Integrand size = 24, antiderivative size = 523 \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=-\frac {x \sqrt {c+a^2 c x^2}}{20 a^3}-\frac {9 \sqrt {c+a^2 c x^2} \arctan (a x)}{20 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a^2}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^3}-\frac {3 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{20 a}-\frac {11 i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{20 a^4 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{2 a^4}+\frac {11 i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}-\frac {11 i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}-\frac {11 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {11 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}} \]
1/2*arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))*c^(1/2)/a^4-11/20*I*c*arctan( (1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^ 2+c)^(1/2)+11/20*I*c*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)) *(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-11/20*I*c*arctan(a*x)*polylog(2 ,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)- 11/20*c*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a ^2*c*x^2+c)^(1/2)+11/20*c*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^ 2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-1/20*x*(a^2*c*x^2+c)^(1/2)/a^3-9/20*arc tan(a*x)*(a^2*c*x^2+c)^(1/2)/a^4+1/10*x^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/ a^2+1/8*x*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/a^3-3/20*x^3*arctan(a*x)^2*(a^ 2*c*x^2+c)^(1/2)/a-2/15*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/a^4+1/15*x^2*arc tan(a*x)^3*(a^2*c*x^2+c)^(1/2)/a^2+1/5*x^4*arctan(a*x)^3*(a^2*c*x^2+c)^(1/ 2)
Time = 1.01 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.50 \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\frac {\sqrt {c+a^2 c x^2} \left (\frac {48 \left (-11 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+10 \text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+11 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-11 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-11 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+11 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}-\left (1+a^2 x^2\right )^2 \left (\frac {48 a x}{\left (1+a^2 x^2\right )^2}+32 \arctan (a x)^3 (-1+5 \cos (2 \arctan (a x)))+6 \arctan (a x) (25+36 \cos (2 \arctan (a x))+11 \cos (4 \arctan (a x)))+\arctan (a x)^2 (6 \sin (2 \arctan (a x))-33 \sin (4 \arctan (a x)))\right )\right )}{960 a^4} \]
(Sqrt[c + a^2*c*x^2]*((48*((-11*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 10*ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (11*I)*ArcTan[a*x]*PolyLog[2, (-I )*E^(I*ArcTan[a*x])] - (11*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 11*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 11*PolyLog[3, I*E^(I*ArcTan[a*x] )]))/Sqrt[1 + a^2*x^2] - (1 + a^2*x^2)^2*((48*a*x)/(1 + a^2*x^2)^2 + 32*Ar cTan[a*x]^3*(-1 + 5*Cos[2*ArcTan[a*x]]) + 6*ArcTan[a*x]*(25 + 36*Cos[2*Arc Tan[a*x]] + 11*Cos[4*ArcTan[a*x]]) + ArcTan[a*x]^2*(6*Sin[2*ArcTan[a*x]] - 33*Sin[4*ArcTan[a*x]]))))/(960*a^4)
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.
| \(\displaystyle \int x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c} \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \int \frac {x^5 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {x^3 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle c \left (-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \int \frac {x \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )+a^2 c \left (-\frac {3 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle c \left (-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )+a^2 c \left (-\frac {3 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )+a^2 c \left (-\frac {3 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle c \left (-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )+a^2 c \left (-\frac {3 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \left (-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )+a^2 c \left (-\frac {3 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle a^2 c \left (-\frac {3 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle a^2 c \left (-\frac {3 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle a^2 c \left (-\frac {3 \int \frac {x^4 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{5 a}-\frac {4 \int \frac {x^3 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{5 a^2}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle a^2 c \left (-\frac {4 \left (-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \int \frac {x \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}-\frac {3 \left (-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}+\frac {x^3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )}{5 a}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle a^2 c \left (-\frac {4 \left (-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \int \frac {x \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}-\frac {3 \left (-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}+\frac {x^3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )}{5 a}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle a^2 c \left (-\frac {4 \left (-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \int \frac {x \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}-\frac {3 \left (-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}+\frac {x^3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )}{5 a}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c \left (-\frac {4 \left (-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \int \frac {x \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}-\frac {3 \left (-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}+\frac {x^3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )}{5 a}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle a^2 c \left (-\frac {4 \left (-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \int \frac {x \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}-\frac {3 \left (-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}+\frac {x^3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )}{5 a}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle a^2 c \left (-\frac {4 \left (-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \int \frac {x \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}-\frac {3 \left (-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}+\frac {x^3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )}{5 a}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle a^2 c \left (-\frac {4 \left (-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \int \frac {x \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{5 a^2}-\frac {3 \left (-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}+\frac {x^3 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )}{5 a}+\frac {x^4 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{5 a^2 c}\right )+c \left (-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^4}{5 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\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)^3}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{3 a^2 c}-\frac {\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}{\sqrt {a^2 c x^2+c}}dx}{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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^4}{5 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\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)^3}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{3 a^2 c}-\frac {\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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^4}{5 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\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)^3}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{3 a^2 c}-\frac {\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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \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)^3 x^4}{5 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\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)^3}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{3 a^2 c}-\frac {\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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^4}{5 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\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)^3}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{5 a^2}\right ) a^2+c \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{3 a^2 c}-\frac {\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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^4}{5 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\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)^3}{3 a^2 c}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{a^2 \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)^3}{3 a^2 c}-\frac {\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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^4}{5 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\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)^3}{3 a^2 c}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-2 \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)\right )}{a^2 \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)^3}{3 a^2 c}-\frac {\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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^4}{5 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\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)^3}{3 a^2 c}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )-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 )\right )}{a^2 \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)^3}{3 a^2 c}-\frac {\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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^4}{5 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2 x^3}{4 a^2 c}-\frac {\int \frac {x^3 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {3 \int \frac {x^2 \arctan (a x)^2}{\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)^3}{3 a^2 c}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \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)^3}{3 a^2 c}-\frac {\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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \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)^3 x^4}{5 a^2 c}-\frac {3 \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 {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{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 {\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}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\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}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \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)^3}{3 a^2 c}-\frac {\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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^4}{5 a^2 c}-\frac {3 \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 {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}}{3 a}-\frac {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{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 {\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}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\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}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \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)^3}{3 a^2 c}-\frac {\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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^4}{5 a^2 c}-\frac {3 \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 {\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 {2 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{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 {\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}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\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}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \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)^3}{3 a^2 c}-\frac {\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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3 x^4}{5 a^2 c}-\frac {3 \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 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{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 {\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}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\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}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \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)^3}{3 a^2 c}-\frac {\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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \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)^3 x^4}{5 a^2 c}-\frac {3 \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 \int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{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 {\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}}\right )}{4 a^2}\right )}{5 a}-\frac {4 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{3 a^2 c}-\frac {\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 a^2 c}-\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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \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)^3}{3 a^2 c}-\frac {\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}}}{a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \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 )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )\) |
3.5.12.3.1 Defintions of rubi rules used
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])
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]
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]
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]]]
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]
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]
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]
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]
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]
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]))
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]
Time = 4.35 (sec) , antiderivative size = 417, normalized size of antiderivative = 0.80
| method | result | size |
| default | \(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (24 a^{4} \arctan \left (a x \right )^{3} x^{4}-18 a^{3} \arctan \left (a x \right )^{2} x^{3}+8 \arctan \left (a x \right )^{3} x^{2} a^{2}+12 a^{2} \arctan \left (a x \right ) x^{2}+15 a \arctan \left (a x \right )^{2} x -16 \arctan \left (a x \right )^{3}-6 a x -54 \arctan \left (a x \right )\right )}{120 a^{4}}+\frac {11 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{120 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {11 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{120 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {i \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2} x^{2}+1}}\) | \(417\) |
1/120/a^4*(c*(a*x-I)*(I+a*x))^(1/2)*(24*a^4*arctan(a*x)^3*x^4-18*a^3*arcta n(a*x)^2*x^3+8*arctan(a*x)^3*x^2*a^2+12*a^2*arctan(a*x)*x^2+15*a*arctan(a* x)^2*x-16*arctan(a*x)^3-6*a*x-54*arctan(a*x))+11/120*(c*(a*x-I)*(I+a*x))^( 1/2)*(I*arctan(a*x)^3-3*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*polylog(3,-I*( 1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^4/(a^2*x^2+1)^(1/2)-11/120*(c*(a*x-I)*(I+a* x))^(1/2)*(I*arctan(a*x)^3-3*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*polylog(3, I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^4/(a^2*x^2+1)^(1/2)-I/a^4*(c*(a*x-I)*(I+ a*x))^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*x^2+1)^(1/2)
\[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{3} \arctan \left (a x\right )^{3} \,d x } \]
\[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int x^{3} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{3}{\left (a x \right )}\, dx \]
\[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{3} \arctan \left (a x\right )^{3} \,d x } \]
Exception generated. \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\text {Exception raised: TypeError} \]
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
Timed out. \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int x^3\,{\mathrm {atan}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c} \,d x \]