Integrand size = 22, antiderivative size = 357 \[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\frac {c \sqrt {c+a^2 c x^2}}{16 a^3}+\frac {\left (c+a^2 c x^2\right )^{3/2}}{72 a^3}-\frac {\left (c+a^2 c x^2\right )^{5/2}}{30 a^3 c}+\frac {c x \sqrt {c+a^2 c x^2} \arctan (a x)}{16 a^2}+\frac {7}{24} c x^3 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {1}{6} a^2 c x^5 \sqrt {c+a^2 c x^2} \arctan (a x)+\frac {i c^2 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{16 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{16 a^3 \sqrt {c+a^2 c x^2}} \]
1/72*(a^2*c*x^2+c)^(3/2)/a^3-1/30*(a^2*c*x^2+c)^(5/2)/a^3/c+1/8*I*c^2*arct an(a*x)*arctan((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2 *c*x^2+c)^(1/2)-1/16*I*c^2*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*( a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)+1/16*I*c^2*polylog(2,I*(1+I*a*x)^ (1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/a^3/(a^2*c*x^2+c)^(1/2)+1/16*c*(a ^2*c*x^2+c)^(1/2)/a^3+1/16*c*x*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^2+7/24*c* x^3*arctan(a*x)*(a^2*c*x^2+c)^(1/2)+1/6*a^2*c*x^5*arctan(a*x)*(a^2*c*x^2+c )^(1/2)
Time = 5.20 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.61 \[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\frac {c \sqrt {c+a^2 c x^2} \left (\frac {3}{4} \left (1+a^2 x^2\right )^{5/2}+\frac {55}{8} \left (1+a^2 x^2\right )^3 \cos (3 \arctan (a x))-\frac {45}{8} \left (1+a^2 x^2\right )^3 \cos (5 \arctan (a x))-90 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+90 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\frac {15}{2} \left (1+a^2 x^2\right )^2 \left (-\frac {2}{\sqrt {1+a^2 x^2}}-6 \cos (3 \arctan (a x))+3 \arctan (a x) \left (-\frac {14 a x}{\sqrt {1+a^2 x^2}}+3 \log \left (1-i e^{i \arctan (a x)}\right )+4 \cos (2 \arctan (a x)) \left (\log \left (1-i e^{i \arctan (a x)}\right )-\log \left (1+i e^{i \arctan (a x)}\right )\right )+\cos (4 \arctan (a x)) \left (\log \left (1-i e^{i \arctan (a x)}\right )-\log \left (1+i e^{i \arctan (a x)}\right )\right )-3 \log \left (1+i e^{i \arctan (a x)}\right )+2 \sin (3 \arctan (a x))\right )\right )+\frac {15}{16} \left (1+a^2 x^2\right )^3 \arctan (a x) \left (\frac {156 a x}{\sqrt {1+a^2 x^2}}+30 \log \left (1-i e^{i \arctan (a x)}\right )+3 \cos (6 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )+45 \cos (2 \arctan (a x)) \left (\log \left (1-i e^{i \arctan (a x)}\right )-\log \left (1+i e^{i \arctan (a x)}\right )\right )+18 \cos (4 \arctan (a x)) \left (\log \left (1-i e^{i \arctan (a x)}\right )-\log \left (1+i e^{i \arctan (a x)}\right )\right )-30 \log \left (1+i e^{i \arctan (a x)}\right )-3 \cos (6 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )-94 \sin (3 \arctan (a x))+6 \sin (5 \arctan (a x))\right )\right )}{1440 a^3 \sqrt {1+a^2 x^2}} \]
(c*Sqrt[c + a^2*c*x^2]*((3*(1 + a^2*x^2)^(5/2))/4 + (55*(1 + a^2*x^2)^3*Co s[3*ArcTan[a*x]])/8 - (45*(1 + a^2*x^2)^3*Cos[5*ArcTan[a*x]])/8 - (90*I)*P olyLog[2, (-I)*E^(I*ArcTan[a*x])] + (90*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] - (15*(1 + a^2*x^2)^2*(-2/Sqrt[1 + a^2*x^2] - 6*Cos[3*ArcTan[a*x]] + 3*Ar cTan[a*x]*((-14*a*x)/Sqrt[1 + a^2*x^2] + 3*Log[1 - I*E^(I*ArcTan[a*x])] + 4*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan [a*x])]) + Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^ (I*ArcTan[a*x])]) - 3*Log[1 + I*E^(I*ArcTan[a*x])] + 2*Sin[3*ArcTan[a*x]]) ))/2 + (15*(1 + a^2*x^2)^3*ArcTan[a*x]*((156*a*x)/Sqrt[1 + a^2*x^2] + 30*L og[1 - I*E^(I*ArcTan[a*x])] + 3*Cos[6*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a *x])] + 45*Cos[2*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^ (I*ArcTan[a*x])]) + 18*Cos[4*ArcTan[a*x]]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])]) - 30*Log[1 + I*E^(I*ArcTan[a*x])] - 3*Cos[6* ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - 94*Sin[3*ArcTan[a*x]] + 6*Sin[ 5*ArcTan[a*x]]))/16))/(1440*a^3*Sqrt[1 + a^2*x^2])
Time = 2.36 (sec) , antiderivative size = 693, normalized size of antiderivative = 1.94, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {5485, 5481, 243, 53, 2009, 5487, 241, 243, 53, 2009, 5425, 5421, 5487, 241, 5425, 5421}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^2 \arctan (a x) \left (a^2 c x^2+c\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle c \int x^2 \sqrt {a^2 c x^2+c} \arctan (a x)dx+a^2 c \int x^4 \sqrt {a^2 c x^2+c} \arctan (a x)dx\) |
| \(\Big \downarrow \) 5481 |
| \(\displaystyle c \left (\frac {1}{4} c \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{4} a c \int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}\right )+a^2 c \left (\frac {1}{6} c \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{6} a c \int \frac {x^5}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c \left (\frac {1}{4} c \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{8} a c \int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2+\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}\right )+a^2 c \left (\frac {1}{6} c \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{12} a c \int \frac {x^4}{\sqrt {a^2 c x^2+c}}dx^2+\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle c \left (\frac {1}{4} c \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{8} a c \int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2+\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}\right )+a^2 c \left (\frac {1}{6} c \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx-\frac {1}{12} a c \int \left (\frac {\left (a^2 c x^2+c\right )^{3/2}}{a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}+\frac {1}{a^4 \sqrt {a^2 c x^2+c}}\right )dx^2+\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^2 c \left (\frac {1}{6} c \int \frac {x^4 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{12} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}\right )\right )+c \left (\frac {1}{4} c \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{8} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}\right )\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle a^2 c \left (\frac {1}{6} c \left (-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{12} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}\right )\right )+c \left (\frac {1}{4} c \left (-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )+\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{8} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}\right )\right )\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle a^2 c \left (\frac {1}{6} c \left (-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {\int \frac {x^3}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{12} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}\right )\right )+c \left (\frac {1}{4} c \left (-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )+\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{8} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}\right )\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle a^2 c \left (\frac {1}{6} c \left (-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {\int \frac {x^2}{\sqrt {a^2 c x^2+c}}dx^2}{8 a}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{12} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}\right )\right )+c \left (\frac {1}{4} c \left (-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )+\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{8} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}\right )\right )\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle a^2 c \left (\frac {1}{6} c \left (-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {\int \left (\frac {\sqrt {a^2 c x^2+c}}{a^2 c}-\frac {1}{a^2 \sqrt {a^2 c x^2+c}}\right )dx^2}{8 a}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{12} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}\right )\right )+c \left (\frac {1}{4} c \left (-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )+\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{8} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a^2 c \left (\frac {1}{6} c \left (-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}\right )+\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{12} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}\right )\right )+c \left (\frac {1}{4} c \left (-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )+\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{8} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle a^2 c \left (\frac {1}{6} c \left (-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}\right )+\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{12} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}\right )\right )+c \left (\frac {1}{4} c \left (-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )+\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{8} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}\right )\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle a^2 c \left (\frac {1}{6} c \left (-\frac {3 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}\right )+\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{12} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}\right )\right )+c \left (\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{8} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}\right )+\frac {1}{4} c \left (-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle a^2 c \left (\frac {1}{6} c \left (-\frac {3 \left (-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )}{4 a^2}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}\right )+\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{12} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}\right )\right )+c \left (\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{8} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}\right )+\frac {1}{4} c \left (-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )\right )\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle a^2 c \left (\frac {1}{6} c \left (-\frac {3 \left (-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}\right )+\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{12} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}\right )\right )+c \left (\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{8} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}\right )+\frac {1}{4} c \left (-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle a^2 c \left (\frac {1}{6} c \left (-\frac {3 \left (-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}+\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}\right )+\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{12} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}\right )\right )+c \left (\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{8} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}\right )+\frac {1}{4} c \left (-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle c \left (\frac {1}{4} x^3 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{8} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}\right )+\frac {1}{4} c \left (-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )\right )+a^2 c \left (\frac {1}{6} x^5 \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {1}{12} a c \left (\frac {2 \left (a^2 c x^2+c\right )^{5/2}}{5 a^6 c^3}-\frac {4 \left (a^2 c x^2+c\right )^{3/2}}{3 a^6 c^2}+\frac {2 \sqrt {a^2 c x^2+c}}{a^6 c}\right )+\frac {1}{6} c \left (\frac {x^3 \arctan (a x) \sqrt {a^2 c x^2+c}}{4 a^2 c}-\frac {\frac {2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^4 c^2}-\frac {2 \sqrt {a^2 c x^2+c}}{a^4 c}}{8 a}-\frac {3 \left (-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x) \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{4 a^2}\right )\right )\) |
c*(-1/8*(a*c*((-2*Sqrt[c + a^2*c*x^2])/(a^4*c) + (2*(c + a^2*c*x^2)^(3/2)) /(3*a^4*c^2))) + (x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/4 + (c*(-1/2*Sqrt[c + a^2*c*x^2]/(a^3*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*a^2*c) - (S qrt[1 + a^2*x^2]*(((-2*I)*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a* x]])/a + (I*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a - (I*Pol yLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a))/(2*a^2*Sqrt[c + a^2*c*x^ 2])))/4) + a^2*c*(-1/12*(a*c*((2*Sqrt[c + a^2*c*x^2])/(a^6*c) - (4*(c + a^ 2*c*x^2)^(3/2))/(3*a^6*c^2) + (2*(c + a^2*c*x^2)^(5/2))/(5*a^6*c^3))) + (x ^5*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/6 + (c*(-1/8*((-2*Sqrt[c + a^2*c*x^2]) /(a^4*c) + (2*(c + a^2*c*x^2)^(3/2))/(3*a^4*c^2))/a + (x^3*Sqrt[c + a^2*c* x^2]*ArcTan[a*x])/(4*a^2*c) - (3*(-1/2*Sqrt[c + a^2*c*x^2]/(a^3*c) + (x*Sq rt[c + a^2*c*x^2]*ArcTan[a*x])/(2*a^2*c) - (Sqrt[1 + a^2*x^2]*(((-2*I)*Arc Tan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/a + (I*PolyLog[2, ((-I)* Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a - (I*PolyLog[2, (I*Sqrt[1 + I*a*x])/S qrt[1 - I*a*x]])/a))/(2*a^2*Sqrt[c + a^2*c*x^2])))/(4*a^2)))/6)
3.3.9.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[-2*I*(a + b*ArcTan[c*x])*(ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/ (c*Sqrt[d])), x] + (Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 + I*c*x]/Sqrt[1 - I*c *x])]/(c*Sqrt[d])), x] - Simp[I*b*(PolyLog[2, I*(Sqrt[1 + I*c*x]/Sqrt[1 - I *c*x])]/(c*Sqrt[d])), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
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_.))*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcTan[c*x ])/(f*(m + 2))), x] + (Simp[d/(m + 2) Int[(f*x)^m*((a + b*ArcTan[c*x])/Sq rt[d + e*x^2]), x], x] - Simp[b*c*(d/(f*(m + 2))) Int[(f*x)^(m + 1)/Sqrt[ d + e*x^2], x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && NeQ[m, -2]
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 = 0.46 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.62
| method | result | size |
| default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (120 \arctan \left (a x \right ) a^{5} x^{5}-24 a^{4} x^{4}+210 \arctan \left (a x \right ) x^{3} a^{3}-38 a^{2} x^{2}+45 x \arctan \left (a x \right ) a +31\right )}{720 a^{3}}+\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{16 a^{3} \sqrt {a^{2} x^{2}+1}}\) | \(221\) |
1/720*c/a^3*(c*(a*x-I)*(I+a*x))^(1/2)*(120*arctan(a*x)*a^5*x^5-24*a^4*x^4+ 210*arctan(a*x)*x^3*a^3-38*a^2*x^2+45*x*arctan(a*x)*a+31)+1/16*c*(c*(a*x-I )*(I+a*x))^(1/2)*(arctan(a*x)*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-arctan(a *x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-I*dilog(1+I*(1+I*a*x)/(a^2*x^2+1)^ (1/2))+I*dilog(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^3/(a^2*x^2+1)^(1/2)
\[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} \arctan \left (a x\right ) \,d x } \]
\[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\int x^{2} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}{\left (a x \right )}\, dx \]
\[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} \arctan \left (a x\right ) \,d x } \]
\[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{2} \arctan \left (a x\right ) \,d x } \]
Timed out. \[ \int x^2 \left (c+a^2 c x^2\right )^{3/2} \arctan (a x) \, dx=\int x^2\,\mathrm {atan}\left (a\,x\right )\,{\left (c\,a^2\,x^2+c\right )}^{3/2} \,d x \]