Integrand size = 24, antiderivative size = 397 \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^{3/2}} \, dx=-\frac {2 a^4 x}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 \sqrt {c+a^2 c x^2}}{3 c^2 x}+\frac {2 a^3 \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{3 c^2 x^2}+\frac {a^4 x \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 c^2 x^3}+\frac {5 a^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{3 c^2 x}+\frac {22 a^3 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 c \sqrt {c+a^2 c x^2}}-\frac {11 i a^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 c \sqrt {c+a^2 c x^2}}+\frac {11 i a^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 c \sqrt {c+a^2 c x^2}} \]
-2*a^4*x/c/(a^2*c*x^2+c)^(1/2)+2*a^3*arctan(a*x)/c/(a^2*c*x^2+c)^(1/2)+a^4 *x*arctan(a*x)^2/c/(a^2*c*x^2+c)^(1/2)+22/3*a^3*arctan(a*x)*arctanh((1+I*a *x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/c/(a^2*c*x^2+c)^(1/2)-11/3*I* a^3*polylog(2,-(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/c/(a^2*c *x^2+c)^(1/2)+11/3*I*a^3*polylog(2,(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x ^2+1)^(1/2)/c/(a^2*c*x^2+c)^(1/2)-1/3*a^2*(a^2*c*x^2+c)^(1/2)/c^2/x-1/3*a* arctan(a*x)*(a^2*c*x^2+c)^(1/2)/c^2/x^2-1/3*arctan(a*x)^2*(a^2*c*x^2+c)^(1 /2)/c^2/x^3+5/3*a^2*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/c^2/x
Time = 2.78 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.68 \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {a^3 \sqrt {1+a^2 x^2} \left (-88 i \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )+\frac {\left (1+a^2 x^2\right )^{3/2} \left (-22+28 \cos (2 \arctan (a x))-6 \cos (4 \arctan (a x))+\arctan (a x)^2 (25-36 \cos (2 \arctan (a x))+3 \cos (4 \arctan (a x)))+\frac {88 i a^3 x^3 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}+\arctan (a x) \left (\frac {66 a x \left (-\log \left (1-e^{i \arctan (a x)}\right )+\log \left (1+e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}+8 \sin (2 \arctan (a x))+22 \left (\log \left (1-e^{i \arctan (a x)}\right )-\log \left (1+e^{i \arctan (a x)}\right )\right ) \sin (3 \arctan (a x))-6 \sin (4 \arctan (a x))\right )\right )}{a^3 x^3}\right )}{24 c \sqrt {c+a^2 c x^2}} \]
(a^3*Sqrt[1 + a^2*x^2]*((-88*I)*PolyLog[2, -E^(I*ArcTan[a*x])] + ((1 + a^2 *x^2)^(3/2)*(-22 + 28*Cos[2*ArcTan[a*x]] - 6*Cos[4*ArcTan[a*x]] + ArcTan[a *x]^2*(25 - 36*Cos[2*ArcTan[a*x]] + 3*Cos[4*ArcTan[a*x]]) + ((88*I)*a^3*x^ 3*PolyLog[2, E^(I*ArcTan[a*x])])/(1 + a^2*x^2)^(3/2) + ArcTan[a*x]*((66*a* x*(-Log[1 - E^(I*ArcTan[a*x])] + Log[1 + E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2 *x^2] + 8*Sin[2*ArcTan[a*x]] + 22*(Log[1 - E^(I*ArcTan[a*x])] - Log[1 + E^ (I*ArcTan[a*x])])*Sin[3*ArcTan[a*x]] - 6*Sin[4*ArcTan[a*x]])))/(a^3*x^3))) /(24*c*Sqrt[c + a^2*c*x^2])
Time = 3.86 (sec) , antiderivative size = 629, normalized size of antiderivative = 1.58, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5501, 5497, 5479, 5493, 5489, 5497, 242, 5493, 5489, 5501, 5433, 208, 5479, 5493, 5489}
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.
| \(\displaystyle \int \frac {\arctan (a x)^2}{x^4 \left (a^2 c x^2+c\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x^4 \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 5497 |
| \(\displaystyle \frac {-\frac {2}{3} a^2 \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx+\frac {2}{3} a \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}}{c}-a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 5479 |
| \(\displaystyle \frac {-\frac {2}{3} a^2 \left (2 a \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {2}{3} a \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}}{c}-a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 5493 |
| \(\displaystyle \frac {-\frac {2}{3} a^2 \left (\frac {2 a \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}\right )+\frac {2}{3} a \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}}{c}-a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 5489 |
| \(\displaystyle -a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx+\frac {\frac {2}{3} a \int \frac {\arctan (a x)}{x^3 \sqrt {a^2 c x^2+c}}dx-\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}+\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}}{c}\) |
| \(\Big \downarrow \) 5497 |
| \(\displaystyle -a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx+\frac {\frac {2}{3} a \left (-\frac {1}{2} a^2 \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx+\frac {1}{2} a \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}\right )-\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}+\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}}{c}\) |
| \(\Big \downarrow \) 242 |
| \(\displaystyle -a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx+\frac {\frac {2}{3} a \left (-\frac {1}{2} a^2 \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}+\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}}{c}\) |
| \(\Big \downarrow \) 5493 |
| \(\displaystyle -a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx+\frac {\frac {2}{3} a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}+\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}}{c}\) |
| \(\Big \downarrow \) 5489 |
| \(\displaystyle -a^2 \int \frac {\arctan (a x)^2}{x^2 \left (a^2 c x^2+c\right )^{3/2}}dx+\frac {-\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}+\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+\frac {2}{3} a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}}{c}\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle -a^2 \left (\frac {\int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \int \frac {\arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx\right )+\frac {-\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}+\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+\frac {2}{3} a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}}{c}\) |
| \(\Big \downarrow \) 5433 |
| \(\displaystyle -a^2 \left (\frac {\int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \left (-2 \int \frac {1}{\left (a^2 c x^2+c\right )^{3/2}}dx+\frac {x \arctan (a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \arctan (a x)}{a c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}+\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+\frac {2}{3} a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}}{c}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -a^2 \left (\frac {\int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \left (\frac {x \arctan (a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {2 x}{c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}+\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+\frac {2}{3} a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}}{c}\) |
| \(\Big \downarrow \) 5479 |
| \(\displaystyle -a^2 \left (\frac {2 a \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}}{c}-a^2 \left (\frac {x \arctan (a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {2 x}{c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}+\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+\frac {2}{3} a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}}{c}\) |
| \(\Big \downarrow \) 5493 |
| \(\displaystyle -a^2 \left (\frac {\frac {2 a \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}}{c}-a^2 \left (\frac {x \arctan (a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {2 x}{c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}+\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+\frac {2}{3} a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}}{c}\) |
| \(\Big \downarrow \) 5489 |
| \(\displaystyle \frac {-\frac {2}{3} a^2 \left (-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}+\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )+\frac {2}{3} a \left (-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{2 c x^2}-\frac {a \sqrt {a^2 c x^2+c}}{2 c x}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 c x^3}}{c}-a^2 \left (-a^2 \left (\frac {x \arctan (a x)^2}{c \sqrt {a^2 c x^2+c}}+\frac {2 \arctan (a x)}{a c \sqrt {a^2 c x^2+c}}-\frac {2 x}{c \sqrt {a^2 c x^2+c}}\right )+\frac {-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}+\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}}{c}\right )\) |
-(a^2*(-(a^2*((-2*x)/(c*Sqrt[c + a^2*c*x^2]) + (2*ArcTan[a*x])/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x]^2)/(c*Sqrt[c + a^2*c*x^2]))) + (-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(c*x)) + (2*a*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x ]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]] + I*PolyLog[2, -(Sqrt[1 + I*a*x ]/Sqrt[1 - I*a*x])] - I*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]]))/Sqrt [c + a^2*c*x^2])/c)) + (-1/3*(Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(c*x^3) - (2*a^2*(-((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(c*x)) + (2*a*Sqrt[1 + a^2* x^2]*(-2*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]] + I*PolyLog[ 2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])] - I*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt [1 - I*a*x]]))/Sqrt[c + a^2*c*x^2]))/3 + (2*a*(-1/2*(a*Sqrt[c + a^2*c*x^2] )/(c*x) - (Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*c*x^2) - (a^2*Sqrt[1 + a^2* x^2]*(-2*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]] + I*PolyLog[ 2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x])] - I*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt [1 - I*a*x]]))/(2*Sqrt[c + a^2*c*x^2])))/3)/c
3.4.46.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_ Symbol] :> Simp[b*p*((a + b*ArcTan[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2])), x] + (Simp[x*((a + b*ArcTan[c*x])^p/(d*Sqrt[d + e*x^2])), x] - Simp[b^2*p*(p - 1) Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[ {a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]
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_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_ Symbol] :> Simp[(-2/Sqrt[d])*(a + b*ArcTan[c*x])*ArcTanh[Sqrt[1 + I*c*x]/Sq rt[1 - I*c*x]], x] + (Simp[I*(b/Sqrt[d])*PolyLog[2, -Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]], x] - Simp[I*(b/Sqrt[d])*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c *x]], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && 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[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 )^(q_), x_Symbol] :> Simp[1/d Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c *x])^p, x], x] - Simp[e/d Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2* q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
Time = 1.26 (sec) , antiderivative size = 318, normalized size of antiderivative = 0.80
| method | result | size |
| default | \(\frac {a^{3} \left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right ) a^{3}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}+\frac {\left (5 x^{2} \arctan \left (a x \right )^{2} a^{2}-a^{2} x^{2}-x \arctan \left (a x \right ) a -\arctan \left (a x \right )^{2}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{3 c^{2} x^{3}}-\frac {11 i a^{3} \left (i \arctan \left (a x \right ) \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-i \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+\operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{3 \sqrt {a^{2} x^{2}+1}\, c^{2}}\) | \(318\) |
1/2*a^3*(arctan(a*x)^2-2+2*I*arctan(a*x))*(a*x-I)*(c*(a*x-I)*(I+a*x))^(1/2 )/(a^2*x^2+1)/c^2+1/2*(c*(a*x-I)*(I+a*x))^(1/2)*(I+a*x)*(arctan(a*x)^2-2-2 *I*arctan(a*x))*a^3/(a^2*x^2+1)/c^2+1/3*(5*x^2*arctan(a*x)^2*a^2-a^2*x^2-x *arctan(a*x)*a-arctan(a*x)^2)*(c*(a*x-I)*(I+a*x))^(1/2)/c^2/x^3-11/3*I*a^3 *(I*arctan(a*x)*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)+1)-I*arctan(a*x)*ln(1-(1+I* a*x)/(a^2*x^2+1)^(1/2))+polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-polylog(2, (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^2
\[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
\[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{4} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
\[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{4}} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^2}{x^4 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^4\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]