Integrand size = 24, antiderivative size = 389 \[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {2}{27 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22}{9 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 a x \arctan (a x)}{9 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {22 a x \arctan (a x)}{9 c^2 \sqrt {c+a^2 c x^2}}+\frac {\arctan (a x)^2}{3 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\arctan (a x)^2}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}}+\frac {2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{c^2 \sqrt {c+a^2 c x^2}} \]
-2/27/c/(a^2*c*x^2+c)^(3/2)-2/9*a*x*arctan(a*x)/c/(a^2*c*x^2+c)^(3/2)+1/3* arctan(a*x)^2/c/(a^2*c*x^2+c)^(3/2)-22/9/c^2/(a^2*c*x^2+c)^(1/2)-22/9*a*x* arctan(a*x)/c^2/(a^2*c*x^2+c)^(1/2)+arctan(a*x)^2/c^2/(a^2*c*x^2+c)^(1/2)- 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^2 /(a^2*c*x^2+c)^(1/2)+2*I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2 ))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)-2*I*arctan(a*x)*polylog(2,(1+ I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1/2)-2*poly log(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c*x^2+c)^(1 /2)+2*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/c^2/(a^2*c* x^2+c)^(1/2)
Time = 0.48 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.63 \[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=\frac {\left (1+a^2 x^2\right )^{3/2} \left (-\frac {270}{\sqrt {1+a^2 x^2}}-\frac {270 a x \arctan (a x)}{\sqrt {1+a^2 x^2}}+\frac {135 \arctan (a x)^2}{\sqrt {1+a^2 x^2}}-2 \cos (3 \arctan (a x))+9 \arctan (a x)^2 \cos (3 \arctan (a x))+108 \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )-108 \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )+216 i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-216 i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-216 \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+216 \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )-6 \arctan (a x) \sin (3 \arctan (a x))\right )}{108 c \left (c \left (1+a^2 x^2\right )\right )^{3/2}} \]
((1 + a^2*x^2)^(3/2)*(-270/Sqrt[1 + a^2*x^2] - (270*a*x*ArcTan[a*x])/Sqrt[ 1 + a^2*x^2] + (135*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] - 2*Cos[3*ArcTan[a*x] ] + 9*ArcTan[a*x]^2*Cos[3*ArcTan[a*x]] + 108*ArcTan[a*x]^2*Log[1 - E^(I*Ar cTan[a*x])] - 108*ArcTan[a*x]^2*Log[1 + E^(I*ArcTan[a*x])] + (216*I)*ArcTa n[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - (216*I)*ArcTan[a*x]*PolyLog[2, E^( I*ArcTan[a*x])] - 216*PolyLog[3, -E^(I*ArcTan[a*x])] + 216*PolyLog[3, E^(I *ArcTan[a*x])] - 6*ArcTan[a*x]*Sin[3*ArcTan[a*x]]))/(108*c*(c*(1 + a^2*x^2 ))^(3/2))
Time = 2.61 (sec) , antiderivative size = 365, normalized size of antiderivative = 0.94, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {5501, 5465, 5431, 5429, 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.
| \(\displaystyle \int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \left (\frac {2 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{5/2}}dx}{3 a}-\frac {\arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5431 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \left (\frac {2 \left (\frac {2 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 c}+\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {\arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5429 |
| \(\displaystyle \frac {\int \frac {\arctan (a x)^2}{x \left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {\arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5501 |
| \(\displaystyle \frac {\frac {\int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{c}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {\arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle \frac {\frac {\int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \left (\frac {2 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {\arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5429 |
| \(\displaystyle \frac {\frac {\int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx}{c}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {\arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5493 |
| \(\displaystyle \frac {\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 x^2+1}}dx}{c \sqrt {a^2 c x^2+c}}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {\arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 5491 |
| \(\displaystyle \frac {\frac {\sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^2}{a x}d\arctan (a x)}{c \sqrt {a^2 c x^2+c}}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {\arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc (\arctan (a x))d\arctan (a x)}{c \sqrt {a^2 c x^2+c}}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{c}-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {\arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle -a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {\arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )+\frac {-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )+\frac {\sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{c \sqrt {a^2 c x^2+c}}}{c}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {\arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )+\frac {-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )+\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{c \sqrt {a^2 c x^2+c}}}{c}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {\arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )+\frac {-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )+\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{c \sqrt {a^2 c x^2+c}}}{c}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{3 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{3 c}+\frac {1}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{3 a}-\frac {\arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}\right )+\frac {-a^2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )+\frac {\sqrt {a^2 x^2+1} \left (-2 \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )}{c \sqrt {a^2 c x^2+c}}}{c}\) |
-(a^2*(-1/3*ArcTan[a*x]^2/(a^2*c*(c + a^2*c*x^2)^(3/2)) + (2*(1/(9*a*c*(c + a^2*c*x^2)^(3/2)) + (x*ArcTan[a*x])/(3*c*(c + a^2*c*x^2)^(3/2)) + (2*(1/ (a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2])))/(3*c )))/(3*a))) + (-(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*ArcTan[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*ArcT an[a*x])])))/(c*Sqrt[c + a^2*c*x^2]))/c
3.4.53.3.1 Defintions of rubi rules used
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_.))*((d_) + (e_.)*(x_)^2)^(q_), x_Symbol ] :> Simp[b*((d + e*x^2)^(q + 1)/(4*c*d*(q + 1)^2)), x] + (-Simp[x*(d + e*x ^2)^(q + 1)*((a + b*ArcTan[c*x])/(2*d*(q + 1))), x] + Simp[(2*q + 3)/(2*d*( q + 1)) Int[(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && LtQ[q, -1] && NeQ[q, -3/2]
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_)/((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_.)*(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 = 1.31 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.19
| method | result | size |
| default | \(-\frac {\left (6 i \arctan \left (a x \right )+9 \arctan \left (a x \right )^{2}-2\right ) \left (i a^{3} x^{3}+3 a^{2} x^{2}-3 i a x -1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{216 \left (a^{2} x^{2}+1\right )^{2} c^{3}}+\frac {5 \left (\arctan \left (a x \right )^{2}-2+2 i \arctan \left (a x \right )\right ) \left (i a x +1\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 c^{3} \left (a^{2} x^{2}+1\right )}-\frac {5 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a x -1\right ) \left (\arctan \left (a x \right )^{2}-2-2 i \arctan \left (a x \right )\right )}{8 c^{3} \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (i a^{3} x^{3}-3 a^{2} x^{2}-3 i a x +1\right ) \left (-6 i \arctan \left (a x \right )+9 \arctan \left (a x \right )^{2}-2\right )}{216 c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}-\frac {\left (\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-\arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, c^{3}}\) | \(461\) |
-1/216*(6*I*arctan(a*x)+9*arctan(a*x)^2-2)*(I*a^3*x^3+3*a^2*x^2-3*I*a*x-1) *(c*(a*x-I)*(I+a*x))^(1/2)/(a^2*x^2+1)^2/c^3+5/8*(arctan(a*x)^2-2+2*I*arct an(a*x))*(1+I*a*x)*(c*(a*x-I)*(I+a*x))^(1/2)/c^3/(a^2*x^2+1)-5/8*(c*(a*x-I )*(I+a*x))^(1/2)*(I*a*x-1)*(arctan(a*x)^2-2-2*I*arctan(a*x))/c^3/(a^2*x^2+ 1)+1/216*(c*(a*x-I)*(I+a*x))^(1/2)*(I*a^3*x^3-3*a^2*x^2-3*I*a*x+1)*(-6*I*a rctan(a*x)+9*arctan(a*x)^2-2)/c^3/(a^4*x^4+2*a^2*x^2+1)-(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^3
\[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \]
integral(sqrt(a^2*c*x^2 + c)*arctan(a*x)^2/(a^6*c^3*x^7 + 3*a^4*c^3*x^5 + 3*a^2*c^3*x^3 + c^3*x), x)
\[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \]
\[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {\arctan \left (a x\right )^{2}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x} \,d x } \]
Timed out. \[ \int \frac {\arctan (a x)^2}{x \left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2}{x\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]