\(\int x^2 (c+a^2 c x^2)^{5/2} \arctan (a x)^3 \, dx\) [428]

Optimal result

Integrand size = 24, antiderivative size = 1019 \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx =\text {Too large to display} \] Output:

13/6720*c^2*(a^2*c*x^2+c)^(1/2)/a^3-3/560*c*(a^2*c*x^2+c)^(3/2)/a^3-1/280* 
(a^2*c*x^2+c)^(5/2)/a^3+43/1344*c^2*x*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a^2+ 
29/560*c^2*x^3*(a^2*c*x^2+c)^(1/2)*arctan(a*x)+1/56*a^2*c^2*x^5*(a^2*c*x^2 
+c)^(1/2)*arctan(a*x)+1373/13440*c^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/a^3 
-737/6720*c^2*x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/a-83/560*a*c^2*x^4*(a^ 
2*c*x^2+c)^(1/2)*arctan(a*x)^2-3/56*a^3*c^2*x^6*(a^2*c*x^2+c)^(1/2)*arctan 
(a*x)^2+5/128*c^2*x*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3/a^2+59/192*c^2*x^3*( 
a^2*c*x^2+c)^(1/2)*arctan(a*x)^3+17/48*a^2*c^2*x^5*(a^2*c*x^2+c)^(1/2)*arc 
tan(a*x)^3+1/8*a^4*c^2*x^7*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3-15/64*I*c^3*( 
a^2*x^2+1)^(1/2)*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^3/(a^2*c*x^2+c 
)^(1/2)-15/128*I*c^3*(a^2*x^2+1)^(1/2)*arctan(a*x)^2*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*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)^3/a^3/(a^2*c*x^2+c)^(1/2)- 
397/1680*I*c^3*(a^2*x^2+1)^(1/2)*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1 
/2))/a^3/(a^2*c*x^2+c)^(1/2)+15/128*I*c^3*(a^2*x^2+1)^(1/2)*arctan(a*x)^2* 
polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^3/(a^2*c*x^2+c)^(1/2)+397/1680* 
I*c^3*(a^2*x^2+1)^(1/2)*polylog(2,I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/a^3/( 
a^2*c*x^2+c)^(1/2)+15/64*c^3*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(3,-I*(1 
+I*a*x)/(a^2*x^2+1)^(1/2))/a^3/(a^2*c*x^2+c)^(1/2)-15/64*c^3*(a^2*x^2+1)^( 
1/2)*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^3/(a^2*c*x^...
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(6517\) vs. \(2(1019)=2038\).

Time = 24.58 (sec) , antiderivative size = 6517, normalized size of antiderivative = 6.40 \[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

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.

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5487

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5423

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5421

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

\(\Big \downarrow \) 5487

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5423

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5421

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

\(\Big \downarrow \) 5487

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 243

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx^2}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{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)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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 {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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 )\)

\(\Big \downarrow \) 53

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\int \left (\frac {\left (a^2 c x^2+c\right )^{3/2}}{a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}+\frac {1}{a^4 \sqrt {a^2 c x^2+c}}\right )dx^2}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{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)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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 {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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 )\)

\(\Big \downarrow \) 2009

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{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)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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 {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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 )\)

\(\Big \downarrow \) 5425

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{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)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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 {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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 )\)

\(\Big \downarrow \) 5421

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{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)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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 {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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 )\)

\(\Big \downarrow \) 5423

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{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)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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 {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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 )\)

\(\Big \downarrow \) 3042

\(\displaystyle c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^7}{8 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^6}{7 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^5}{6 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}}{12 a}-\frac {5 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{6 a^2}\right )}{7 a}-\frac {6 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}\right )}{7 a^2}\right )}{8 a}-\frac {7 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{6 a^2}\right )}{8 a^2}\right ) a^2+c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^3 \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 )}{6 a^2}\right )\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^3 \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 )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3 x^5}{6 a^2 c}-\frac {\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{5 a^2}}{2 a}-\frac {5 \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^3 \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 )}{6 a^2}\right ) a^2+c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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 {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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 )\)

Input:

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

Output:

$Aborted
 

Maple [A] (verified)

Time = 14.62 (sec) , antiderivative size = 566, normalized size of antiderivative = 0.56

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (1680 \arctan \left (a x \right )^{3} a^{7} x^{7}-720 a^{6} x^{6} \arctan \left (a x \right )^{2}+4760 \arctan \left (a x \right )^{3} a^{5} x^{5}+240 \arctan \left (a x \right ) a^{5} x^{5}-1992 a^{4} \arctan \left (a x \right )^{2} x^{4}+4130 \arctan \left (a x \right )^{3} a^{3} x^{3}-48 a^{4} x^{4}+696 \arctan \left (a x \right ) x^{3} a^{3}-1474 \arctan \left (a x \right )^{2} x^{2} a^{2}+525 x \arctan \left (a x \right )^{3} a -168 a^{2} x^{2}+430 \arctan \left (a x \right ) a x +1373 \arctan \left (a x \right )^{2}-94\right )}{13440 a^{3}}-\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (525 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-525 \arctan \left (a x \right )^{3} \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} \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+1575 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3176 \arctan \left (a x \right ) \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 (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3176 \arctan \left (a x \right ) \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 (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3150 i \operatorname {polylog}\left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3150 i \operatorname {polylog}\left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3176 i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3176 i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{13440 a^{3} \sqrt {a^{2} x^{2}+1}}\)	\(566\)

Input:

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

Output:

1/13440*c^2/a^3*(c*(a*x-I)*(a*x+I))^(1/2)*(1680*arctan(a*x)^3*a^7*x^7-720* 
a^6*x^6*arctan(a*x)^2+4760*arctan(a*x)^3*a^5*x^5+240*arctan(a*x)*a^5*x^5-1 
992*a^4*arctan(a*x)^2*x^4+4130*arctan(a*x)^3*a^3*x^3-48*a^4*x^4+696*arctan 
(a*x)*x^3*a^3-1474*arctan(a*x)^2*x^2*a^2+525*x*arctan(a*x)^3*a-168*a^2*x^2 
+430*arctan(a*x)*a*x+1373*arctan(a*x)^2-94)-1/13440*c^2*(c*(a*x-I)*(a*x+I) 
)^(1/2)*(525*arctan(a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-525*arctan( 
a*x)^3*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-1575*I*arctan(a*x)^2*polylog(2, 
I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+1575*I*arctan(a*x)^2*polylog(2,-I*(1+I*a*x) 
/(a^2*x^2+1)^(1/2))+3176*arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3 
150*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3176*arctan(a*x)* 
ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3150*arctan(a*x)*polylog(3,-I*(1+I*a*x 
)/(a^2*x^2+1)^(1/2))+3150*I*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3150* 
I*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+3176*I*dilog(1+I*(1+I*a*x)/(a^ 
2*x^2+1)^(1/2))-3176*I*dilog(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^3/(a^2*x^ 
2+1)^(1/2)
 

Fricas [F]

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

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3 \, dx=\sqrt {c}\, c^{2} \left (\left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} x^{6}d x \right ) a^{4}+2 \left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} x^{4}d x \right ) a^{2}+\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} x^{2}d x \right ) \] Input:

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

Output:

sqrt(c)*c**2*(int(sqrt(a**2*x**2 + 1)*atan(a*x)**3*x**6,x)*a**4 + 2*int(sq 
rt(a**2*x**2 + 1)*atan(a*x)**3*x**4,x)*a**2 + int(sqrt(a**2*x**2 + 1)*atan 
(a*x)**3*x**2,x))