\(\int \frac {(c+a^2 c x^2)^{3/2} \arctan (a x)^2}{x^4} \, dx\) [322]

Optimal result

Integrand size = 24, antiderivative size = 579 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{x^4} \, dx=-\frac {a^2 c \sqrt {c+a^2 c x^2}}{3 x}-\frac {a c \sqrt {c+a^2 c x^2} \arctan (a x)}{3 x^2}-\frac {a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{x}-\frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{3 x^3}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}-\frac {14 a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {2 i a^3 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {7 i a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {2 a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {2 a^3 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \] Output:

-1/3*a^2*c*(a^2*c*x^2+c)^(1/2)/x-1/3*a*c*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/x 
^2-a^2*c*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/x-1/3*(a^2*c*x^2+c)^(3/2)*arcta 
n(a*x)^2/x^3-2*I*a^3*c^2*(a^2*x^2+1)^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^(1 
/2))*arctan(a*x)^2/(a^2*c*x^2+c)^(1/2)-14/3*a^3*c^2*(a^2*x^2+1)^(1/2)*arct 
an(a*x)*arctanh((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^2+c)^(1/2)+2*I*a 
^3*c^2*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1 
/2))/(a^2*c*x^2+c)^(1/2)-2*I*a^3*c^2*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog 
(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)+7/3*I*a^3*c^2*(a^2*x 
^2+1)^(1/2)*polylog(2,-(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^2+c)^(1/2 
)-7/3*I*a^3*c^2*(a^2*x^2+1)^(1/2)*polylog(2,(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2 
))/(a^2*c*x^2+c)^(1/2)-2*a^3*c^2*(a^2*x^2+1)^(1/2)*polylog(3,-I*(1+I*a*x)/ 
(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)+2*a^3*c^2*(a^2*x^2+1)^(1/2)*polylog 
(3,I*(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 = 4.32 (sec) , antiderivative size = 453, normalized size of antiderivative = 0.78 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}{x^4} \, dx=\frac {a^3 c^2 \sqrt {1+a^2 x^2} \left (8 i \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-24 \left (\frac {\sqrt {1+a^2 x^2} \arctan (a x)^2}{a x}-2 \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )-\arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )+\arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )+2 \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )-2 i \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+2 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+2 i \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-\frac {2 \left (1+a^2 x^2\right )^{3/2} \left (2+4 \arctan (a x)^2-2 \cos (2 \arctan (a x))+\frac {4 i a^3 x^3 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}+\arctan (a x) \left (2 \sin (2 \arctan (a x))+\frac {\left (\log \left (1-e^{i \arctan (a x)}\right )-\log \left (1+e^{i \arctan (a x)}\right )\right ) \left (-3 a x+\sqrt {1+a^2 x^2} \sin (3 \arctan (a x))\right )}{\sqrt {1+a^2 x^2}}\right )\right )}{a^3 x^3}\right )}{24 \sqrt {c+a^2 c x^2}} \] Input:

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

Output:

(a^3*c^2*Sqrt[1 + a^2*x^2]*((8*I)*PolyLog[2, -E^(I*ArcTan[a*x])] - 24*((Sq 
rt[1 + a^2*x^2]*ArcTan[a*x]^2)/(a*x) - 2*ArcTan[a*x]*Log[1 - E^(I*ArcTan[a 
*x])] - ArcTan[a*x]^2*Log[1 - I*E^(I*ArcTan[a*x])] + ArcTan[a*x]^2*Log[1 + 
 I*E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]*Log[1 + E^(I*ArcTan[a*x])] - (2*I)*P 
olyLog[2, -E^(I*ArcTan[a*x])] - (2*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*Arc 
Tan[a*x])] + (2*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + (2*I)*Pol 
yLog[2, E^(I*ArcTan[a*x])] + 2*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] - 2*Poly 
Log[3, I*E^(I*ArcTan[a*x])]) - (2*(1 + a^2*x^2)^(3/2)*(2 + 4*ArcTan[a*x]^2 
 - 2*Cos[2*ArcTan[a*x]] + ((4*I)*a^3*x^3*PolyLog[2, E^(I*ArcTan[a*x])])/(1 
 + a^2*x^2)^(3/2) + ArcTan[a*x]*(2*Sin[2*ArcTan[a*x]] + ((Log[1 - E^(I*Arc 
Tan[a*x])] - Log[1 + E^(I*ArcTan[a*x])])*(-3*a*x + Sqrt[1 + a^2*x^2]*Sin[3 
*ArcTan[a*x]]))/Sqrt[1 + a^2*x^2])))/(a^3*x^3)))/(24*Sqrt[c + a^2*c*x^2])
 

Rubi [A] (verified)

Time = 4.65 (sec) , antiderivative size = 575, normalized size of antiderivative = 0.99, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.792, Rules used = {5485, 5479, 5481, 242, 5485, 5425, 5423, 3042, 4669, 3011, 2720, 5479, 5493, 5489, 5497, 242, 5493, 5489, 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.

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5479

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

\(\Big \downarrow \) 5481

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

\(\Big \downarrow \) 242

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5423

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5479

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

\(\Big \downarrow \) 5493

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

\(\Big \downarrow \) 5489

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

\(\Big \downarrow \) 5497

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

\(\Big \downarrow \) 242

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

\(\Big \downarrow \) 5493

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

\(\Big \downarrow \) 5489

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

\(\Big \downarrow \) 7143

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

Input:

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

Output:

c*(-1/3*((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(c*x^3) + (2*a*(-((a*Sqrt[c 
+ a^2*c*x^2])/x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/x^2 - c*(-1/2*(a*Sqrt 
[c + a^2*c*x^2])/(c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*c*x^2) - (a^ 
2*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x 
]] + I*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])] - I*PolyLog[2, Sqrt[ 
1 + I*a*x]/Sqrt[1 - I*a*x]]))/(2*Sqrt[c + a^2*c*x^2]))))/3) + a^2*c*(c*(-( 
(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(c*x)) + (2*a*Sqrt[1 + a^2*x^2]*(-2*Ar 
cTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]] + I*PolyLog[2, -(Sqrt[1 
 + I*a*x]/Sqrt[1 - I*a*x])] - I*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x] 
]))/Sqrt[c + a^2*c*x^2]) + (a*c*Sqrt[1 + a^2*x^2]*((-2*I)*ArcTan[E^(I*ArcT 
an[a*x])]*ArcTan[a*x]^2 + 2*(I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x 
])] - PolyLog[3, (-I)*E^(I*ArcTan[a*x])]) - 2*(I*ArcTan[a*x]*PolyLog[2, I* 
E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])])))/Sqrt[c + a^2*c*x^2 
])
 

Defintions of rubi rules used

rule 242

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ 
(m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x 
] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
 

rule 2720

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

rule 3011

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

rule 3042

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

rule 4669

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

rule 5423

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

rule 5425

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

rule 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 5481

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

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 5489

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_ 
Symbol] :> Simp[(-2/Sqrt[d])*(a + b*ArcTan[c*x])*ArcTanh[Sqrt[1 + I*c*x]/Sq 
rt[1 - I*c*x]], x] + (Simp[I*(b/Sqrt[d])*PolyLog[2, -Sqrt[1 + I*c*x]/Sqrt[1 
 - I*c*x]], x] - Simp[I*(b/Sqrt[d])*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c 
*x]], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && 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 = 2.67 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.59

method	result	size
default	\(-\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (4 \arctan \left (a x \right )^{2} x^{2} a^{2}+a^{2} x^{2}+\arctan \left (a x \right ) a x +\arctan \left (a x \right )^{2}\right )}{3 x^{3}}-\frac {c \,a^{3} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+7 \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-7 i \operatorname {dilog}\left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-7 i \operatorname {dilog}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{3 \sqrt {a^{2} x^{2}+1}}\)	\(343\)

Input:

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

Output:

-1/3*c*(c*(a*x-I)*(a*x+I))^(1/2)*(4*arctan(a*x)^2*x^2*a^2+a^2*x^2+arctan(a 
*x)*a*x+arctan(a*x)^2)/x^3-1/3*c*a^3*(c*(a*x-I)*(a*x+I))^(1/2)*(3*arctan(a 
*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*arctan(a*x)^2*ln(1-I*(1+I*a*x) 
/(a^2*x^2+1)^(1/2))-6*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/ 
2))+6*I*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+7*arctan(a*x) 
*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))-7*I*dilog(1+(1+I*a*x)/(a^2*x^2+1)^(1/2) 
)-7*I*dilog((1+I*a*x)/(a^2*x^2+1)^(1/2))+6*polylog(3,-I*(1+I*a*x)/(a^2*x^2 
+1)^(1/2))-6*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^2*x^2+1)^(1/2)
 

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

sqrt(c)*c*(int((sqrt(a**2*x**2 + 1)*atan(a*x)**2)/x**4,x) + int((sqrt(a**2 
*x**2 + 1)*atan(a*x)**2)/x**2,x)*a**2)