\(\int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx\) [308]

Optimal result

Integrand size = 24, antiderivative size = 436 \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\frac {x \sqrt {c+a^2 c x^2}}{12 a^2}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{6 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{6 a^3}-\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}} \] Output:

1/12*x*(a^2*c*x^2+c)^(1/2)/a^2+1/12*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a^3-1/ 
6*x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a+1/8*x*(a^2*c*x^2+c)^(1/2)*arctan(a 
*x)^2/a^2+1/4*x^3*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2+1/4*I*c*(a^2*x^2+1)^(1 
/2)*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2/a^3/(a^2*c*x^2+c)^(1 
/2)-1/6*c^(1/2)*arctanh(a*c^(1/2)*x/(a^2*c*x^2+c)^(1/2))/a^3-1/4*I*c*(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^3/(a^ 
2*c*x^2+c)^(1/2)+1/4*I*c*(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^3/(a^2*c*x^2+c)^(1/2)+1/4*c*(a^2*x^2+1)^(1/2)*poly 
log(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^3/(a^2*c*x^2+c)^(1/2)-1/4*c*(a^2*x 
^2+1)^(1/2)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^3/(a^2*c*x^2+c)^(1/ 
2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.91 (sec) , antiderivative size = 267, normalized size of antiderivative = 0.61 \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=\frac {\sqrt {c+a^2 c x^2} \left (8 \left (-2 \coth ^{-1}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+3 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-3 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+3 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )+3 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )-3 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )+\left (1+a^2 x^2\right )^{3/2} \left (\arctan (a x) \left (2+6 \sqrt {1+a^2 x^2} \cos (3 \arctan (a x))\right )-3 \arctan (a x)^2 \left (-7 a x+\sqrt {1+a^2 x^2} \sin (3 \arctan (a x))\right )+2 \left (a x+\sqrt {1+a^2 x^2} \sin (3 \arctan (a x))\right )\right )\right )}{96 a^3 \sqrt {1+a^2 x^2}} \] Input:

Integrate[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]
 

Output:

(Sqrt[c + a^2*c*x^2]*(8*(-2*ArcCoth[(a*x)/Sqrt[1 + a^2*x^2]] + (3*I)*ArcTa 
n[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - (3*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^( 
I*ArcTan[a*x])] + (3*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] + 3*Po 
lyLog[3, (-I)*E^(I*ArcTan[a*x])] - 3*PolyLog[3, I*E^(I*ArcTan[a*x])]) + (1 
 + a^2*x^2)^(3/2)*(ArcTan[a*x]*(2 + 6*Sqrt[1 + a^2*x^2]*Cos[3*ArcTan[a*x]] 
) - 3*ArcTan[a*x]^2*(-7*a*x + Sqrt[1 + a^2*x^2]*Sin[3*ArcTan[a*x]]) + 2*(a 
*x + Sqrt[1 + a^2*x^2]*Sin[3*ArcTan[a*x]]))))/(96*a^3*Sqrt[1 + a^2*x^2])
 

Rubi [A] (verified)

Time = 5.94 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.62, number of steps used = 26, number of rules used = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.042, Rules used = {5485, 5487, 5425, 5423, 3042, 4669, 3011, 2720, 5465, 224, 219, 5487, 262, 224, 219, 5425, 5423, 3042, 4669, 3011, 2720, 5465, 224, 219, 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.

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5487

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5423

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5487

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5423

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4669

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

\(\Big \downarrow \) 3011

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

\(\Big \downarrow \) 2720

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 7143

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

Input:

Int[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]
 

Output:

a^2*c*((x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(4*a^2*c) - ((x^2*Sqrt[c + 
a^2*c*x^2]*ArcTan[a*x])/(3*a^2*c) - ((x*Sqrt[c + a^2*c*x^2])/(2*a^2*c) - A 
rcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(2*a^3*Sqrt[c]))/(3*a) - (2*((Sq 
rt[c + a^2*c*x^2]*ArcTan[a*x])/(a^2*c) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^ 
2*c*x^2]]/(a^2*Sqrt[c])))/(3*a^2))/(2*a) - (3*((x*Sqrt[c + a^2*c*x^2]*ArcT 
an[a*x]^2)/(2*a^2*c) - ((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^2*c) - ArcTan 
h[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^2*Sqrt[c]))/a - (Sqrt[1 + a^2*x^2] 
*((-2*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 2*(I*ArcTan[a*x]*PolyLo 
g[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 
])])))/(2*a^3*Sqrt[c + a^2*c*x^2])))/(4*a^2)) + c*((x*Sqrt[c + a^2*c*x^2]* 
ArcTan[a*x]^2)/(2*a^2*c) - ((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^2*c) - Ar 
cTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^2*Sqrt[c]))/a - (Sqrt[1 + a^2* 
x^2]*((-2*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 2*(I*ArcTan[a*x]*Po 
lyLog[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])])))/(2*a^3*Sqrt[c + a^2*c*x^2]))
 

Defintions of rubi rules used

rule 219

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

rule 224

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

rule 262

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

rule 2720

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

rule 3011

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

rule 3042

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

rule 4669

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

rule 5423

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

rule 5425

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

rule 5465

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

rule 5485

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

rule 5487

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

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 = 2.12 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.69

method	result	size
default	\(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (6 a^{3} \arctan \left (a x \right )^{2} x^{3}-4 x^{2} a^{2} \arctan \left (a x \right )+3 a \arctan \left (a x \right )^{2} x +2 a x +2 \arctan \left (a x \right )\right )}{24 a^{3}}+\frac {i \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (3 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+8 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{24 a^{3} \sqrt {a^{2} x^{2}+1}}\)	\(302\)

Input:

int(x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/24/a^3*(c*(a*x-I)*(a*x+I))^(1/2)*(6*a^3*arctan(a*x)^2*x^3-4*x^2*a^2*arct 
an(a*x)+3*a*arctan(a*x)^2*x+2*a*x+2*arctan(a*x))+1/24*I*(c*(a*x-I)*(a*x+I) 
)^(1/2)*(3*I*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*arctan( 
a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*arctan(a*x)*polylog(2,I*(1+I* 
a*x)/(a^2*x^2+1)^(1/2))-6*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^( 
1/2))+6*I*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*polylog(3,-I*(1+I*a 
*x)/(a^2*x^2+1)^(1/2))+8*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^3/(a^2*x^2 
+1)^(1/2)
 

Fricas [F]

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

integrate(x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2,x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2,x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2,x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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