3.5.20 \(\int x^3 (c+a^2 c x^2)^{3/2} \arctan (a x)^3 \, dx\) [420]

3.5.20.1 Optimal result

Integrand size = 24, antiderivative size = 652 \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\frac {c x \sqrt {c+a^2 c x^2}}{420 a^3}-\frac {c x^3 \sqrt {c+a^2 c x^2}}{140 a}-\frac {163 c \sqrt {c+a^2 c x^2} \arctan (a x)}{840 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{60 a^2}+\frac {1}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {9 c x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{112 a^3}-\frac {23 c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{280 a}-\frac {1}{14} a c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {51 i c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{35 a^4}+\frac {c x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{35 a^2}+\frac {8}{35} c x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {1}{7} a^2 c x^6 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {23 c^{3/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{120 a^4}+\frac {51 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {51 i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}-\frac {51 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}}+\frac {51 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{280 a^4 \sqrt {c+a^2 c x^2}} \]

output

23/120*c^(3/2)*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a^4+51/280*I*c^2*a 
rctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4 
/(a^2*c*x^2+c)^(1/2)-51/280*I*c^2*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arct 
an(a*x)^2*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-51/280*I*c^2*arctan(a* 
x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x 
^2+c)^(1/2)-51/280*c^2*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+ 
1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)+51/280*c^2*polylog(3,I*(1+I*a*x)/(a^2*x^2 
+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)+1/420*c*x*(a^2*c*x^2+ 
c)^(1/2)/a^3-1/140*c*x^3*(a^2*c*x^2+c)^(1/2)/a-163/840*c*arctan(a*x)*(a^2* 
c*x^2+c)^(1/2)/a^4+1/60*c*x^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^2+1/35*c*x 
^4*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+9/112*c*x*arctan(a*x)^2*(a^2*c*x^2+c)^( 
1/2)/a^3-23/280*c*x^3*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a-1/14*a*c*x^5*arc 
tan(a*x)^2*(a^2*c*x^2+c)^(1/2)-2/35*c*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/a^ 
4+1/35*c*x^2*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/a^2+8/35*c*x^4*arctan(a*x)^ 
3*(a^2*c*x^2+c)^(1/2)+1/7*a^2*c*x^6*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)
 

3.5.20.2 Mathematica [A] (verified)

Time = 2.78 (sec) , antiderivative size = 538, normalized size of antiderivative = 0.83 \[ \int x^3 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3 \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (64 \left (309 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-259 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-309 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+309 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+309 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-309 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )+2688 \left (-11 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+10 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+11 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-11 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-11 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+11 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-56 \left (1+a^2 x^2\right )^{5/2} \left (\frac {48 a x}{\left (1+a^2 x^2\right )^2}+32 \arctan (a x)^3 (-1+5 \cos (2 \arctan (a x)))+6 \arctan (a x) (25+36 \cos (2 \arctan (a x))+11 \cos (4 \arctan (a x)))+\arctan (a x)^2 (6 \sin (2 \arctan (a x))-33 \sin (4 \arctan (a x)))\right )+\left (1+a^2 x^2\right )^{7/2} \left (64 \arctan (a x)^3 (57-28 \cos (2 \arctan (a x))+35 \cos (4 \arctan (a x)))+\frac {8 \arctan (a x) (647+764 \cos (2 \arctan (a x))+309 \cos (4 \arctan (a x)))}{1+a^2 x^2}+4 (101 \sin (2 \arctan (a x))+88 \sin (4 \arctan (a x))+25 \sin (6 \arctan (a x)))-3 \arctan (a x)^2 (211 \sin (2 \arctan (a x))-60 \sin (4 \arctan (a x))+103 \sin (6 \arctan (a x)))\right )\right )}{53760 a^4 \sqrt {1+a^2 x^2}} \]

input

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

output

(c*Sqrt[c + a^2*c*x^2]*(64*((309*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^ 
2 - 259*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] - (309*I)*ArcTan[a*x]*PolyLog[2, 
(-I)*E^(I*ArcTan[a*x])] + (309*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x 
])] + 309*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 309*PolyLog[3, I*E^(I*ArcTa 
n[a*x])]) + 2688*((-11*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 10*Arc 
Tanh[(a*x)/Sqrt[1 + a^2*x^2]] + (11*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*Ar 
cTan[a*x])] - (11*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 11*Poly 
Log[3, (-I)*E^(I*ArcTan[a*x])] + 11*PolyLog[3, I*E^(I*ArcTan[a*x])]) - 56* 
(1 + a^2*x^2)^(5/2)*((48*a*x)/(1 + a^2*x^2)^2 + 32*ArcTan[a*x]^3*(-1 + 5*C 
os[2*ArcTan[a*x]]) + 6*ArcTan[a*x]*(25 + 36*Cos[2*ArcTan[a*x]] + 11*Cos[4* 
ArcTan[a*x]]) + ArcTan[a*x]^2*(6*Sin[2*ArcTan[a*x]] - 33*Sin[4*ArcTan[a*x] 
])) + (1 + a^2*x^2)^(7/2)*(64*ArcTan[a*x]^3*(57 - 28*Cos[2*ArcTan[a*x]] + 
35*Cos[4*ArcTan[a*x]]) + (8*ArcTan[a*x]*(647 + 764*Cos[2*ArcTan[a*x]] + 30 
9*Cos[4*ArcTan[a*x]]))/(1 + a^2*x^2) + 4*(101*Sin[2*ArcTan[a*x]] + 88*Sin[ 
4*ArcTan[a*x]] + 25*Sin[6*ArcTan[a*x]]) - 3*ArcTan[a*x]^2*(211*Sin[2*ArcTa 
n[a*x]] - 60*Sin[4*ArcTan[a*x]] + 103*Sin[6*ArcTan[a*x]]))))/(53760*a^4*Sq 
rt[1 + a^2*x^2])
 

3.5.20.3 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.

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

\(\Big \downarrow \) 5485

\(\displaystyle a^2 c \int x^5 \sqrt {a^2 c x^2+c} \arctan (a x)^3dx+c \int x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3dx\)

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5487

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5423

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5487

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5423

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5423

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5487

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

input

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

output

$Aborted
 

3.5.20.3.1 Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 5423

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[1/(c*Sqrt[d])   Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[ 
c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Gt 
Q[d, 0]
 

rule 5425

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]   Int[(a + b*ArcTan[c*x])^ 
p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] & 
& IGtQ[p, 0] &&  !GtQ[d, 0]
 

rule 5465

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

rule 5485

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

rule 5487

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) 
+ (e_.)*(x_)^2], x_Symbol] :> Simp[f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*((a + b* 
ArcTan[c*x])^p/(c^2*d*m)), x] + (-Simp[b*f*(p/(c*m))   Int[(f*x)^(m - 1)*(( 
a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Simp[f^2*((m - 1)/(c^ 
2*m))   Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/Sqrt[d + e*x^2]), x], x]) 
/; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && GtQ[m, 1]
 

3.5.20.4 Maple [A] (verified)

Time = 4.04 (sec) , antiderivative size = 469, normalized size of antiderivative = 0.72

method	result	size
default	\(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (240 \arctan \left (a x \right )^{3} a^{6} x^{6}-120 a^{5} \arctan \left (a x \right )^{2} x^{5}+384 a^{4} \arctan \left (a x \right )^{3} x^{4}+48 \arctan \left (a x \right ) a^{4} x^{4}-138 a^{3} \arctan \left (a x \right )^{2} x^{3}+48 \arctan \left (a x \right )^{3} x^{2} a^{2}-12 a^{3} x^{3}+28 a^{2} \arctan \left (a x \right ) x^{2}+135 a \arctan \left (a x \right )^{2} x -96 \arctan \left (a x \right )^{3}+4 a x -326 \arctan \left (a x \right )\right )}{1680 a^{4}}+\frac {17 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{560 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {17 c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{560 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {23 i c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{60 a^{4} \sqrt {a^{2} x^{2}+1}}\)	\(469\)

input

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

output

1/1680*c/a^4*(c*(a*x-I)*(I+a*x))^(1/2)*(240*arctan(a*x)^3*a^6*x^6-120*a^5* 
arctan(a*x)^2*x^5+384*a^4*arctan(a*x)^3*x^4+48*arctan(a*x)*a^4*x^4-138*a^3 
*arctan(a*x)^2*x^3+48*arctan(a*x)^3*x^2*a^2-12*a^3*x^3+28*a^2*arctan(a*x)* 
x^2+135*a*arctan(a*x)^2*x-96*arctan(a*x)^3+4*a*x-326*arctan(a*x))+17/560*c 
*(c*(a*x-I)*(I+a*x))^(1/2)*(I*arctan(a*x)^3-3*arctan(a*x)^2*ln(1+I*(1+I*a* 
x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^( 
1/2))-6*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^4/(a^2*x^2+1)^(1/2)-1 
7/560*c*(c*(a*x-I)*(I+a*x))^(1/2)*(I*arctan(a*x)^3-3*arctan(a*x)^2*ln(1-I* 
(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^ 
2+1)^(1/2))-6*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^4/(a^2*x^2+1)^(1 
/2)-23/60*I*c/a^4*(c*(a*x-I)*(I+a*x))^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^( 
1/2))/(a^2*x^2+1)^(1/2)
 

3.5.20.5 Fricas [F]

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

input

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

output

integral((a^2*c*x^5 + c*x^3)*sqrt(a^2*c*x^2 + c)*arctan(a*x)^3, x)
 

3.5.20.6 Sympy [F]

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

input

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

output

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

3.5.20.7 Maxima [F]

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

input

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

output

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

3.5.20.8 Giac [F(-2)]

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

input

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

output

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

3.5.20.9 Mupad [F(-1)]

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

input

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

output

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