Integrand size = 24, antiderivative size = 422 \[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {2 a^2}{c \sqrt {c+a^2 c x^2}}+\frac {2 a^3 x \arctan (a x)}{c \sqrt {c+a^2 c x^2}}-\frac {a \sqrt {c+a^2 c x^2} \arctan (a x)}{c^2 x}-\frac {a^2 \arctan (a x)^2}{c \sqrt {c+a^2 c x^2}}-\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 c^2 x^2}+\frac {3 a^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {a^2 \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )}{c^{3/2}}-\frac {3 i a^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}+\frac {3 a^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}}-\frac {3 a^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c \sqrt {c+a^2 c x^2}} \]
-a^2*arctanh((a^2*c*x^2+c)^(1/2)/c^(1/2))/c^(3/2)+2*a^2/c/(a^2*c*x^2+c)^(1 /2)+2*a^3*x*arctan(a*x)/c/(a^2*c*x^2+c)^(1/2)-a^2*arctan(a*x)^2/c/(a^2*c*x ^2+c)^(1/2)+3*a^2*arctan(a*x)^2*arctanh((1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2* x^2+1)^(1/2)/c/(a^2*c*x^2+c)^(1/2)-3*I*a^2*arctan(a*x)*polylog(2,-(1+I*a*x )/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/c/(a^2*c*x^2+c)^(1/2)+3*I*a^2*arcta n(a*x)*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/c/(a^2*c*x ^2+c)^(1/2)+3*a^2*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2 )/c/(a^2*c*x^2+c)^(1/2)-3*a^2*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2* x^2+1)^(1/2)/c/(a^2*c*x^2+c)^(1/2)-a*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/c^2/x -1/2*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/c^2/x^2
Time = 1.93 (sec) , antiderivative size = 371, normalized size of antiderivative = 0.88 \[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\frac {a^2 \left (16+16 a x \arctan (a x)-8 \arctan (a x)^2-2 a x \arctan (a x) \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-\sqrt {1+a^2 x^2} \arctan (a x)^2 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )-12 \sqrt {1+a^2 x^2} \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )+12 \sqrt {1+a^2 x^2} \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )+8 \sqrt {1+a^2 x^2} \log \left (\tan \left (\frac {1}{2} \arctan (a x)\right )\right )-24 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )+24 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )+24 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )-24 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )+\sqrt {1+a^2 x^2} \arctan (a x)^2 \sec ^2\left (\frac {1}{2} \arctan (a x)\right )-4 \sqrt {1+a^2 x^2} \arctan (a x) \tan \left (\frac {1}{2} \arctan (a x)\right )\right )}{8 c \sqrt {c+a^2 c x^2}} \]
(a^2*(16 + 16*a*x*ArcTan[a*x] - 8*ArcTan[a*x]^2 - 2*a*x*ArcTan[a*x]*Csc[Ar cTan[a*x]/2]^2 - Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Csc[ArcTan[a*x]/2]^2 - 12 *Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 - E^(I*ArcTan[a*x])] + 12*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + 8*Sqrt[1 + a^2*x^2]*Lo g[Tan[ArcTan[a*x]/2]] - (24*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, -E ^(I*ArcTan[a*x])] + (24*I)*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*PolyLog[2, E^(I*A rcTan[a*x])] + 24*Sqrt[1 + a^2*x^2]*PolyLog[3, -E^(I*ArcTan[a*x])] - 24*Sq rt[1 + a^2*x^2]*PolyLog[3, E^(I*ArcTan[a*x])] + Sqrt[1 + a^2*x^2]*ArcTan[a *x]^2*Sec[ArcTan[a*x]/2]^2 - 4*Sqrt[1 + a^2*x^2]*ArcTan[a*x]*Tan[ArcTan[a* x]/2]))/(8*c*Sqrt[c + a^2*c*x^2])
Time = 4.26 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.05, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5501, 5497, 5479, 243, 73, 221, 5493, 5491, 3042, 4671, 3011, 2720, 5501, 5465, 5429, 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.
\(\displaystyle \int \frac {\arctan (a x)^2}{x^3 \left (a^2 c x^2+c\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 5501 |
\(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx\) |
\(\Big \downarrow \) 5497 |
\(\displaystyle \frac {-\frac {1}{2} a^2 \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+a \int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}-a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx\) |
\(\Big \downarrow \) 5479 |
\(\displaystyle \frac {-\frac {1}{2} a^2 \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+a \left (a \int \frac {1}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}-a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {-\frac {1}{2} a^2 \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+a \left (\frac {1}{2} a \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx^2-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}-a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {-\frac {1}{2} a^2 \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+a \left (\frac {\int \frac {1}{\frac {x^4}{a^2 c}-\frac {1}{a^2}}d\sqrt {a^2 c x^2+c}}{a c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}-a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {-\frac {1}{2} a^2 \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}-a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx\) |
\(\Big \downarrow \) 5493 |
\(\displaystyle \frac {-\frac {a^2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}-a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx\) |
\(\Big \downarrow \) 5491 |
\(\displaystyle \frac {-\frac {a^2 \sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^2}{a x}d\arctan (a x)}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}-a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {-\frac {a^2 \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc (\arctan (a x))d\arctan (a x)}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}-a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle -a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx+\frac {-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx+\frac {-\frac {a^2 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -a^2 \int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx+\frac {-\frac {a^2 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}\) |
\(\Big \downarrow \) 5501 |
\(\displaystyle -a^2 \left (\frac {\int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx\right )+\frac {-\frac {a^2 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}\) |
\(\Big \downarrow \) 5465 |
\(\displaystyle -a^2 \left (\frac {\int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \left (\frac {2 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {a^2 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}\) |
\(\Big \downarrow \) 5429 |
\(\displaystyle -a^2 \left (\frac {\int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {a^2 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}\) |
\(\Big \downarrow \) 5493 |
\(\displaystyle -a^2 \left (\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 x^2+1}}dx}{c \sqrt {a^2 c x^2+c}}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {a^2 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}\) |
\(\Big \downarrow \) 5491 |
\(\displaystyle -a^2 \left (\frac {\sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^2}{a x}d\arctan (a x)}{c \sqrt {a^2 c x^2+c}}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {a^2 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -a^2 \left (\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc (\arctan (a x))d\arctan (a x)}{c \sqrt {a^2 c x^2+c}}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )\right )+\frac {-\frac {a^2 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle \frac {-\frac {a^2 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}-a^2 \left (-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )+\frac {\sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{c \sqrt {a^2 c x^2+c}}\right )\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle \frac {-\frac {a^2 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}-a^2 \left (-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )+\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{c \sqrt {a^2 c x^2+c}}\right )\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \frac {-\frac {a^2 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}-a^2 \left (-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )+\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{c \sqrt {a^2 c x^2+c}}\right )\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )}{2 \sqrt {a^2 c x^2+c}}+a \left (-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 c x^2}}{c}-a^2 \left (-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )+\frac {\sqrt {a^2 x^2+1} \left (-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )}{c \sqrt {a^2 c x^2+c}}\right )\) |
(-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]))/c - a^2*(-(a^2*(-(ArcTan[a*x]^2 /(a^2*c*Sqrt[c + a^2*c*x^2])) + (2*(1/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTa n[a*x])/(c*Sqrt[c + a^2*c*x^2])))/a)) + (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])])))/(c*Sqrt[c + a^2*c*x^2]))
3.4.45.3.1 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_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbo l] :> Simp[b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTan[c*x])/(d*Sqr t[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ .), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 1))), x] - Simp[b*(p/(2*c*(q + 1))) Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) ^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && NeQ[q, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(q_.), x_Symbol] :> Simp[(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[((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]
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 = 1.09 (sec) , antiderivative size = 376, normalized size of antiderivative = 0.89
method | result | size |
default | \(-\frac {a^{2} \left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right ) a^{2}}{2 \left (a^{2} x^{2}+1\right ) c^{2}}-\frac {\left (2 a x +\arctan \left (a x \right )\right ) \arctan \left (a x \right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 c^{2} x^{2}}+\frac {a^{2} \left (3 \arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-4 \,\operatorname {arctanh}\left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{2 \sqrt {a^{2} x^{2}+1}\, c^{2}}\) | \(376\) |
-1/2*a^2*(arctan(a*x)^2-2+2*I*arctan(a*x))*(1+I*a*x)*(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-1)*(arctan(a*x)^ 2-2-2*I*arctan(a*x))*a^2/(a^2*x^2+1)/c^2-1/2*(2*a*x+arctan(a*x))*arctan(a* x)*(c*(a*x-I)*(I+a*x))^(1/2)/c^2/x^2+1/2*a^2*(3*arctan(a*x)^2*ln((1+I*a*x) /(a^2*x^2+1)^(1/2)+1)-3*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6* I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*poly log(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-4*arctanh((1+I*a*x)/(a^2*x^2+1)^(1/2))+ 6*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*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^2
\[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
\[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
\[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^2}{x^3 \left (c+a^2 c x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]