Integrand size = 24, antiderivative size = 396 \[ \int \frac {\arctan (a x)^3}{x^4 \sqrt {c+a^2 c x^2}} \, dx=-\frac {a^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{c x}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 c x^2}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 c x^3}+\frac {2 a^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 c x}+\frac {5 a^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {a^3 \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {5 i a^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {5 i a^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {5 a^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {5 a^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]
-a^3*arctanh((a^2*c*x^2+c)^(1/2)/c^(1/2))/c^(1/2)+5*a^3*arctan(a*x)^2*arct anh((1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-5*I *a^3*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2) /(a^2*c*x^2+c)^(1/2)+5*I*a^3*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^( 1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+5*a^3*polylog(3,-(1+I*a*x)/(a^ 2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-5*a^3*polylog(3,(1+I *a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-a^2*arctan( a*x)*(a^2*c*x^2+c)^(1/2)/c/x-1/2*a*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/c/x^2 -1/3*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/c/x^3+2/3*a^2*arctan(a*x)^3*(a^2*c* x^2+c)^(1/2)/c/x
Time = 4.92 (sec) , antiderivative size = 343, normalized size of antiderivative = 0.87 \[ \int \frac {\arctan (a x)^3}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\frac {a^3 \sqrt {c \left (1+a^2 x^2\right )} \left (-12 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right )+10 \arctan (a x)^3 \cot \left (\frac {1}{2} \arctan (a x)\right )-3 \arctan (a x)^2 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-\frac {a x \arctan (a x)^3 \csc ^4\left (\frac {1}{2} \arctan (a x)\right )}{2 \sqrt {1+a^2 x^2}}-60 \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )+60 \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )+24 \log \left (\tan \left (\frac {1}{2} \arctan (a x)\right )\right )-120 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )+120 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+120 \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )-120 \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )+3 \arctan (a x)^2 \sec ^2\left (\frac {1}{2} \arctan (a x)\right )-\frac {8 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^3 \sin ^4\left (\frac {1}{2} \arctan (a x)\right )}{a^3 x^3}-12 \arctan (a x) \tan \left (\frac {1}{2} \arctan (a x)\right )+10 \arctan (a x)^3 \tan \left (\frac {1}{2} \arctan (a x)\right )\right )}{24 c \sqrt {1+a^2 x^2}} \]
(a^3*Sqrt[c*(1 + a^2*x^2)]*(-12*ArcTan[a*x]*Cot[ArcTan[a*x]/2] + 10*ArcTan [a*x]^3*Cot[ArcTan[a*x]/2] - 3*ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 - (a*x*A rcTan[a*x]^3*Csc[ArcTan[a*x]/2]^4)/(2*Sqrt[1 + a^2*x^2]) - 60*ArcTan[a*x]^ 2*Log[1 - E^(I*ArcTan[a*x])] + 60*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 24*Log[Tan[ArcTan[a*x]/2]] - (120*I)*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTa n[a*x])] + (120*I)*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] + 120*PolyLog [3, -E^(I*ArcTan[a*x])] - 120*PolyLog[3, E^(I*ArcTan[a*x])] + 3*ArcTan[a*x ]^2*Sec[ArcTan[a*x]/2]^2 - (8*(1 + a^2*x^2)^(3/2)*ArcTan[a*x]^3*Sin[ArcTan [a*x]/2]^4)/(a^3*x^3) - 12*ArcTan[a*x]*Tan[ArcTan[a*x]/2] + 10*ArcTan[a*x] ^3*Tan[ArcTan[a*x]/2]))/(24*c*Sqrt[1 + a^2*x^2])
Time = 3.40 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.06, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.833, Rules used = {5497, 5479, 5493, 5491, 3042, 4671, 3011, 2720, 5497, 5479, 243, 73, 221, 5493, 5491, 3042, 4671, 3011, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\arctan (a x)^3}{x^4 \sqrt {a^2 c x^2+c}} \, dx\) |
| \(\Big \downarrow \) 5497 |
| \(\displaystyle -\frac {2}{3} a^2 \int \frac {\arctan (a x)^3}{x^2 \sqrt {a^2 c x^2+c}}dx+a \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 5479 |
| \(\displaystyle -\frac {2}{3} a^2 \left (3 a \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}\right )+a \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 5493 |
| \(\displaystyle -\frac {2}{3} a^2 \left (\frac {3 a \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}\right )+a \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 5491 |
| \(\displaystyle -\frac {2}{3} a^2 \left (\frac {3 a \sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^2}{a x}d\arctan (a x)}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}\right )+a \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2}{3} a^2 \left (\frac {3 a \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc (\arctan (a x))d\arctan (a x)}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}\right )+a \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+a \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+a \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+a \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 5497 |
| \(\displaystyle -\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+a \left (-\frac {1}{2} a^2 \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+a \int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 5479 |
| \(\displaystyle -\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+a \left (-\frac {1}{2} a^2 \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+a \left (a \int \frac {1}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+a \left (-\frac {1}{2} a^2 \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+a \left (\frac {1}{2} a \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx^2-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+a \left (-\frac {1}{2} a^2 \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+a \left (\frac {\int \frac {1}{\frac {x^4}{a^2 c}-\frac {1}{a^2}}d\sqrt {a^2 c x^2+c}}{a c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+a \left (-\frac {1}{2} a^2 \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 5493 |
| \(\displaystyle -\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 5491 |
| \(\displaystyle -\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^2}{a x}d\arctan (a x)}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc (\arctan (a x))d\arctan (a x)}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{c x}+\frac {3 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 c x^3}\) |
-1/3*(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(c*x^3) - (2*a^2*(-((Sqrt[c + a^2 *c*x^2]*ArcTan[a*x]^3)/(c*x)) + (3*a*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]^2*A rcTanh[E^(I*ArcTan[a*x])] + 2*(I*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x]) ] - PolyLog[3, -E^(I*ArcTan[a*x])]) - 2*(I*ArcTan[a*x]*PolyLog[2, E^(I*Arc Tan[a*x])] - PolyLog[3, E^(I*ArcTan[a*x])])))/Sqrt[c + a^2*c*x^2]))/3 + a* (-1/2*(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(c*x^2) + a*(-((Sqrt[c + a^2*c*x ^2]*ArcTan[a*x])/(c*x)) - (a*ArcTanh[Sqrt[c + a^2*c*x^2]/Sqrt[c]])/Sqrt[c] ) - (a^2*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])] + 2*(I*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - PolyLog[3, -E^(I*ArcTan[ a*x])]) - 2*(I*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - PolyLog[3, E^(I *ArcTan[a*x])])))/(2*Sqrt[c + a^2*c*x^2]))
3.5.43.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[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_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Simp[b*c*(p/(f*(m + 1))) Int[(f*x) ^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2] ), x_Symbol] :> Simp[1/Sqrt[d] Subst[Int[(a + b*x)^p*Csc[x], x], x, ArcTa n[c*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_)*Sqrt[(d_) + (e_.)*(x_)^2 ]), x_Symbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2] Int[(a + b*ArcTan [c*x])^p/(x*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_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Ar cTan[c*x])^p/(d*f*(m + 1))), x] + (-Simp[b*c*(p/(f*(m + 1))) Int[(f*x)^(m + 1)*((a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Simp[c^2*((m + 2)/(f^2*(m + 1))) 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] && LtQ[m, -1] && NeQ[m, -2]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 4.21 (sec) , antiderivative size = 324, normalized size of antiderivative = 0.82
| method | result | size |
| default | \(\frac {\left (4 x^{2} \arctan \left (a x \right )^{2} a^{2}-6 a^{2} x^{2}-3 x \arctan \left (a x \right ) a -2 \arctan \left (a x \right )^{2}\right ) \arctan \left (a x \right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 c \,x^{3}}-\frac {2 a^{3} \operatorname {arctanh}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c}+\frac {5 a^{3} \left (\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-\arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, c}\) | \(324\) |
1/6*(4*x^2*arctan(a*x)^2*a^2-6*a^2*x^2-3*x*arctan(a*x)*a-2*arctan(a*x)^2)* arctan(a*x)*(c*(a*x-I)*(I+a*x))^(1/2)/c/x^3-2*a^3*arctanh((1+I*a*x)/(a^2*x ^2+1)^(1/2))*(c*(a*x-I)*(I+a*x))^(1/2)/(a^2*x^2+1)^(1/2)/c+5/2*a^3*(arctan (a*x)^2*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-arctan(a*x)^2*ln(1-(1+I*a*x)/(a^ 2*x^2+1)^(1/2))-2*I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+2* I*arctan(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+2*polylog(3,-(1+I*a*x )/(a^2*x^2+1)^(1/2))-2*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2)))*(c*(a*x-I)* (I+a*x))^(1/2)/(a^2*x^2+1)^(1/2)/c
\[ \int \frac {\arctan (a x)^3}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c} x^{4}} \,d x } \]
\[ \int \frac {\arctan (a x)^3}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{4} \sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
\[ \int \frac {\arctan (a x)^3}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c} x^{4}} \,d x } \]
\[ \int \frac {\arctan (a x)^3}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^3}{x^4 \sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3}{x^4\,\sqrt {c\,a^2\,x^2+c}} \,d x \]