Integrand size = 24, antiderivative size = 747 \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=-\frac {\sqrt {c+a^2 c x^2}}{4 a^3}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)}{4 a^2}+\frac {\sqrt {c+a^2 c x^2} \arctan (a x)^2}{8 a^3}-\frac {x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{4 a}+\frac {x \sqrt {c+a^2 c x^2} \arctan (a x)^3}{8 a^2}+\frac {1}{4} x^3 \sqrt {c+a^2 c x^2} \arctan (a x)^3+\frac {i c \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a^3 \sqrt {c+a^2 c x^2}}-\frac {i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{2 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 c \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}+\frac {3 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}}-\frac {3 i c \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{4 a^3 \sqrt {c+a^2 c x^2}} \] Output:
-1/4*(a^2*c*x^2+c)^(1/2)/a^3+1/4*x*(a^2*c*x^2+c)^(1/2)*arctan(a*x)/a^2+1/8 *(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/a^3-1/4*x^2*(a^2*c*x^2+c)^(1/2)*arctan( a*x)^2/a+1/8*x*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3/a^2+1/4*x^3*(a^2*c*x^2+c) ^(1/2)*arctan(a*x)^3+I*c*(a^2*x^2+1)^(1/2)*arctan(a*x)*arctan((1+I*a*x)^(1 /2)/(1-I*a*x)^(1/2))/a^3/(a^2*c*x^2+c)^(1/2)+3/8*I*c*(a^2*x^2+1)^(1/2)*arc tan(a*x)^2*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^3/(a^2*c*x^2+c)^(1/2 )-1/2*I*c*(a^2*x^2+1)^(1/2)*polylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/ a^3/(a^2*c*x^2+c)^(1/2)+1/2*I*c*(a^2*x^2+1)^(1/2)*polylog(2,I*(1+I*a*x)^(1 /2)/(1-I*a*x)^(1/2))/a^3/(a^2*c*x^2+c)^(1/2)+3/4*I*c*(a^2*x^2+1)^(1/2)*pol ylog(4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^3/(a^2*c*x^2+c)^(1/2)-3/8*I*c*(a^ 2*x^2+1)^(1/2)*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^3 /(a^2*c*x^2+c)^(1/2)+3/4*c*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog(3,-I*(1+I *a*x)/(a^2*x^2+1)^(1/2))/a^3/(a^2*c*x^2+c)^(1/2)-3/4*c*(a^2*x^2+1)^(1/2)*a rctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^3/(a^2*c*x^2+c)^(1/2 )-3/4*I*c*(a^2*x^2+1)^(1/2)*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a^3/( a^2*c*x^2+c)^(1/2)+1/4*I*c*(a^2*x^2+1)^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^ (1/2))*arctan(a*x)^3/a^3/(a^2*c*x^2+c)^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1844\) vs. \(2(747)=1494\).
Time = 12.20 (sec) , antiderivative size = 1844, normalized size of antiderivative = 2.47 \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx =\text {Too large to display} \] Input:
Integrate[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]
Output:
((Sqrt[c*(1 + a^2*x^2)]*(-1 + ArcTan[a*x]^2))/(4*Sqrt[1 + a^2*x^2]) + (Sqr t[c*(1 + a^2*x^2)]*(-(ArcTan[a*x]*(Log[1 - I*E^(I*ArcTan[a*x])] - Log[1 + I*E^(I*ArcTan[a*x])])) - I*(PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - PolyLog[2 , I*E^(I*ArcTan[a*x])])))/(2*Sqrt[1 + a^2*x^2]) + (Sqrt[c*(1 + a^2*x^2)]*( -1/8*(Pi^3*Log[Cot[(Pi/2 - ArcTan[a*x])/2]]) - (3*Pi^2*((Pi/2 - ArcTan[a*x ])*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x] ))]) + I*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/2 - ArcTan[a*x]))])))/4 + (3*Pi*((Pi/2 - ArcTan[a*x])^2*(Log[1 - E^(I*(Pi/2 - ArcTan[a*x]))] - Log[1 + E^(I*(Pi/2 - ArcTan[a*x]))]) + (2*I)*(Pi/2 - Ar cTan[a*x])*(PolyLog[2, -E^(I*(Pi/2 - ArcTan[a*x]))] - PolyLog[2, E^(I*(Pi/ 2 - ArcTan[a*x]))]) + 2*(-PolyLog[3, -E^(I*(Pi/2 - ArcTan[a*x]))] + PolyLo g[3, E^(I*(Pi/2 - ArcTan[a*x]))])))/2 - 8*((I/64)*(Pi/2 - ArcTan[a*x])^4 + (I/4)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^4 - ((Pi/2 - ArcTan[a*x])^3*Log[ 1 + E^(I*(Pi/2 - ArcTan[a*x]))])/8 - (Pi^3*(I*(Pi/2 + (-1/2*Pi + ArcTan[a* x])/2) - Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2))]))/8 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^3*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan [a*x])/2))] + ((3*I)/8)*(Pi/2 - ArcTan[a*x])^2*PolyLog[2, -E^(I*(Pi/2 - Ar cTan[a*x]))] + (3*Pi^2*((I/2)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)^2 - (Pi/2 + (-1/2*Pi + ArcTan[a*x])/2)*Log[1 + E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a *x])/2))] + (I/2)*PolyLog[2, -E^((2*I)*(Pi/2 + (-1/2*Pi + ArcTan[a*x])/...
Time = 10.22 (sec) , antiderivative size = 1306, normalized size of antiderivative = 1.75, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {5485, 5487, 5425, 5423, 3042, 4669, 3011, 5465, 5425, 5421, 5487, 5425, 5423, 3042, 4669, 3011, 5465, 5425, 5421, 5487, 241, 5425, 5421, 7163, 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.
| \(\displaystyle \int x^2 \arctan (a x)^3 \sqrt {a^2 c x^2+c} \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle c \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx+a^2 c \int \frac {x^4 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle c \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )+a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )+a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-3 \int \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+3 \int \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {3 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {3 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle a^2 c \left (-\frac {3 \int \frac {x^2 \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{4 a^2}-\frac {3 \int \frac {x^3 \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{4 a}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle a^2 c \left (-\frac {3 \left (-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{4 a}-\frac {3 \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )}{4 a^2}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle a^2 c \left (-\frac {3 \left (-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{4 a}-\frac {3 \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )}{4 a^2}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle a^2 c \left (-\frac {3 \left (-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{4 a}-\frac {3 \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )}{4 a^2}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c \left (-\frac {3 \left (-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{4 a}-\frac {3 \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )}{4 a^2}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )+c \left (-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle c \left (-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )+a^2 c \left (-\frac {3 \left (-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{4 a}-\frac {3 \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-3 \int \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+3 \int \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )}{4 a^2}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle c \left (-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}-\frac {3 \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )+a^2 c \left (-\frac {3 \left (-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}-\frac {2 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{3 a^2}+\frac {x^2 \arctan (a x)^2 \sqrt {a^2 c x^2+c}}{3 a^2 c}\right )}{4 a}-\frac {3 \left (-\frac {3 \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \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)^3\right )}{2 a^3 \sqrt {a^2 c x^2+c}}+\frac {x \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 a^2 c}\right )}{4 a^2}+\frac {x^3 \arctan (a x)^3 \sqrt {a^2 c x^2+c}}{4 a^2 c}\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \int \frac {x^2 \arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\) |
| \(\Big \downarrow \) 5487 |
| \(\displaystyle c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {x}{\sqrt {a^2 c x^2+c}}dx}{2 a}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{2 a^2}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle c \left (\frac {x^3 \sqrt {a^2 c x^2+c} \arctan (a x)^3}{4 a^2 c}-\frac {3 \left (\frac {x^2 \sqrt {a^2 c x^2+c} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)}{2 a^2 c}-\frac {\sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 a^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a^3 c}\right )}{3 a}-\frac {2 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{3 a^2}\right )}{4 a}-\frac {3 \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )}{4 a^2}\right ) a^2+c \left (\frac {x \sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 a^2 c}-\frac {3 \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )}{2 a}-\frac {\sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )\right )}{2 a^3 \sqrt {a^2 c x^2+c}}\right )\) |
Input:
Int[x^2*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3,x]
Output:
c*((x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(2*a^2*c) - (3*((Sqrt[c + a^2*c*x ^2]*ArcTan[a*x]^2)/(a^2*c) - (2*Sqrt[1 + a^2*x^2]*(((-2*I)*ArcTan[a*x]*Arc Tan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/a + (I*PolyLog[2, ((-I)*Sqrt[1 + I*a *x])/Sqrt[1 - I*a*x]])/a - (I*PolyLog[2, (I*Sqrt[1 + I*a*x])/Sqrt[1 - I*a* x]])/a))/(a*Sqrt[c + a^2*c*x^2])))/(2*a) - (Sqrt[1 + a^2*x^2]*((-2*I)*ArcT an[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 + 3*(I*ArcTan[a*x]^2*PolyLog[2, (-I)*E ^(I*ArcTan[a*x])] - (2*I)*((-I)*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a* x])] + PolyLog[4, (-I)*E^(I*ArcTan[a*x])])) - 3*(I*ArcTan[a*x]^2*PolyLog[2 , I*E^(I*ArcTan[a*x])] - (2*I)*((-I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTan[ a*x])] + PolyLog[4, I*E^(I*ArcTan[a*x])]))))/(2*a^3*Sqrt[c + a^2*c*x^2])) + a^2*c*((x^3*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^3)/(4*a^2*c) - (3*((x^2*Sqrt [c + a^2*c*x^2]*ArcTan[a*x]^2)/(3*a^2*c) - (2*(-1/2*Sqrt[c + a^2*c*x^2]/(a ^3*c) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/(2*a^2*c) - (Sqrt[1 + a^2*x^2] *(((-2*I)*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/a + (I*Poly Log[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a - (I*PolyLog[2, (I*Sqrt[ 1 + I*a*x])/Sqrt[1 - I*a*x]])/a))/(2*a^2*Sqrt[c + a^2*c*x^2])))/(3*a) - (2 *((Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/(a^2*c) - (2*Sqrt[1 + a^2*x^2]*(((-2 *I)*ArcTan[a*x]*ArcTan[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]])/a + (I*PolyLog[2, ((-I)*Sqrt[1 + I*a*x])/Sqrt[1 - I*a*x]])/a - (I*PolyLog[2, (I*Sqrt[1 + I* a*x])/Sqrt[1 - I*a*x]])/a))/(a*Sqrt[c + a^2*c*x^2])))/(3*a^2)))/(4*a) -...
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
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_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[-2*I*(a + b*ArcTan[c*x])*(ArcTan[Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]]/ (c*Sqrt[d])), x] + (Simp[I*b*(PolyLog[2, (-I)*(Sqrt[1 + I*c*x]/Sqrt[1 - I*c *x])]/(c*Sqrt[d])), x] - Simp[I*b*(PolyLog[2, I*(Sqrt[1 + I*c*x]/Sqrt[1 - I *c*x])]/(c*Sqrt[d])), x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 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_)*((d_) + (e_. )*(x_)^2)^(q_.), x_Symbol] :> Simp[d Int[(f*x)^m*(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[c^2*(d/f^2) Int[(f*x)^(m + 2)*(d + e*x^2 )^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && IGtQ[p, 0] && (RationalQ[m] || (EqQ[p, 1] && IntegerQ[q]))
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]
Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. )*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F])) Int[(e + f*x) ^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c , d, e, f, n, p}, x] && GtQ[m, 0]
Time = 5.00 (sec) , antiderivative size = 460, normalized size of antiderivative = 0.62
| method | result | size |
| default | \(\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 \arctan \left (a x \right )^{3} a^{3} x^{3}-2 \arctan \left (a x \right )^{2} x^{2} a^{2}+x \arctan \left (a x \right )^{3} a +2 \arctan \left (a x \right ) a x +\arctan \left (a x \right )^{2}-2\right )}{8 a^{3}}-\frac {\sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right )^{3} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right )^{3} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-4 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {polylog}\left (4, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-4 i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {polylog}\left (4, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{8 a^{3} \sqrt {a^{2} x^{2}+1}}\) | \(460\) |
Input:
int(x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3,x,method=_RETURNVERBOSE)
Output:
1/8/a^3*(c*(a*x-I)*(a*x+I))^(1/2)*(2*arctan(a*x)^3*a^3*x^3-2*arctan(a*x)^2 *x^2*a^2+x*arctan(a*x)^3*a+2*arctan(a*x)*a*x+arctan(a*x)^2-2)-1/8*(c*(a*x- I)*(a*x+I))^(1/2)*(arctan(a*x)^3*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-arcta n(a*x)^3*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*arctan(a*x)^2*polylog(2,I *(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*I*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^ 2*x^2+1)^(1/2))+4*arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*arctan (a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-4*arctan(a*x)*ln(1+I*(1+I*a *x)/(a^2*x^2+1)^(1/2))-6*arctan(a*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1 /2))+6*I*polylog(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+4*I*dilog(1+I*(1+I*a*x)/ (a^2*x^2+1)^(1/2))-4*I*dilog(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*polylog( 4,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/a^3/(a^2*x^2+1)^(1/2)
\[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{3} \,d x } \] Input:
integrate(x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3,x, algorithm="fricas")
Output:
integral(sqrt(a^2*c*x^2 + c)*x^2*arctan(a*x)^3, x)
\[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int x^{2} \sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {atan}^{3}{\left (a x \right )}\, dx \] Input:
integrate(x**2*(a**2*c*x**2+c)**(1/2)*atan(a*x)**3,x)
Output:
Integral(x**2*sqrt(c*(a**2*x**2 + 1))*atan(a*x)**3, x)
\[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{3} \,d x } \] Input:
integrate(x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3,x, algorithm="maxima")
Output:
integrate(sqrt(a^2*c*x^2 + c)*x^2*arctan(a*x)^3, x)
\[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int { \sqrt {a^{2} c x^{2} + c} x^{2} \arctan \left (a x\right )^{3} \,d x } \] Input:
integrate(x^2*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3,x, algorithm="giac")
Output:
integrate(sqrt(a^2*c*x^2 + c)*x^2*arctan(a*x)^3, x)
Timed out. \[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\int x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,\sqrt {c\,a^2\,x^2+c} \,d x \] Input:
int(x^2*atan(a*x)^3*(c + a^2*c*x^2)^(1/2),x)
Output:
int(x^2*atan(a*x)^3*(c + a^2*c*x^2)^(1/2), x)
\[ \int x^2 \sqrt {c+a^2 c x^2} \arctan (a x)^3 \, dx=\sqrt {c}\, \left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3} x^{2}d x \right ) \] Input:
int(x^2*(a^2*c*x^2+c)^(1/2)*atan(a*x)^3,x)
Output:
sqrt(c)*int(sqrt(a**2*x**2 + 1)*atan(a*x)**3*x**2,x)