\(\int \frac {(c+a^2 c x^2)^{3/2} \arctan (a x)^2}{x^3} \, dx\) [321]

Optimal result

Integrand size = 24, antiderivative size = 567 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{x^3} \, dx=-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)}{x}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {4 i a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-a^2 c^{3/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {3 a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \] Output:

-a*c*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/x+a^2*c*(a^2*c*x^2+c)^(1/2)*arctan(a* 
x)^2-1/2*c*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/x^2+4*I*a^2*c^2*(a^2*x^2+1)^( 
1/2)*arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^2+c)^(1/ 
2)-3*a^2*c^2*(a^2*x^2+1)^(1/2)*arctan(a*x)^2*arctanh((1+I*a*x)/(a^2*x^2+1) 
^(1/2))/(a^2*c*x^2+c)^(1/2)-a^2*c^(3/2)*arctanh((a^2*c*x^2+c)^(1/2)/c^(1/2 
))+3*I*a^2*c^2*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2 
+1)^(1/2))/(a^2*c*x^2+c)^(1/2)-3*I*a^2*c^2*(a^2*x^2+1)^(1/2)*arctan(a*x)*p 
olylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)-2*I*a^2*c^2*(a^2 
*x^2+1)^(1/2)*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^2+c)^ 
(1/2)+2*I*a^2*c^2*(a^2*x^2+1)^(1/2)*polylog(2,I*(1+I*a*x)^(1/2)/(1-I*a*x)^ 
(1/2))/(a^2*c*x^2+c)^(1/2)-3*a^2*c^2*(a^2*x^2+1)^(1/2)*polylog(3,-(1+I*a*x 
)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)+3*a^2*c^2*(a^2*x^2+1)^(1/2)*polyl 
og(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 2.17 (sec) , antiderivative size = 455, normalized size of antiderivative = 0.80 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{x^3} \, dx=\frac {a^2 c \sqrt {c+a^2 c x^2} \left (-4 \arctan (a x)-4 \arctan (a x) \cot ^2\left (\frac {1}{2} \arctan (a x)\right )+4 a x \arctan (a x)^2 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-\arctan (a x)^2 \cot \left (\frac {1}{2} \arctan (a x)\right ) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+12 \arctan (a x)^2 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1-e^{i \arctan (a x)}\right )-16 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1-i e^{i \arctan (a x)}\right )+16 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1+i e^{i \arctan (a x)}\right )-12 \arctan (a x)^2 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (1+e^{i \arctan (a x)}\right )+8 \cot \left (\frac {1}{2} \arctan (a x)\right ) \log \left (\tan \left (\frac {1}{2} \arctan (a x)\right )\right )+24 i \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-16 i \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+16 i \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-24 i \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-24 \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+24 \cot \left (\frac {1}{2} \arctan (a x)\right ) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )+\arctan (a x)^2 \csc \left (\frac {1}{2} \arctan (a x)\right ) \sec \left (\frac {1}{2} \arctan (a x)\right )\right ) \tan \left (\frac {1}{2} \arctan (a x)\right )}{8 \sqrt {1+a^2 x^2}} \] Input:

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

Output:

(a^2*c*Sqrt[c + a^2*c*x^2]*(-4*ArcTan[a*x] - 4*ArcTan[a*x]*Cot[ArcTan[a*x] 
/2]^2 + 4*a*x*ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 - ArcTan[a*x]^2*Cot[ArcTa 
n[a*x]/2]*Csc[ArcTan[a*x]/2]^2 + 12*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2]*Log[1 
 - E^(I*ArcTan[a*x])] - 16*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 - I*E^(I*A 
rcTan[a*x])] + 16*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 + I*E^(I*ArcTan[a*x 
])] - 12*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2]*Log[1 + E^(I*ArcTan[a*x])] + 8*C 
ot[ArcTan[a*x]/2]*Log[Tan[ArcTan[a*x]/2]] + (24*I)*ArcTan[a*x]*Cot[ArcTan[ 
a*x]/2]*PolyLog[2, -E^(I*ArcTan[a*x])] - (16*I)*Cot[ArcTan[a*x]/2]*PolyLog 
[2, (-I)*E^(I*ArcTan[a*x])] + (16*I)*Cot[ArcTan[a*x]/2]*PolyLog[2, I*E^(I* 
ArcTan[a*x])] - (24*I)*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*PolyLog[2, E^(I*ArcT 
an[a*x])] - 24*Cot[ArcTan[a*x]/2]*PolyLog[3, -E^(I*ArcTan[a*x])] + 24*Cot[ 
ArcTan[a*x]/2]*PolyLog[3, E^(I*ArcTan[a*x])] + ArcTan[a*x]^2*Csc[ArcTan[a* 
x]/2]*Sec[ArcTan[a*x]/2])*Tan[ArcTan[a*x]/2])/(8*Sqrt[1 + a^2*x^2])
 

Rubi [A] (verified)

Time = 6.21 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.18, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.958, Rules used = {5485, 5485, 5465, 5425, 5421, 5493, 5491, 3042, 4671, 3011, 2720, 5497, 5479, 243, 73, 221, 5493, 5491, 3042, 4671, 3011, 2720, 7143}

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.

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5421

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

\(\Big \downarrow \) 5493

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

\(\Big \downarrow \) 5491

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5497

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

\(\Big \downarrow \) 5479

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

\(\Big \downarrow \) 243

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

\(\Big \downarrow \) 5493

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

\(\Big \downarrow \) 5491

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4671

\(\displaystyle a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (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 )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \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 )}{a \sqrt {a^2 c x^2+c}}\right )\right )+c \left (\frac {a^2 c \sqrt {a^2 x^2+1} \left (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 )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (-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)-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )\right )\)

\(\Big \downarrow \) 3011

\(\displaystyle a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (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 )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \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 )}{a \sqrt {a^2 c x^2+c}}\right )\right )+c \left (\frac {a^2 c \sqrt {a^2 x^2+1} \left (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 )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (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 )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )\right )\)

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

a^2*c*(a^2*c*((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(a^2*c) - (2*Sqrt[1 + a^ 
2*x^2]*(((-2*I)*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/a + ( 
I*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a - (I*PolyLog[2, (I 
*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a))/(a*Sqrt[c + a^2*c*x^2])) + (c*Sqrt 
[1 + a^2*x^2]*(-2*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])] + 2*(I*ArcTan[a 
*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - PolyLog[3, -E^(I*ArcTan[a*x])]) - 2*( 
I*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - PolyLog[3, E^(I*ArcTan[a*x]) 
])))/Sqrt[c + a^2*c*x^2]) + c*(c*(-1/2*(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2) 
/(c*x^2) + a*(-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(c*x)) - (a*ArcTanh[Sqrt 
[c + a^2*c*x^2]/Sqrt[c]])/Sqrt[c]) - (a^2*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x 
]^2*ArcTanh[E^(I*ArcTan[a*x])] + 2*(I*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[ 
a*x])] - PolyLog[3, -E^(I*ArcTan[a*x])]) - 2*(I*ArcTan[a*x]*PolyLog[2, E^( 
I*ArcTan[a*x])] - PolyLog[3, E^(I*ArcTan[a*x])])))/(2*Sqrt[c + a^2*c*x^2]) 
) + (a^2*c*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])] 
+ 2*(I*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - PolyLog[3, -E^(I*ArcTa 
n[a*x])]) - 2*(I*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - PolyLog[3, E^ 
(I*ArcTan[a*x])])))/Sqrt[c + a^2*c*x^2])
 

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 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 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 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 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]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

Maple [A] (verified)

Time = 2.56 (sec) , antiderivative size = 412, normalized size of antiderivative = 0.73

method	result	size
default	\(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right ) \left (2 x^{2} a^{2} \arctan \left (a x \right )-2 a x -\arctan \left (a x \right )\right )}{2 x^{2}}-\frac {c \,a^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-4 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-4 i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-2 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right )+2 \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 \sqrt {a^{2} x^{2}+1}}\)	\(412\)

Input:

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

Output:

1/2*c*(c*(a*x-I)*(a*x+I))^(1/2)*arctan(a*x)*(2*x^2*a^2*arctan(a*x)-2*a*x-a 
rctan(a*x))/x^2-1/2*c*a^2*(c*(a*x-I)*(a*x+I))^(1/2)*(3*arctan(a*x)^2*ln(1+ 
(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1 
/2))-6*I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a* 
x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-4*arctan(a*x)*ln(1+I*(1+I*a*x)/( 
a^2*x^2+1)^(1/2))+4*arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+4*I*di 
log(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-4*I*dilog(1-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+(1+I*a*x)/(a^2*x^2+1)^(1/ 
2))+6*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*polylog(3,(1+I*a*x)/(a^2*x 
^2+1)^(1/2)))/(a^2*x^2+1)^(1/2)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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