Integrand size = 24, antiderivative size = 661 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=\frac {1}{3} a^2 c^2 \sqrt {c+a^2 c x^2}-\frac {a c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{x}-\frac {1}{3} a^3 c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)+2 a^2 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2-\frac {c^2 \sqrt {c+a^2 c x^2} \arctan (a x)^2}{2 x^2}+\frac {1}{3} a^2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {26 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {5 a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x)^2 \text {arctanh}\left (e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-a^2 c^{5/2} \text {arctanh}\left (\frac {\sqrt {c+a^2 c x^2}}{\sqrt {c}}\right )+\frac {5 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {5 i a^2 c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}-\frac {13 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}+\frac {13 i a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{3 \sqrt {c+a^2 c x^2}}-\frac {5 a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}}+\frac {5 a^2 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )}{\sqrt {c+a^2 c x^2}} \]
1/3*a^2*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^2-a^2*c^(5/2)*arctanh((a^2*c*x^2 +c)^(1/2)/c^(1/2))+26/3*I*a^2*c^3*arctan(a*x)*arctan((1+I*a*x)^(1/2)/(1-I* a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-5*a^2*c^3*arctan(a*x)^2* arctanh((1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2) +5*I*a^2*c^3*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+ 1)^(1/2)/(a^2*c*x^2+c)^(1/2)-5*I*a^2*c^3*arctan(a*x)*polylog(2,(1+I*a*x)/( a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-13/3*I*a^2*c^3*pol ylog(2,-I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c) ^(1/2)+13/3*I*a^2*c^3*polylog(2,I*(1+I*a*x)^(1/2)/(1-I*a*x)^(1/2))*(a^2*x^ 2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)-5*a^2*c^3*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^ (1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+5*a^2*c^3*polylog(3,(1+I*a*x) /(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/(a^2*c*x^2+c)^(1/2)+1/3*a^2*c^2*(a^2 *c*x^2+c)^(1/2)-a*c^2*arctan(a*x)*(a^2*c*x^2+c)^(1/2)/x-1/3*a^3*c^2*x*arct an(a*x)*(a^2*c*x^2+c)^(1/2)+2*a^2*c^2*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)-1/ 2*c^2*arctan(a*x)^2*(a^2*c*x^2+c)^(1/2)/x^2
Time = 7.59 (sec) , antiderivative size = 761, normalized size of antiderivative = 1.15 \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=2 a^2 c^2 \sqrt {c \left (1+a^2 x^2\right )} \left (\arctan (a x)^2+\frac {\arctan (a x)^2 \left (\log \left (1-e^{i \arctan (a x)}\right )-\log \left (1+e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}-\frac {2 \left (\arctan (a x) \left (\log \left (1-i e^{i \arctan (a x)}\right )-\log \left (1+i e^{i \arctan (a x)}\right )\right )+i \left (\operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )\right )\right )}{\sqrt {1+a^2 x^2}}+\frac {2 i \arctan (a x) \left (\operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}+\frac {2 \left (-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )}{\sqrt {1+a^2 x^2}}\right )+\frac {1}{12} a^2 c^2 \left (1+a^2 x^2\right ) \sqrt {c \left (1+a^2 x^2\right )} \left (2+4 \arctan (a x)^2+2 \cos (2 \arctan (a x))-\frac {3 \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}-\arctan (a x) \cos (3 \arctan (a x)) \log \left (1-i e^{i \arctan (a x)}\right )+\frac {3 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )}{\sqrt {1+a^2 x^2}}+\arctan (a x) \cos (3 \arctan (a x)) \log \left (1+i e^{i \arctan (a x)}\right )-\frac {4 i \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}+\frac {4 i \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{\left (1+a^2 x^2\right )^{3/2}}-2 \arctan (a x) \sin (2 \arctan (a x))\right )+\frac {a^2 c^2 \sqrt {c \left (1+a^2 x^2\right )} \left (-4 \arctan (a x) \cot \left (\frac {1}{2} \arctan (a x)\right )-\arctan (a x)^2 \csc ^2\left (\frac {1}{2} \arctan (a x)\right )+4 \arctan (a x)^2 \left (\log \left (1-e^{i \arctan (a x)}\right )-\log \left (1+e^{i \arctan (a x)}\right )\right )+8 \log \left (\tan \left (\frac {1}{2} \arctan (a x)\right )\right )+8 i \arctan (a x) \left (\operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )\right )+8 \left (-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )+\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )+\arctan (a x)^2 \sec ^2\left (\frac {1}{2} \arctan (a x)\right )-4 \arctan (a x) \tan \left (\frac {1}{2} \arctan (a x)\right )\right )}{8 \sqrt {1+a^2 x^2}} \]
2*a^2*c^2*Sqrt[c*(1 + a^2*x^2)]*(ArcTan[a*x]^2 + (ArcTan[a*x]^2*(Log[1 - E ^(I*ArcTan[a*x])] - Log[1 + E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] - (2*(A rcTan[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])])) )/Sqrt[1 + a^2*x^2] + ((2*I)*ArcTan[a*x]*(PolyLog[2, -E^(I*ArcTan[a*x])] - PolyLog[2, E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2] + (2*(-PolyLog[3, -E^(I *ArcTan[a*x])] + PolyLog[3, E^(I*ArcTan[a*x])]))/Sqrt[1 + a^2*x^2]) + (a^2 *c^2*(1 + a^2*x^2)*Sqrt[c*(1 + a^2*x^2)]*(2 + 4*ArcTan[a*x]^2 + 2*Cos[2*Ar cTan[a*x]] - (3*ArcTan[a*x]*Log[1 - I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2 ] - ArcTan[a*x]*Cos[3*ArcTan[a*x]]*Log[1 - I*E^(I*ArcTan[a*x])] + (3*ArcTa n[a*x]*Log[1 + I*E^(I*ArcTan[a*x])])/Sqrt[1 + a^2*x^2] + ArcTan[a*x]*Cos[3 *ArcTan[a*x]]*Log[1 + I*E^(I*ArcTan[a*x])] - ((4*I)*PolyLog[2, (-I)*E^(I*A rcTan[a*x])])/(1 + a^2*x^2)^(3/2) + ((4*I)*PolyLog[2, I*E^(I*ArcTan[a*x])] )/(1 + a^2*x^2)^(3/2) - 2*ArcTan[a*x]*Sin[2*ArcTan[a*x]]))/12 + (a^2*c^2*S qrt[c*(1 + a^2*x^2)]*(-4*ArcTan[a*x]*Cot[ArcTan[a*x]/2] - ArcTan[a*x]^2*Cs c[ArcTan[a*x]/2]^2 + 4*ArcTan[a*x]^2*(Log[1 - E^(I*ArcTan[a*x])] - Log[1 + E^(I*ArcTan[a*x])]) + 8*Log[Tan[ArcTan[a*x]/2]] + (8*I)*ArcTan[a*x]*(Poly Log[2, -E^(I*ArcTan[a*x])] - PolyLog[2, E^(I*ArcTan[a*x])]) + 8*(-PolyLog[ 3, -E^(I*ArcTan[a*x])] + PolyLog[3, E^(I*ArcTan[a*x])]) + ArcTan[a*x]^2*Se c[ArcTan[a*x]/2]^2 - 4*ArcTan[a*x]*Tan[ArcTan[a*x]/2]))/(8*Sqrt[1 + a^2...
Time = 10.82 (sec) , antiderivative size = 1215, normalized size of antiderivative = 1.84, number of steps used = 29, number of rules used = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.167, Rules used = {5485, 5485, 5465, 5413, 5425, 5421, 5485, 5465, 5425, 5421, 5493, 5491, 3042, 4671, 3011, 2720, 5497, 5479, 243, 73, 221, 5493, 5491, 3042, 4671, 3011, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}}{x^3} \, dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{x}dx+c \int \frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{x^3}dx\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle a^2 c \left (a^2 c \int x \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^3}dx\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle a^2 c \left (a^2 c \left (\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {2 \int \sqrt {a^2 c x^2+c} \arctan (a x)dx}{3 a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^3}dx\right )\) |
| \(\Big \downarrow \) 5413 |
| \(\displaystyle a^2 c \left (a^2 c \left (\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {2 \left (\frac {1}{2} c \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^3}dx\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle a^2 c \left (a^2 c \left (\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {2 \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right )+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x}dx\right )+c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^3}dx\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle c \left (a^2 c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x}dx+c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x^3}dx\right )+a^2 c \left (c \int \frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{x}dx+a^2 c \left (\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {2 \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right )\right )\) |
| \(\Big \downarrow \) 5485 |
| \(\displaystyle c \left (a^2 c \left (a^2 c \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx\right )+c \left (a^2 c \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx\right )\right )+a^2 c \left (c \left (a^2 c \int \frac {x \arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {2 \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right )\right )\) |
| \(\Big \downarrow \) 5465 |
| \(\displaystyle c \left (a^2 c \left (a^2 c \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}\right )+c \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx\right )+c \left (a^2 c \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx\right )\right )+a^2 c \left (c \left (a^2 c \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \int \frac {\arctan (a x)}{\sqrt {a^2 c x^2+c}}dx}{a}\right )+c \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {2 \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right )\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle c \left (a^2 c \left (a^2 c \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )+c \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx\right )+c \left (a^2 c \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx\right )\right )+a^2 c \left (c \left (a^2 c \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)}{\sqrt {a^2 x^2+1}}dx}{a \sqrt {a^2 c x^2+c}}\right )+c \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {2 \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right )\right )\) |
| \(\Big \downarrow \) 5421 |
| \(\displaystyle a^2 c \left (c \left (c \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+a^2 c \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {2 \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right )\right )+c \left (c \left (a^2 c \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+c \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (c \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx+a^2 c \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 5493 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right ) a^2+\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right ) a^2+\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 x^2+1}}dx}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 x^2+1}}dx a^2}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx\right )\right )\) |
| \(\Big \downarrow \) 5491 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right ) a^2+\frac {c \sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^2}{a x}d\arctan (a x)}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \left (\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right ) a^2+\frac {c \sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^2}{a x}d\arctan (a x)}{\sqrt {a^2 c x^2+c}}\right ) a^2+c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^2}{a x}d\arctan (a x) a^2}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 c \left (c \left (\frac {c \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc (\arctan (a x))d\arctan (a x)}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )\right )+a^2 c \left (\frac {\arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}}{3 a^2 c}-\frac {2 \left (\frac {c \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x) \sqrt {a^2 c x^2+c}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right )\right )+c \left (c \left (\frac {a^2 c \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc (\arctan (a x))d\arctan (a x)}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx\right )+a^2 c \left (\frac {c \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc (\arctan (a x))d\arctan (a x)}{\sqrt {a^2 c x^2+c}}+a^2 c \left (\frac {\arctan (a x)^2 \sqrt {a^2 c x^2+c}}{a^2 c}-\frac {2 \sqrt {a^2 x^2+1} \left (-\frac {2 i \arctan (a x) \arctan \left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a}+\frac {i \operatorname {PolyLog}\left (2,-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}-\frac {i \operatorname {PolyLog}\left (2,\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a}\right )}{a \sqrt {a^2 c x^2+c}}\right )\right )\right )\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-2 \int \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-2 \int \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)\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)^2-2 \int \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\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)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\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)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \int \frac {\arctan (a x)^2}{x^3 \sqrt {a^2 c x^2+c}}dx\right )\right )\) |
| \(\Big \downarrow \) 5497 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\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)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {1}{2} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx a^2+\int \frac {\arctan (a x)}{x^2 \sqrt {a^2 c x^2+c}}dx a-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 c x^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5479 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\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)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {1}{2} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx a^2+\left (a \int \frac {1}{x \sqrt {a^2 c x^2+c}}dx-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{c x}\right ) a-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 c x^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\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)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {1}{2} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx a^2+\left (\frac {1}{2} a \int \frac {1}{x^2 \sqrt {a^2 c x^2+c}}dx^2-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{c x}\right ) a-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 c x^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\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)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {1}{2} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx a^2+\left (\frac {\int \frac {1}{\frac {x^4}{a^2 c}-\frac {1}{a^2}}d\sqrt {a^2 c x^2+c}}{a c}-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{c x}\right ) a-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 c x^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\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)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {1}{2} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 c x^2+c}}dx a^2+\left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) a-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 c x^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5493 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\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)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{x \sqrt {a^2 x^2+1}}dx a^2}{2 \sqrt {a^2 c x^2+c}}+\left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) a-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 c x^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 5491 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\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)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 x^2+1} \int \frac {\sqrt {a^2 x^2+1} \arctan (a x)^2}{a x}d\arctan (a x) a^2}{2 \sqrt {a^2 c x^2+c}}+\left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) a-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 c x^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\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)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc (\arctan (a x))d\arctan (a x) a^2}{2 \sqrt {a^2 c x^2+c}}+\left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) a-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 c x^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\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)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2-2 \int \arctan (a x) \log \left (1-e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+e^{i \arctan (a x)}\right )d\arctan (a x)\right ) a^2}{2 \sqrt {a^2 c x^2+c}}+\left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) a-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 c x^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\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)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )d\arctan (a x)\right )\right ) a^2}{2 \sqrt {a^2 c x^2+c}}+\left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) a-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 c x^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\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)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )\right ) a^2}{2 \sqrt {a^2 c x^2+c}}+\left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) a-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 c x^2}\right )\right )\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle c \left (c \left (\frac {\left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2}{3 a^2 c}-\frac {2 \left (\frac {1}{2} x \sqrt {a^2 c x^2+c} \arctan (a x)+\frac {c \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 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {a^2 c x^2+c}}{2 a}\right )}{3 a}\right ) a^2+c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )}{\sqrt {a^2 c x^2+c}}\right )\right ) a^2+c \left (c \left (c \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 ) a^2+\frac {c \sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right )}{\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)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right ) a^2}{\sqrt {a^2 c x^2+c}}+c \left (-\frac {\sqrt {a^2 x^2+1} \left (-2 \text {arctanh}\left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,e^{i \arctan (a x)}\right )\right )\right ) a^2}{2 \sqrt {a^2 c x^2+c}}+\left (-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)}{c x}-\frac {a \text {arctanh}\left (\frac {\sqrt {a^2 c x^2+c}}{\sqrt {c}}\right )}{\sqrt {c}}\right ) a-\frac {\sqrt {a^2 c x^2+c} \arctan (a x)^2}{2 c x^2}\right )\right )\right )\) |
a^2*c*(a^2*c*(((c + a^2*c*x^2)^(3/2)*ArcTan[a*x]^2)/(3*a^2*c) - (2*(-1/2*S qrt[c + a^2*c*x^2]/a + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x])/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*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))/(2*Sqrt[c + a^2*c*x^2])))/(3*a) ) + c*(a^2*c*((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])) + (c*Sqrt [1 + a^2*x^2]*(-2*ArcTan[a*x]^2*ArcTanh[E^(I*ArcTan[a*x])] + 2*(I*ArcTan[a *x]*PolyLog[2, -E^(I*ArcTan[a*x])] - PolyLog[3, -E^(I*ArcTan[a*x])]) - 2*( I*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x])] - PolyLog[3, E^(I*ArcTan[a*x]) ])))/Sqrt[c + a^2*c*x^2])) + c*(a^2*c*(a^2*c*((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])) + (c*Sqrt[1 + a^2*x^2]*(-2*ArcTan[a*x]^2*ArcTanh[E^ (I*ArcTan[a*x])] + 2*(I*ArcTan[a*x]*PolyLog[2, -E^(I*ArcTan[a*x])] - PolyL og[3, -E^(I*ArcTan[a*x])]) - 2*(I*ArcTan[a*x]*PolyLog[2, E^(I*ArcTan[a*x]) ] - PolyLog[3, E^(I*ArcTan[a*x])])))/Sqrt[c + a^2*c*x^2]) + c*(c*(-1/2*...
3.4.29.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbo l] :> Simp[(-b)*((d + e*x^2)^q/(2*c*q*(2*q + 1))), x] + (Simp[x*(d + e*x^2) ^q*((a + b*ArcTan[c*x])/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x]), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 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[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_.))^(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]
Time = 3.09 (sec) , antiderivative size = 454, normalized size of antiderivative = 0.69
| method | result | size |
| default | \(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (2 a^{4} \arctan \left (a x \right )^{2} x^{4}-2 \arctan \left (a x \right ) x^{3} a^{3}+14 x^{2} \arctan \left (a x \right )^{2} a^{2}+2 a^{2} x^{2}-6 x \arctan \left (a x \right ) a -3 \arctan \left (a x \right )^{2}\right )}{6 x^{2}}-\frac {c^{2} a^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (15 \arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )-15 \arctan \left (a x \right )^{2} \ln \left (1-\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-30 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )+30 i \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-26 \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+26 \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+26 i \operatorname {dilog}\left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-26 i \operatorname {dilog}\left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-6 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}-1\right )+6 \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}+1\right )+30 \operatorname {polylog}\left (3, -\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )-30 \operatorname {polylog}\left (3, \frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{6 \sqrt {a^{2} x^{2}+1}}\) | \(454\) |
1/6*c^2*(c*(a*x-I)*(I+a*x))^(1/2)*(2*a^4*arctan(a*x)^2*x^4-2*arctan(a*x)*x ^3*a^3+14*x^2*arctan(a*x)^2*a^2+2*a^2*x^2-6*x*arctan(a*x)*a-3*arctan(a*x)^ 2)/x^2-1/6*c^2*a^2*(c*(a*x-I)*(I+a*x))^(1/2)*(15*arctan(a*x)^2*ln((1+I*a*x )/(a^2*x^2+1)^(1/2)+1)-15*arctan(a*x)^2*ln(1-(1+I*a*x)/(a^2*x^2+1)^(1/2))- 30*I*arctan(a*x)*polylog(2,-(1+I*a*x)/(a^2*x^2+1)^(1/2))+30*I*arctan(a*x)* polylog(2,(1+I*a*x)/(a^2*x^2+1)^(1/2))-26*arctan(a*x)*ln(1+I*(1+I*a*x)/(a^ 2*x^2+1)^(1/2))+26*arctan(a*x)*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+26*I*di log(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-26*I*dilog(1-I*(1+I*a*x)/(a^2*x^2+1)^ (1/2))-6*ln((1+I*a*x)/(a^2*x^2+1)^(1/2)-1)+6*ln((1+I*a*x)/(a^2*x^2+1)^(1/2 )+1)+30*polylog(3,-(1+I*a*x)/(a^2*x^2+1)^(1/2))-30*polylog(3,(1+I*a*x)/(a^ 2*x^2+1)^(1/2)))/(a^2*x^2+1)^(1/2)
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}}{x^{3}} \,d x } \]
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=\int \frac {\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}{x^{3}}\, dx \]
\[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=\int { \frac {{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2}}{x^{3}} \,d x } \]
Exception generated. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2}{x^3} \, dx=\int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}}{x^3} \,d x \]