3.5.35 \(\int \frac {(c+a^2 c x^2)^{5/2} \arctan (a x)^3}{x^4} \, dx\) [435]

3.5.35.1 Optimal result

Integrand size = 24, antiderivative size = 1061 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x^4} \, dx=-\frac {a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {3}{2} a^3 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}-\frac {2 a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{x}+\frac {1}{2} a^4 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{3 x^3}-\frac {5 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{\sqrt {c+a^2 c x^2}}-\frac {6 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {13 a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-a^3 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {13 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {15 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {15 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {13 i a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {13 a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {15 a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {15 a^3 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {13 a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {15 i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {15 i a^3 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]

output

-1/3*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x^3-a^3*c^(5/2)*arctanh((a^2*c*x^ 
2+c)^(1/2)/c^(1/2))+15/2*I*a^3*c^3*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a 
^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-15*I*a^3*c^3*polylo 
g(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)- 
13*a^3*c^3*arctan(a*x)^2*arctanh((1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^ 
(1/2)/(a^2*c*x^2+c)^(1/2)-15/2*I*a^3*c^3*arctan(a*x)^2*polylog(2,I*(1+I*a* 
x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+3*I*a^3*c^3*po 
lylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c 
)^(1/2)+13*I*a^3*c^3*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*( 
a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+15*I*a^3*c^3*polylog(4,I*(1+I*a*x)/(a 
^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-13*I*a^3*c^3*arctan 
(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+ 
c)^(1/2)-5*I*a^3*c^3*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^3*(a^ 
2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-13*a^3*c^3*polylog(3,-(1+I*a*x)/(a^2*x^ 
2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-15*a^3*c^3*arctan(a*x)*p 
olylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^( 
1/2)+15*a^3*c^3*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2* 
x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+13*a^3*c^3*polylog(3,(1+I*a*x)/(a^2*x^2+1 
)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-6*I*a^3*c^3*arctan(a*x)*arc 
tan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1...
 

3.5.35.2 Mathematica [A] (warning: unable to verify)

Time = 10.26 (sec) , antiderivative size = 1771, normalized size of antiderivative = 1.67 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x^4} \, dx =\text {Too large to display} \]

input

Integrate[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x^4,x]
 

output

((-1/2*I)*a^3*c^2*Sqrt[c*(1 + a^2*x^2)]*(12*ArcTan[E^(I*ArcTan[a*x])]*ArcT 
an[a*x] - (3*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + I*a*x*Sqrt[1 + a^2*x^2]* 
ArcTan[a*x]^3 + 2*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 - 3*(2 + ArcTan[ 
a*x]^2)*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + 3*(2 + ArcTan[a*x]^2)*PolyLog 
[2, I*E^(I*ArcTan[a*x])] - (6*I)*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a 
*x])] + (6*I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[a*x])] + 6*PolyLog[4, ( 
-I)*E^(I*ArcTan[a*x])] - 6*PolyLog[4, I*E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2* 
x^2] + (a^3*c^2*Sqrt[c*(1 + a^2*x^2)]*Csc[ArcTan[a*x]/2]*(((-7*I)*a*Pi^4*x 
)/Sqrt[1 + a^2*x^2] - ((8*I)*a*Pi^3*x*ArcTan[a*x])/Sqrt[1 + a^2*x^2] + ((2 
4*I)*a*Pi^2*x*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] - 64*ArcTan[a*x]^3 - ((32*I 
)*a*Pi*x*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] + ((16*I)*a*x*ArcTan[a*x]^4)/Sqr 
t[1 + a^2*x^2] + (48*a*Pi^2*x*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])])/Sq 
rt[1 + a^2*x^2] - (96*a*Pi*x*ArcTan[a*x]^2*Log[1 - I/E^(I*ArcTan[a*x])])/S 
qrt[1 + a^2*x^2] - (8*a*Pi^3*x*Log[1 + I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2* 
x^2] + (64*a*x*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^ 
2] + (192*a*x*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] 
+ (8*a*Pi^3*x*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (48*a*Pi^2 
*x*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + (96*a*Pi* 
x*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (64*a*x* 
ArcTan[a*x]^3*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] - (192*a*...
 

3.5.35.3 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.

\(\displaystyle \int \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{5/2}}{x^4} \, dx\)

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5415

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5421

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

\(\Big \downarrow \) 5423

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 5479

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5423

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 5479

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

\(\Big \downarrow \) 5493

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

\(\Big \downarrow \) 5491

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5497

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

\(\Big \downarrow \) 5479

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 5493

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

\(\Big \downarrow \) 5491

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

input

Int[((c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^3)/x^4,x]
 

output

$Aborted
 

3.5.35.3.1 Defintions of rubi rules used

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

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 4671

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

rule 5415

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

rule 5421

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] 
 :> Simp[-2*I*(a + b*ArcTan[c*x])*(ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/ 
(c*Sqrt[d])), x] + (Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 + I*c*x]/Sqrt[1 - I*c 
*x])]/(c*Sqrt[d])), x] - Simp[I*b*(PolyLog[2, I*(Sqrt[1 + I*c*x]/Sqrt[1 - I 
*c*x])]/(c*Sqrt[d])), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && 
GtQ[d, 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 5479

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + 
 b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Simp[b*c*(p/(f*(m + 1)))   Int[(f*x) 
^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, 
c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] 
&& NeQ[m, -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 5491

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

rule 5493

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2 
]), x_Symbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]   Int[(a + b*ArcTan 
[c*x])^p/(x*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 5497

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) 
+ (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Ar 
cTan[c*x])^p/(d*f*(m + 1))), x] + (-Simp[b*c*(p/(f*(m + 1)))   Int[(f*x)^(m 
 + 1)*((a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Simp[c^2*((m 
+ 2)/(f^2*(m + 1)))   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] 
 && LtQ[m, -1] && NeQ[m, -2]
 

3.5.35.4 Maple [A] (verified)

Time = 7.50 (sec) , antiderivative size = 694, normalized size of antiderivative = 0.65

method	result	size
default	\(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right ) \left (3 a^{4} \arctan \left (a x \right )^{2} x^{4}-9 \arctan \left (a x \right ) x^{3} a^{3}-14 x^{2} \arctan \left (a x \right )^{2} a^{2}-6 a^{2} x^{2}-3 x \arctan \left (a x \right ) a -2 \arctan \left (a x \right )^{2}\right )}{6 x^{3}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (5 \arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-5 \arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-13 \arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+13 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-26 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+30 i \operatorname {polylog}\left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-30 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+30 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+15 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+2 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right )-2 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-26 \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+26 \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-15 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-30 i \operatorname {polylog}\left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+26 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) c^{2} a^{3}}{2 \sqrt {a^{2} x^{2}+1}}\)	\(694\)

input

int((a^2*c*x^2+c)^(5/2)*arctan(a*x)^3/x^4,x,method=_RETURNVERBOSE)
 

output

1/6*c^2*(c*(a*x-I)*(I+a*x))^(1/2)*arctan(a*x)*(3*a^4*arctan(a*x)^2*x^4-9*a 
rctan(a*x)*x^3*a^3-14*x^2*arctan(a*x)^2*a^2-6*a^2*x^2-3*x*arctan(a*x)*a-2* 
arctan(a*x)^2)/x^3+1/2*(c*(a*x-I)*(I+a*x))^(1/2)/(a^2*x^2+1)^(1/2)*(5*arct 
an(a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-5*arctan(a*x)^3*ln(1+I*(1+I* 
a*x)/(a^2*x^2+1)^(1/2))-13*arctan(a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1) 
+13*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-26*I*arctan(a*x)*polyl 
og(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+30*I*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^( 
1/2))+6*arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-30*arctan(a*x)*pol 
ylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+30*arctan(a*x)*polylog(3,I*(1+I*a*x 
)/(a^2*x^2+1)^(1/2))-6*arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+15* 
I*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*polylog(2,-I 
*(1+I*a*x)/(a^2*x^2+1)^(1/2))+2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)-2*ln((1+ 
I*a*x)/(a^2*x^2+1)^(1/2)+1)-26*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+26* 
polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*polylog(2,I*(1+I*a*x)/(a^2*x^2+ 
1)^(1/2))-15*I*arctan(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-30*I 
*polylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+26*I*arctan(a*x)*polylog(2,-(1+ 
I*a*x)/(a^2*x^2+1)^(1/2)))*c^2*a^3
 

3.5.35.5 Fricas [F]

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

input

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^3/x^4,x, algorithm="fricas")
 

output

integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*sqrt(a^2*c*x^2 + c)*arctan(a* 
x)^3/x^4, x)
 

3.5.35.6 Sympy [F]

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

input

integrate((a**2*c*x**2+c)**(5/2)*atan(a*x)**3/x**4,x)
 

output

Integral((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**3/x**4, x)
 

3.5.35.7 Maxima [F]

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

input

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^3/x^4,x, algorithm="maxima")
 

output

integrate((a^2*c*x^2 + c)^(5/2)*arctan(a*x)^3/x^4, x)
 

3.5.35.8 Giac [F(-2)]

Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x^4} \, dx=\text {Exception raised: TypeError} \]

input

integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^3/x^4,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
 

3.5.35.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3}{x^4} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x^4} \,d x \]

input

int((atan(a*x)^3*(c + a^2*c*x^2)^(5/2))/x^4,x)
 

output

int((atan(a*x)^3*(c + a^2*c*x^2)^(5/2))/x^4, x)