Integrand size = 24, antiderivative size = 523 \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=-\frac {x \sqrt {c+a^2 c x^2}}{20 a^3}-\frac {9 \sqrt {c+a^2 c x^2} \arctan (a x)}{20 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{10 a^2}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^3}-\frac {3 x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{20 a}-\frac {11 i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{20 a^4 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{15 a^4}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{15 a^2}+\frac {1}{5} x^4 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{2 a^4}+\frac {11 i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}-\frac {11 i c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}-\frac {11 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}}+\frac {11 c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{20 a^4 \sqrt {c+a^2 c x^2}} \] Output:
-1/20*x*(a^2*c*x^2+c)^(1/2)/a^3-9/20*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a^4+1 /10*x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a^2+1/8*x*(a^2*c*x^2+c)^(1/2)*arct an(a*x)^2/a^3-3/20*x^3*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/a-11/20*I*c*(a^2* x^2+1)^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2/a^4/(a^2*c* x^2+c)^(1/2)-2/15*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3/a^4+1/15*x^2*(a^2*c*x^ 2+c)^(1/2)*arctan(a*x)^3/a^2+1/5*x^4*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3+1/2 *c^(1/2)*arctanh(a*c^(1/2)*x/(a^2*c*x^2+c)^(1/2))/a^4+11/20*I*c*(a^2*x^2+1 )^(1/2)*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^4/(a^2*c*x ^2+c)^(1/2)-11/20*I*c*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(2,I*(1+I*a*x)/ (a^2*x^2+1)^(1/2))/a^4/(a^2*c*x^2+c)^(1/2)-11/20*c*(a^2*x^2+1)^(1/2)*polyl og(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^4/(a^2*c*x^2+c)^(1/2)+11/20*c*(a^2* x^2+1)^(1/2)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^4/(a^2*c*x^2+c)^(1 /2)
Time = 0.90 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.50 \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\frac {\sqrt {c+a^2 c x^2} \left (\frac {48 \left (10 \coth ^{-1}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-11 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+11 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-11 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-11 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+11 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}-\left (1+a^2 x^2\right )^2 \left (\frac {48 a x}{\left (1+a^2 x^2\right )^2}+32 \arctan (a x)^3 (-1+5 \cos (2 \arctan (a x)))+6 \arctan (a x) (25+36 \cos (2 \arctan (a x))+11 \cos (4 \arctan (a x)))+\arctan (a x)^2 (6 \sin (2 \arctan (a x))-33 \sin (4 \arctan (a x)))\right )\right )}{960 a^4} \] Input:
Integrate[x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]
Output:
(Sqrt[c + a^2*c*x^2]*((48*(10*ArcCoth[(a*x)/Sqrt[1 + a^2*x^2]] - (11*I)*Ar cTan[E^(I*ArcTan[a*x])]*ArcTan[a*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*Arc Tan[a*x]^3*(-1 + 5*Cos[2*ArcTan[a*x]]) + 6*ArcTan[a*x]*(25 + 36*Cos[2*ArcT an[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 )\) |
Input:
Int[x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]
Output:
$Aborted
Time = 4.68 (sec) , antiderivative size = 417, normalized size of antiderivative = 0.80
| method | result | size |
| default | \(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (24 \arctan \left (a x \right )^{3} a^{4} x^{4}-18 a^{3} \arctan \left (a x \right )^{2} x^{3}+8 \arctan \left (a x \right )^{3} a^{2} x^{2}+12 x^{2} a^{2} \arctan \left (a x \right )+15 a \arctan \left (a x \right )^{2} x -16 \arctan \left (a x \right )^{3}-6 a x -54 \arctan \left (a x \right )\right )}{120 a^{4}}+\frac {11 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{120 a^{4} \sqrt {a^{2} x^{2}+1}}+\frac {11 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (-i \arctan \left (a x \right )^{3}+3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{120 a^{4} \sqrt {a^{2} x^{2}+1}}-\frac {i \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{4} \sqrt {a^{2} x^{2}+1}}\) | \(417\) |
Input:
int(x^3*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3,x,method=_RETURNVERBOSE)
Output:
1/120/a^4*(c*(a*x-I)*(a*x+I))^(1/2)*(24*arctan(a*x)^3*a^4*x^4-18*a^3*arcta n(a*x)^2*x^3+8*arctan(a*x)^3*a^2*x^2+12*x^2*a^2*arctan(a*x)+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)*(a*x+I))^( 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)*(a*x+ I))^(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)*(a *x+I))^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*x^2+1)^(1/2)
\[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{3} \arctan \left (a x\right )^{3} \,d x } \] Input:
integrate(x^3*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3,x, algorithm="fricas")
Output:
integral(sqrt(a^2*c*x^2 + c)*x^3*arctan(a*x)^3, x)
\[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int x^{3} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{3}{\left (a x \right )}\, dx \] Input:
integrate(x**3*(a**2*c*x**2+c)**(1/2)*atan(a*x)**3,x)
Output:
Integral(x**3*sqrt(c*(a**2*x**2 + 1))*atan(a*x)**3, x)
\[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{3} \arctan \left (a x\right )^{3} \,d x } \] Input:
integrate(x^3*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3,x, algorithm="maxima")
Output:
integrate(sqrt(a^2*c*x^2 + c)*x^3*arctan(a*x)^3, x)
Exception generated. \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int x^3\,{\mathrm {atan}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c} \,d x \] Input:
int(x^3*atan(a*x)^3*(c + a^2*c*x^2)^(1/2),x)
Output:
int(x^3*atan(a*x)^3*(c + a^2*c*x^2)^(1/2), x)
\[ \int x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\sqrt {c}\, \left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} x^{3}d x \right ) \] Input:
int(x^3*(a^2*c*x^2+c)^(1/2)*atan(a*x)^3,x)
Output:
sqrt(c)*int(sqrt(a**2*x**2 + 1)*atan(a*x)**3*x**3,x)