Integrand size = 24, antiderivative size = 408 \[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \arctan (a x)}{a^4 c}-\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 a^3 c}-\frac {5 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{a^4 \sqrt {c+a^2 c x^2}}-\frac {2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^4 c}+\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3}{3 a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{a^4 \sqrt {c}}+\frac {5 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}-\frac {5 i \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}-\frac {5 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}}+\frac {5 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^4 \sqrt {c+a^2 c x^2}} \]
-arctanh(a*x*c^(1/2)/(a^2*c*x^2+c)^(1/2))/a^4/c^(1/2)-5*I*arctan((1+I*a*x) /(a^2*x^2+1)^(1/2))*arctan(a*x)^2*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2 )+5*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1 /2)/a^4/(a^2*c*x^2+c)^(1/2)-5*I*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2 +1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)-5*polylog(3,-I*(1+I*a *x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2)+5*polylog (3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^4/(a^2*c*x^2+c)^(1/2 )+arctan(a*x)*(a^2*c*x^2+c)^(1/2)/a^4/c-1/2*x*arctan(a*x)^2*(a^2*c*x^2+c)^ (1/2)/a^3/c-2/3*arctan(a*x)^3*(a^2*c*x^2+c)^(1/2)/a^4/c+1/3*x^2*arctan(a*x )^3*(a^2*c*x^2+c)^(1/2)/a^2/c
Time = 0.78 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.54 \[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {c+a^2 c x^2} \left (\frac {12 \left (-5 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-\text {arctanh}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )+5 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-5 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-5 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+5 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}-\left (1+a^2 x^2\right ) \arctan (a x) \left (-6+2 \arctan (a x)^2+6 \left (-1+\arctan (a x)^2\right ) \cos (2 \arctan (a x))+3 \arctan (a x) \sin (2 \arctan (a x))\right )\right )}{12 a^4 c} \]
(Sqrt[c + a^2*c*x^2]*((12*((-5*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 - ArcTanh[(a*x)/Sqrt[1 + a^2*x^2]] + (5*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^( I*ArcTan[a*x])] - (5*I)*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 5*Po lyLog[3, (-I)*E^(I*ArcTan[a*x])] + 5*PolyLog[3, I*E^(I*ArcTan[a*x])]))/Sqr t[1 + a^2*x^2] - (1 + a^2*x^2)*ArcTan[a*x]*(-6 + 2*ArcTan[a*x]^2 + 6*(-1 + ArcTan[a*x]^2)*Cos[2*ArcTan[a*x]] + 3*ArcTan[a*x]*Sin[2*ArcTan[a*x]])))/( 12*a^4*c)
Time = 3.21 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.13, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.792, Rules used = {5487, 5465, 5425, 5423, 3042, 4669, 3011, 2720, 5487, 5425, 5423, 3042, 4669, 3011, 2720, 5465, 224, 219, 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.
| \(\displaystyle \int \frac {x^3 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}} \, dx\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle -\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \int \frac {x \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle -\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {\int \frac {x^2 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\int \frac {x \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}-\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\int \frac {1}{\sqrt {a^2 c x^2+c}}dx}{a}}{a}-\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}}{a}}{a}-\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {-\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a^3 \sqrt {a^2 c x^2+c}}-\frac {\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}}{a}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -\frac {2 \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )}{3 a^2}+\frac {x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{3 a^2 c}-\frac {-\frac {\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {\text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a^2 \sqrt {c}}}{a}+\frac {x \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a^3 \sqrt {a^2 c x^2+c}}}{a}\) |
(x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(3*a^2*c) - ((x*Sqrt[c + a^2*c*x^2 ]*ArcTan[a*x]^2)/(2*a^2*c) - ((Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(a^2*c) - ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]]/(a^2*Sqrt[c]))/a - (Sqrt[1 + a^ 2*x^2]*((-2*I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 2*(I*ArcTan[a*x]* PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[3, (-I)*E^(I*ArcTan[a*x])]) - 2*(I*ArcTan[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcT an[a*x])])))/(2*a^3*Sqrt[c + a^2*c*x^2]))/a - (2*((Sqrt[c + a^2*c*x^2]*Arc Tan[a*x]^3)/(a^2*c) - (3*Sqrt[1 + a^2*x^2]*((-2*I)*ArcTan[E^(I*ArcTan[a*x] )]*ArcTan[a*x]^2 + 2*(I*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - P olyLog[3, (-I)*E^(I*ArcTan[a*x])]) - 2*(I*ArcTan[a*x]*PolyLog[2, I*E^(I*Ar cTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])])))/(a^2*Sqrt[c + a^2*c*x^2]) ))/(3*a^2)
3.5.36.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[1/(c*Sqrt[d]) Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[ c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Gt Q[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S ymbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2] Int[(a + b*ArcTan[c*x])^ p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] & & IGtQ[p, 0] && !GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(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_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*((a + b* ArcTan[c*x])^p/(c^2*d*m)), x] + (-Simp[b*f*(p/(c*m)) Int[(f*x)^(m - 1)*(( a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Simp[f^2*((m - 1)/(c^ 2*m)) Int[(f*x)^(m - 2)*((a + b*ArcTan[c*x])^p/Sqrt[d + e*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && GtQ[m, 1]
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 = 3.97 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.94
| method | result | size |
| default | \(\frac {\left (2 x^{2} \arctan \left (a x \right )^{2} a^{2}-3 x \arctan \left (a x \right ) a -4 \arctan \left (a x \right )^{2}+6\right ) \arctan \left (a x \right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 c \,a^{4}}+\frac {5 \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 \sqrt {a^{2} x^{2}+1}\, a^{4} c}-\frac {5 \left (i \arctan \left (a x \right )^{3}-3 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{6 \sqrt {a^{2} x^{2}+1}\, a^{4} c}+\frac {2 i \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{4} c}\) | \(382\) |
1/6*(2*x^2*arctan(a*x)^2*a^2-3*x*arctan(a*x)*a-4*arctan(a*x)^2+6)*arctan(a *x)*(c*(a*x-I)*(I+a*x))^(1/2)/c/a^4+5/6*(I*arctan(a*x)^3-3*arctan(a*x)^2*l n(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(2,-I*(1+I*a*x)/ (a^2*x^2+1)^(1/2))-6*polylog(3,-I*(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)/a^4/c-5/6*(I*arctan(a*x)^3-3*arctan(a*x) ^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*I*arctan(a*x)*polylog(2,I*(1+I*a* x)/(a^2*x^2+1)^(1/2))-6*polylog(3,I*(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)/a^4/c+2*I*arctan((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)/a^4/c
\[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
\[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^{3} \operatorname {atan}^{3}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
\[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {x^{3} \arctan \left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
Exception generated. \[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {x^3 \arctan (a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {x^3\,{\mathrm {atan}\left (a\,x\right )}^3}{\sqrt {c\,a^2\,x^2+c}} \,d x \]