\(\int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx\) [307]

Optimal result

Integrand size = 24, antiderivative size = 385 \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=-\frac {11 \sqrt {c+a^2 c x^2}}{60 a^4}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{30 a^4 c}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{12 a^3}-\frac {x^3 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {11 i c \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{30 a^4 \sqrt {c+a^2 c x^2}}+\frac {11 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {c+a^2 c x^2}}-\frac {11 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{60 a^4 \sqrt {c+a^2 c x^2}} \] Output:

-11/60*(a^2*c*x^2+c)^(1/2)/a^4+1/30*(a^2*c*x^2+c)^(3/2)/a^4/c+1/12*x*(a^2* 
c*x^2+c)^(1/2)*arctan(a*x)/a^3-1/10*x^3*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a- 
2/15*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/a^4+1/15*x^2*(a^2*c*x^2+c)^(1/2)*ar 
ctan(a*x)^2/a^2+1/5*x^4*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2-11/30*I*c*(a^2*x 
^2+1)^(1/2)*arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/a^4/(a^2*c 
*x^2+c)^(1/2)+11/60*I*c*(a^2*x^2+1)^(1/2)*polylog(2,-I*(1+I*a*x)^(1/2)/(1- 
I*a*x)^(1/2))/a^4/(a^2*c*x^2+c)^(1/2)-11/60*I*c*(a^2*x^2+1)^(1/2)*polylog( 
2,I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/a^4/(a^2*c*x^2+c)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 0.91 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.94 \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2 \, dx=-\frac {\left (1+a^2 x^2\right )^2 \sqrt {c \left (1+a^2 x^2\right )} \left (50-32 \arctan (a x)^2+72 \cos (2 \arctan (a x))+160 \arctan (a x)^2 \cos (2 \arctan (a x))+22 \cos (4 \arctan (a x))-\frac {110 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-55 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )-11 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )+\frac {110 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+55 \arctan (a x) \cos (3 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )+11 \arctan (a x) \cos (5 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )-\frac {176 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{5/2}}+\frac {176 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{5/2}}+4 \arctan (a x) \sin (2 \arctan (a x))-22 \arctan (a x) \sin (4 \arctan (a x))\right )}{960 a^4} \] Input:

Integrate[x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]
 

Output:

-1/960*((1 + a^2*x^2)^2*Sqrt[c*(1 + a^2*x^2)]*(50 - 32*ArcTan[a*x]^2 + 72* 
Cos[2*ArcTan[a*x]] + 160*ArcTan[a*x]^2*Cos[2*ArcTan[a*x]] + 22*Cos[4*ArcTa 
n[a*x]] - (110*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] 
 - 55*ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] - 11*Arc 
Tan[a*x]*Cos[5*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (110*ArcTan[a*x 
]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + 55*ArcTan[a*x]*Cos[3*A 
rcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] + 11*ArcTan[a*x]*Cos[5*ArcTan[a*x 
]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((176*I)*PolyLog[2, (-I)*E^(I*ArcTan[a*x 
])])/(1 + a^2*x^2)^(5/2) + ((176*I)*PolyLog[2, I*E^(I*ArcTan[a*x])])/(1 + 
a^2*x^2)^(5/2) + 4*ArcTan[a*x]*Sin[2*ArcTan[a*x]] - 22*ArcTan[a*x]*Sin[4*A 
rcTan[a*x]]))/a^4
 

Rubi [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1189\) vs. \(2(385)=770\).

Time = 5.97 (sec) , antiderivative size = 1189, normalized size of antiderivative = 3.09, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.792, Rules used = {5485, 5487, 5465, 5425, 5421, 5487, 241, 243, 53, 2009, 5425, 5421, 5465, 5425, 5421, 5487, 241, 5425, 5421}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 5485

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

\(\Big \downarrow \) 5487

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

\(\Big \downarrow \) 5465

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

\(\Big \downarrow \) 5425

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

\(\Big \downarrow \) 5421

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

\(\Big \downarrow \) 5487

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 243

\(\displaystyle a^2 c \left (-\frac {4 \left (-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}-\frac {2 \left (-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )}{5 a}+\frac {x^4 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )\)

\(\Big \downarrow \) 53

\(\displaystyle a^2 c \left (-\frac {4 \left (-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}-\frac {2 \left (-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {\int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2}{8 a}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )}{5 a}+\frac {x^4 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle a^2 c \left (-\frac {4 \left (-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}-\frac {2 \left (-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}\right )}{5 a}+\frac {x^4 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )\)

\(\Big \downarrow \) 5425

\(\displaystyle a^2 c \left (-\frac {4 \left (-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}-\frac {2 \left (-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}\right )}{5 a}+\frac {x^4 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )\)

\(\Big \downarrow \) 5421

\(\displaystyle a^2 c \left (-\frac {4 \left (-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}-\frac {2 \left (-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}\right )}{5 a}+\frac {x^4 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {2 \left (-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )\)

\(\Big \downarrow \) 5465

\(\displaystyle a^2 c \left (-\frac {4 \left (-\frac {2 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}-\frac {2 \left (-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}\right )}{5 a}+\frac {x^4 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {2 \left (-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )\)

\(\Big \downarrow \) 5425

\(\displaystyle a^2 c \left (-\frac {4 \left (-\frac {2 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}-\frac {2 \left (-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}\right )}{5 a}+\frac {x^4 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {2 \left (-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )\)

\(\Big \downarrow \) 5421

\(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\)

\(\Big \downarrow \) 5487

\(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \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 )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{3 a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\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 )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \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 )}{3 a^2}\right )\)

\(\Big \downarrow \) 241

\(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \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 )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\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 )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \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 )}{3 a^2}\right )\)

\(\Big \downarrow \) 5425

\(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \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 )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\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 )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \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 )}{3 a^2}\right )\)

\(\Big \downarrow \) 5421

\(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^4}{5 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x) x^3}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\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 )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\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 )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \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 )}{3 a^2}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\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 )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \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 )}{3 a^2}\right )\)

Input:

Int[x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2,x]
 

Output:

c*((x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*a^2*c) - (2*(-1/2*Sqrt[c + a 
^2*c*x^2]/(a^3*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*a^2*c) - (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))/(2*a^2*Sqrt[c + a^2*c*x^2])) 
)/(3*a) - (2*((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])))/(3*a^2)) 
 + a^2*c*((x^4*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(5*a^2*c) - (2*(-1/8*((- 
2*Sqrt[c + a^2*c*x^2])/(a^4*c) + (2*(c + a^2*c*x^2)^(3/2))/(3*a^4*c^2))/a 
+ (x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(4*a^2*c) - (3*(-1/2*Sqrt[c + a^2* 
c*x^2]/(a^3*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*a^2*c) - (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))/(2*a^2*Sqrt[c + a^2*c*x^2])))/( 
4*a^2)))/(5*a) - (4*((x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*a^2*c) - ( 
2*(-1/2*Sqrt[c + a^2*c*x^2]/(a^3*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/ 
(2*a^2*c) - (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*...
 

Defintions of rubi rules used

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 241

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

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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

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 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]
 

Maple [A] (verified)

Time = 2.02 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.61

method	result	size
default	\(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (12 a^{4} \arctan \left (a x \right )^{2} x^{4}-6 \arctan \left (a x \right ) x^{3} a^{3}+4 \arctan \left (a x \right )^{2} x^{2} a^{2}+2 a^{2} x^{2}+5 \arctan \left (a x \right ) a x -8 \arctan \left (a x \right )^{2}-9\right )}{60 a^{4}}-\frac {11 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{60 a^{4} \sqrt {a^{2} x^{2}+1}}\)	\(235\)

Input:

int(x^3*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/60/a^4*(c*(a*x-I)*(a*x+I))^(1/2)*(12*a^4*arctan(a*x)^2*x^4-6*arctan(a*x) 
*x^3*a^3+4*arctan(a*x)^2*x^2*a^2+2*a^2*x^2+5*arctan(a*x)*a*x-8*arctan(a*x) 
^2-9)-11/60*(c*(a*x-I)*(a*x+I))^(1/2)*(arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x 
^2+1)^(1/2))-arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+I*dilog(1-I*( 
1+I*a*x)/(a^2*x^2+1)^(1/2))-I*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^4/ 
(a^2*x^2+1)^(1/2)
                                                                                    
                                                                                    
 

Fricas [F]

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

integrate(x^3*(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^3*arctan(a*x)^2, x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(x^3*(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^3*arctan(a*x)^2, x)
 

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

int(x^3*(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**3,x)