Integrand size = 24, antiderivative size = 919 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=-\frac {3 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x}+\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{\sqrt {c+a^2 c x^2}}+a^2 c \sqrt {c+a^2 c x^2} \arctan (a x)^3-\frac {c \sqrt {c+a^2 c x^2} \arctan (a x)^3}{2 x^2}-\frac {3 a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \text {arctanh}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 i a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}-\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 i a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {9 i a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{2 \sqrt {c+a^2 c x^2}}+\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {3 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{\sqrt {c+a^2 c x^2}}-\frac {9 a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {6 a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 a^2 c^2 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {9 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {9 i a^2 c^2 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \] Output:
-3/2*a*c*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^2/x+9/2*I*a^2*c^2*(a^2*x^2+1)^(1/ 2)*arctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/ 2)+a^2*c*(a^2*c*x^2+c)^(1/2)*arctan(a*x)^3-1/2*c*(a^2*c*x^2+c)^(1/2)*arcta n(a*x)^3/x^2-3*a^2*c^2*(a^2*x^2+1)^(1/2)*arctan(a*x)^3*arctanh((1+I*a*x)/( a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)-6*a^2*c^2*(a^2*x^2+1)^(1/2)*arctan(a *x)*arctanh((1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))/(a^2*c*x^2+c)^(1/2)-9*I*a^2*c ^2*(a^2*x^2+1)^(1/2)*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c) ^(1/2)-3*I*a^2*c^2*(a^2*x^2+1)^(1/2)*polylog(2,(1+I*a*x)^(1/2)/(1-I*a*x)^( 1/2))/(a^2*c*x^2+c)^(1/2)-9/2*I*a^2*c^2*(a^2*x^2+1)^(1/2)*arctan(a*x)^2*po lylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)-6*I*a^2*c^2*(a^2* x^2+1)^(1/2)*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c* x^2+c)^(1/2)+3*I*a^2*c^2*(a^2*x^2+1)^(1/2)*polylog(2,-(1+I*a*x)^(1/2)/(1-I *a*x)^(1/2))/(a^2*c*x^2+c)^(1/2)+6*I*a^2*c^2*(a^2*x^2+1)^(1/2)*arctan((1+I *a*x)/(a^2*x^2+1)^(1/2))*arctan(a*x)^2/(a^2*c*x^2+c)^(1/2)-9*a^2*c^2*(a^2* x^2+1)^(1/2)*arctan(a*x)*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^ 2+c)^(1/2)+6*a^2*c^2*(a^2*x^2+1)^(1/2)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^ (1/2))/(a^2*c*x^2+c)^(1/2)-6*a^2*c^2*(a^2*x^2+1)^(1/2)*polylog(3,I*(1+I*a* x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)+9*a^2*c^2*(a^2*x^2+1)^(1/2)*arct an(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^2+c)^(1/2)+9*I*a^2 *c^2*(a^2*x^2+1)^(1/2)*polylog(4,(1+I*a*x)/(a^2*x^2+1)^(1/2))/(a^2*c*x^...
Time = 5.99 (sec) , antiderivative size = 691, normalized size of antiderivative = 0.75 \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx =\text {Too large to display} \] Input:
Integrate[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^3,x]
Output:
(a^2*c*Sqrt[c + a^2*c*x^2]*(-12*ArcTan[a*x]^2 - (3*I)*Pi^4*Cot[ArcTan[a*x] /2] + (6*I)*ArcTan[a*x]^4*Cot[ArcTan[a*x]/2] - 12*ArcTan[a*x]^2*Cot[ArcTan [a*x]/2]^2 + 8*a*x*ArcTan[a*x]^3*Csc[ArcTan[a*x]/2]^2 - 2*ArcTan[a*x]^3*Co t[ArcTan[a*x]/2]*Csc[ArcTan[a*x]/2]^2 + 24*ArcTan[a*x]^3*Cot[ArcTan[a*x]/2 ]*Log[1 - E^((-I)*ArcTan[a*x])] + 48*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 - E^(I*ArcTan[a*x])] - 48*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2]*Log[1 - I*E^(I* ArcTan[a*x])] + 48*ArcTan[a*x]^2*Cot[ArcTan[a*x]/2]*Log[1 + I*E^(I*ArcTan[ a*x])] - 48*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*Log[1 + E^(I*ArcTan[a*x])] - 24 *ArcTan[a*x]^3*Cot[ArcTan[a*x]/2]*Log[1 + E^(I*ArcTan[a*x])] + (72*I)*ArcT an[a*x]^2*Cot[ArcTan[a*x]/2]*PolyLog[2, E^((-I)*ArcTan[a*x])] + (24*I)*(2 + 3*ArcTan[a*x]^2)*Cot[ArcTan[a*x]/2]*PolyLog[2, -E^(I*ArcTan[a*x])] - (96 *I)*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] + (9 6*I)*ArcTan[a*x]*Cot[ArcTan[a*x]/2]*PolyLog[2, I*E^(I*ArcTan[a*x])] - (48* I)*Cot[ArcTan[a*x]/2]*PolyLog[2, E^(I*ArcTan[a*x])] + 144*ArcTan[a*x]*Cot[ ArcTan[a*x]/2]*PolyLog[3, E^((-I)*ArcTan[a*x])] - 144*ArcTan[a*x]*Cot[ArcT an[a*x]/2]*PolyLog[3, -E^(I*ArcTan[a*x])] + 96*Cot[ArcTan[a*x]/2]*PolyLog[ 3, (-I)*E^(I*ArcTan[a*x])] - 96*Cot[ArcTan[a*x]/2]*PolyLog[3, I*E^(I*ArcTa n[a*x])] - (144*I)*Cot[ArcTan[a*x]/2]*PolyLog[4, E^((-I)*ArcTan[a*x])] - ( 144*I)*Cot[ArcTan[a*x]/2]*PolyLog[4, -E^(I*ArcTan[a*x])] + 2*ArcTan[a*x]^3 *Csc[ArcTan[a*x]/2]*Sec[ArcTan[a*x]/2])*Tan[ArcTan[a*x]/2])/(16*Sqrt[1 ...
Time = 9.24 (sec) , antiderivative size = 925, normalized size of antiderivative = 1.01, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {5485, 5485, 5465, 5425, 5423, 3042, 4669, 3011, 2720, 5493, 5491, 3042, 4671, 3011, 5497, 5479, 5493, 5489, 5491, 3042, 4671, 3011, 7143, 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.
| \(\displaystyle \int \frac {\arctan (a x)^3 \left (a^2 c x^2+c\right )^{3/2}}{x^3} \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{x^3}dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \left (a^2 c \int \frac {x \arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx\right )+c \left (a^2 c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^3}{x^3 \sqrt {a^2 c x^2+c}}dx\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle a^2 c \left (a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx}{a}\right )+c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx\right )+c \left (a^2 c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^3}{x^3 \sqrt {a^2 c x^2+c}}dx\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle a^2 c \left (a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )+c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx\right )+c \left (a^2 c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^3}{x^3 \sqrt {a^2 c x^2+c}}dx\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle a^2 c \left (a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{a^2 \sqrt {a^2 c x^2+c}}\right )+c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx\right )+c \left (a^2 c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^3}{x^3 \sqrt {a^2 c x^2+c}}dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c \left (c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )+c \left (a^2 c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^3}{x^3 \sqrt {a^2 c x^2+c}}dx\right )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle c \left (a^2 c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^3}{x^3 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle c \left (a^2 c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^3}{x^3 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle c \left (a^2 c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^3}{x^3 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (c \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 5493 |
| \(\displaystyle c \left (\frac {a^2 c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^3}{x^3 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 5491 |
| \(\displaystyle c \left (\frac {a^2 c \sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^3}{a x}d\arctan (a x)}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^3}{x^3 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^3}{a x}d\arctan (a x)}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \left (\frac {a^2 c \sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc (\arctan (a x))d\arctan (a x)}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^3}{x^3 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc (\arctan (a x))d\arctan (a x)}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (-3 \int \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+3 \int \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )+c \left (c \int \frac {\arctan (a x)^3}{x^3 \sqrt {a^2 c x^2+c}}dx+\frac {a^2 c \sqrt {a^2 x^2+1} \left (-3 \int \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+3 \int \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )+c \left (c \int \frac {\arctan (a x)^3}{x^3 \sqrt {a^2 c x^2+c}}dx+\frac {a^2 c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\) |
| \(\Big \downarrow \) 5497 |
| \(\displaystyle a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )+c \left (c \left (\frac {3}{2} a \int \frac {\arctan (a x)^2}{x^2 \sqrt {a^2 c x^2+c}}dx-\frac {1}{2} a^2 \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\) |
| \(\Big \downarrow \) 5479 |
| \(\displaystyle a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{a^2 \sqrt {a^2 c x^2+c}}\right )\right )+c \left (c \left (\frac {3}{2} a \left (2 a \int \frac {\arctan (a x)}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}\right )-\frac {1}{2} a^2 \int \frac {\arctan (a x)^3}{x \sqrt {a^2 c x^2+c}}dx-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\) |
| \(\Big \downarrow \) 5493 |
| \(\displaystyle a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}+a^2 c \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 )\right )+c \left (c \left (\frac {3}{2} a \left (\frac {2 a \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{c x}\right )-\frac {a^2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{x \sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}-\frac {\arctan (a x)^3 \sqrt {a^2 c x^2+c}}{2 c x^2}\right )+\frac {a^2 c \sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \arctan (a x)^3 \text {arctanh}\left (e^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\) |
| \(\Big \downarrow \) 5489 |
| \(\displaystyle c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+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 )\right )}{a^2 \sqrt {a^2 c x^2+c}}\right ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 c x^2}+\frac {3}{2} a \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )-\frac {a^2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{x \sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 5491 |
| \(\displaystyle c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+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 )\right )}{a^2 \sqrt {a^2 c x^2+c}}\right ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 c x^2}+\frac {3}{2} a \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )-\frac {a^2 \sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^3}{a x}d\arctan (a x)}{2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+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 )\right )}{a^2 \sqrt {a^2 c x^2+c}}\right ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 c x^2}+\frac {3}{2} a \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )-\frac {a^2 \sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc (\arctan (a x))d\arctan (a x)}{2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+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 )\right )}{a^2 \sqrt {a^2 c x^2+c}}\right ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 c x^2}+\frac {3}{2} a \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3-3 \int \arctan (a x)^2 \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+3 \int \arctan (a x)^2 \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)\right )}{2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+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 )\right )}{a^2 \sqrt {a^2 c x^2+c}}\right ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 c x^2}+\frac {3}{2} a \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+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 )\right )}{a^2 \sqrt {a^2 c x^2+c}}\right ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 c x^2}+\frac {3}{2} a \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 7163 |
| \(\displaystyle c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+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 )\right )}{a^2 \sqrt {a^2 c x^2+c}}\right ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 c x^2}+\frac {3}{2} a \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )\right )}{2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+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 )\right )}{a^2 \sqrt {a^2 c x^2+c}}\right ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 c x^2}+\frac {3}{2} a \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )\right )}{2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{a^2 c}-\frac {3 \sqrt {a^2 x^2+1} \left (-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+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 )\right )}{a^2 \sqrt {a^2 c x^2+c}}\right ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )\right )}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^3}{2 c x^2}+\frac {3}{2} a \left (\frac {2 a \sqrt {a^2 x^2+1} \left (-2 \arctan (a x) \text {arctanh}\left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )+i \operatorname {PolyLog}\left (2,-\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )-i \operatorname {PolyLog}\left (2,\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )\right )}{\sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{c x}\right )-\frac {a^2 \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^3+3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,-e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )\right )}{2 \sqrt {a^2 c x^2+c}}\right )\right )\) |
Input:
Int[((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^3)/x^3,x]
Output:
a^2*c*(a^2*c*((Sqrt[c + a^2*c*x^2]*ArcTan[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])] - 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])])))/(a^2*Sqrt[c + a^2*c*x^2])) + (c*Sqrt[1 + a^2*x^2]*(-2*ArcTan[ a*x]^3*ArcTanh[E^(I*ArcTan[a*x])] + 3*(I*ArcTan[a*x]^2*PolyLog[2, -E^(I*Ar cTan[a*x])] - (2*I)*((-I)*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan[a*x])] + Pol yLog[4, -E^(I*ArcTan[a*x])])) - 3*(I*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[ a*x])] - (2*I)*((-I)*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x])] + PolyLog[4 , E^(I*ArcTan[a*x])]))))/Sqrt[c + a^2*c*x^2]) + c*((a^2*c*Sqrt[1 + a^2*x^2 ]*(-2*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])] + 3*(I*ArcTan[a*x]^2*PolyLo g[2, -E^(I*ArcTan[a*x])] - (2*I)*((-I)*ArcTan[a*x]*PolyLog[3, -E^(I*ArcTan [a*x])] + PolyLog[4, -E^(I*ArcTan[a*x])])) - 3*(I*ArcTan[a*x]^2*PolyLog[2, E^(I*ArcTan[a*x])] - (2*I)*((-I)*ArcTan[a*x]*PolyLog[3, E^(I*ArcTan[a*x]) ] + PolyLog[4, E^(I*ArcTan[a*x])]))))/Sqrt[c + a^2*c*x^2] + c*(-1/2*(Sqrt[ c + a^2*c*x^2]*ArcTan[a*x]^3)/(c*x^2) + (3*a*(-((Sqrt[c + a^2*c*x^2]*ArcTa n[a*x]^2)/(c*x)) + (2*a*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]*ArcTanh[Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]] + I*PolyLog[2, -(Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]) ] - I*PolyLog[2, Sqrt[1 + I*a*x]/Sqrt[1 - I*a*x]]))/Sqrt[c + a^2*c*x^2]))/ 2 - (a^2*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]^3*ArcTanh[E^(I*ArcTan[a*x])]...
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[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_.))^(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[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Simp[b*c*(p/(f*(m + 1))) Int[(f*x) ^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] && NeQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((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_.))/((x_)*Sqrt[(d_) + (e_.)*(x_)^2]), x_ Symbol] :> Simp[(-2/Sqrt[d])*(a + b*ArcTan[c*x])*ArcTanh[Sqrt[1 + I*c*x]/Sq rt[1 - I*c*x]], x] + (Simp[I*(b/Sqrt[d])*PolyLog[2, -Sqrt[1 + I*c*x]/Sqrt[1 - I*c*x]], x] - Simp[I*(b/Sqrt[d])*PolyLog[2, Sqrt[1 + I*c*x]/Sqrt[1 - I*c *x]], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2] ), x_Symbol] :> Simp[1/Sqrt[d] Subst[Int[(a + b*x)^p*Csc[x], x], x, ArcTa n[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && GtQ[d, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((x_)*Sqrt[(d_) + (e_.)*(x_)^2 ]), x_Symbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2] Int[(a + b*ArcTan [c*x])^p/(x*Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[ e, c^2*d] && IGtQ[p, 0] && !GtQ[d, 0]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Ar cTan[c*x])^p/(d*f*(m + 1))), x] + (-Simp[b*c*(p/(f*(m + 1))) Int[(f*x)^(m + 1)*((a + b*ArcTan[c*x])^(p - 1)/Sqrt[d + e*x^2]), x], x] - Simp[c^2*((m + 2)/(f^2*(m + 1))) Int[(f*x)^(m + 2)*((a + b*ArcTan[c*x])^p/Sqrt[d + e*x ^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[p, 0] && LtQ[m, -1] && NeQ[m, -2]
Int[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 = 5.76 (sec) , antiderivative size = 592, normalized size of antiderivative = 0.64
| method | result | size |
| default | \(\frac {c \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \arctan \left (a x \right )^{2} \left (2 x^{2} a^{2} \arctan \left (a x \right )-3 a x -\arctan \left (a x \right )\right )}{2 x^{2}}-\frac {3 c \,a^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (\arctan \left (a x \right )^{3} \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-\arctan \left (a x \right )^{3} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 i \operatorname {polylog}\left (4, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 i \operatorname {polylog}\left (4, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-2 \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+4 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 )+3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+2 \arctan \left (a x \right ) \ln \left (1+\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 \arctan \left (a x \right ) \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-6 \arctan \left (a x \right ) \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-3 i \arctan \left (a x \right )^{2} \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-2 i \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-4 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 )+2 i \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+4 \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-4 \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{2 \sqrt {a^{2} x^{2}+1}}\) | \(592\) |
Input:
int((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x^3,x,method=_RETURNVERBOSE)
Output:
1/2*c*(c*(a*x-I)*(a*x+I))^(1/2)*arctan(a*x)^2*(2*x^2*a^2*arctan(a*x)-3*a*x -arctan(a*x))/x^2-3/2*c*a^2*(c*(a*x-I)*(a*x+I))^(1/2)*(arctan(a*x)^3*ln(1+ (1+I*a*x)/(a^2*x^2+1)^(1/2))-arctan(a*x)^3*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2 ))+6*I*polylog(4,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-6*I*polylog(4,(1+I*a*x)/(a^ 2*x^2+1)^(1/2))+2*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*arct an(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+4*I*arctan(a*x)*polylog(2,-I *(1+I*a*x)/(a^2*x^2+1)^(1/2))+3*I*arctan(a*x)^2*polylog(2,(1+I*a*x)/(a^2*x ^2+1)^(1/2))+2*arctan(a*x)*ln(1+(1+I*a*x)/(a^2*x^2+1)^(1/2))+6*arctan(a*x) *polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*arctan(a*x)*ln(1-(1+I*a*x)/(a^2 *x^2+1)^(1/2))-6*arctan(a*x)*polylog(3,(1+I*a*x)/(a^2*x^2+1)^(1/2))-3*I*ar ctan(a*x)^2*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-2*I*polylog(2,-(1+I*a* x)/(a^2*x^2+1)^(1/2))-4*I*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1 /2))+2*I*polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))+4*polylog(3,I*(1+I*a*x)/(a ^2*x^2+1)^(1/2))-4*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2)))/(a^2*x^2+1)^ (1/2)
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}}{x^{3}} \,d x } \] Input:
integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x^3,x, algorithm="fricas")
Output:
integral((a^2*c*x^2 + c)^(3/2)*arctan(a*x)^3/x^3, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}}{x^{3}}\, dx \] Input:
integrate((a**2*c*x**2+c)**(3/2)*atan(a*x)**3/x**3,x)
Output:
Integral((c*(a**2*x**2 + 1))**(3/2)*atan(a*x)**3/x**3, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}}{x^{3}} \,d x } \] Input:
integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x^3,x, algorithm="maxima")
Output:
integrate((a^2*c*x^2 + c)^(3/2)*arctan(a*x)^3/x^3, x)
Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a^2*c*x^2+c)^(3/2)*arctan(a*x)^3/x^3,x, algorithm="giac")
Output:
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. \[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2}}{x^3} \,d x \] Input:
int((atan(a*x)^3*(c + a^2*c*x^2)^(3/2))/x^3,x)
Output:
int((atan(a*x)^3*(c + a^2*c*x^2)^(3/2))/x^3, x)
\[ \int \frac {\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3}{x^3} \, dx=\sqrt {c}\, c \left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3}}{x^{3}}d x +\left (\int \frac {\sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{3}}{x}d x \right ) a^{2}\right ) \] Input:
int((a^2*c*x^2+c)^(3/2)*atan(a*x)^3/x^3,x)
Output:
sqrt(c)*c*(int((sqrt(a**2*x**2 + 1)*atan(a*x)**3)/x**3,x) + int((sqrt(a**2 *x**2 + 1)*atan(a*x)**3)/x,x)*a**2)