a^2*x^2)^4*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 23296*(1 + a^2*x^2)^3*Si 
n[5*ArcTan[a*x]] + 7678*(1 + a^2*x^2)^4*Sin[5*ArcTan[a*x]] + 20160*(1 + a^ 
2*x^2)^3*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + 41685*(1 + a^2*x^2)^4*ArcTa...
 

Rubi [F]

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.

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5485

\(\displaystyle c \left (c \int x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+a^2 c \int x^4 \sqrt {a^2 c x^2+c} \arctan (a x)^2dx\right )+a^2 c \left (a^2 c \int x^6 \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+c \int x^4 \sqrt {a^2 c x^2+c} \arctan (a x)^2dx\right )\)

\(\Big \downarrow \) 5485

\(\displaystyle c \left (c \left (c \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+a^2 c \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (a^2 c \int \frac {x^6 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx\right )\right )+a^2 c \left (a^2 c \left (a^2 c \int \frac {x^8 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {x^6 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx\right )+c \left (a^2 c \int \frac {x^6 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx\right )\right )\)

\(\Big \downarrow \) 5487

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\int \frac {x^7 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {7 \int \frac {x^6 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )\right )\right )\)

\(\Big \downarrow \) 5425

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\int \frac {x^7 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {7 \int \frac {x^6 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 5423

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\int \frac {x^7 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {7 \int \frac {x^6 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\int \frac {x^7 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {7 \int \frac {x^6 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 4669

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\int \frac {x^7 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {7 \int \frac {x^6 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\int \frac {x^7 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {7 \int \frac {x^6 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\int \frac {x^7 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {7 \int \frac {x^6 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 5465

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\int \frac {x^7 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {7 \int \frac {x^6 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 224

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\int \frac {x^7 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {7 \int \frac {x^6 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{a}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 219

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\int \frac {x^7 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {7 \int \frac {x^6 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 5487

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\int \frac {x^6}{\sqrt {a^2 c x^2+c}}dx}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 262

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 224

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{2 a^2}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{2 a^2}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{2 a^2}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 219

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 262

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 224

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{2 a^2}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{2 a^2}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{2 a^2}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 219

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 262

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 224

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 219

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 5425

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 5423

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 4669

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 2720

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

\(\Big \downarrow \) 5465

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^7}{8 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^6}{7 a^2 c}-\frac {\frac {x^5 \sqrt {a^2 c x^2+c}}{6 a^2 c}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}\right )}{6 a^2}}{7 a}-\frac {6 \int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{7 a^2}}{4 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\int \frac {x^5 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {5 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^4}{5 a^2 c}-\frac {\frac {x^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}\right )}{4 a^2}}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{3 a}-\frac {5 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^2}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a^3 \sqrt {c}}}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}}{2 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\right )\right )\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 7.59 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.59

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (5040 \arctan \left (a x \right )^{2} a^{7} x^{7}-1440 a^{6} \arctan \left (a x \right ) x^{6}+14280 a^{5} \arctan \left (a x \right )^{2} x^{5}+240 a^{5} x^{5}-3984 x^{4} \arctan \left (a x \right ) a^{4}+12390 a^{3} \arctan \left (a x \right )^{2} x^{3}+696 a^{3} x^{3}-2948 x^{2} a^{2} \arctan \left (a x \right )+1575 a \arctan \left (a x \right )^{2} x +430 a x +2746 \arctan \left (a x \right )\right )}{40320 a^{3}}-\frac {i c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (1575 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-1575 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3150 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3150 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3150 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3150 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6352 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{40320 a^{3} \sqrt {a^{2} x^{2}+1}}\)	\(376\)

Input:

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

Output:

1/40320*c^2/a^3*(c*(a*x-I)*(a*x+I))^(1/2)*(5040*arctan(a*x)^2*a^7*x^7-1440 
*a^6*arctan(a*x)*x^6+14280*a^5*arctan(a*x)^2*x^5+240*a^5*x^5-3984*x^4*arct 
an(a*x)*a^4+12390*a^3*arctan(a*x)^2*x^3+696*a^3*x^3-2948*x^2*a^2*arctan(a* 
x)+1575*a*arctan(a*x)^2*x+430*a*x+2746*arctan(a*x))-1/40320*I*c^2*(c*(a*x- 
I)*(a*x+I))^(1/2)*(1575*I*arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2) 
)-1575*I*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3150*arctan(a*x 
)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3150*arctan(a*x)*polylog(2,I*( 
1+I*a*x)/(a^2*x^2+1)^(1/2))+3150*I*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2 
))-3150*I*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6352*arctan((1+I*a*x)/( 
a^2*x^2+1)^(1/2)))/a^3/(a^2*x^2+1)^(1/2)
 

Fricas [F]

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

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

Output:

integral((a^4*c^2*x^6 + 2*a^2*c^2*x^4 + c^2*x^2)*sqrt(a^2*c*x^2 + c)*arcta 
n(a*x)^2, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\sqrt {c}\, c^{2} \left (\left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{2} x^{6}d x \right ) a^{4}+2 \left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{2} x^{4}d x \right ) a^{2}+\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{2} x^{2}d x \right ) \] Input:

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

Output:

sqrt(c)*c**2*(int(sqrt(a**2*x**2 + 1)*atan(a*x)**2*x**6,x)*a**4 + 2*int(sq 
rt(a**2*x**2 + 1)*atan(a*x)**2*x**4,x)*a**2 + int(sqrt(a**2*x**2 + 1)*atan 
(a*x)**2*x**2,x))