Integrand size = 21, antiderivative size = 632 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{2 d^2 x}-\frac {a+b \arcsin (c x)}{2 d^2 x^2}-\frac {e (a+b \arcsin (c x))}{2 d^2 \left (d+e x^2\right )}+\frac {b c e \arctan \left (\frac {\sqrt {c^2 d+e} x}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e}}+\frac {e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{d^3}-\frac {2 e (a+b \arcsin (c x)) \log \left (1-e^{2 i \arcsin (c x)}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{d^3}+\frac {i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3} \]
1/2*(-a-b*arcsin(c*x))/d^2/x^2-1/2*e*(a+b*arcsin(c*x))/d^2/(e*x^2+d)-2*e*( a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3+e*(a+b*arcsin(c*x) )*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)) )/d^3+e*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d )^(1/2)-(c^2*d+e)^(1/2)))/d^3+e*(a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1) ^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/d^3+e*(a+b*arcsin(c*x))* ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/ d^3+I*b*e*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3-I*b*e*polylog(2,-(I* c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/d^3-I*b* e*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(c^2*d+e)^( 1/2)))/d^3-I*b*e*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^( 1/2)+(c^2*d+e)^(1/2)))/d^3-I*b*e*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1 /2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/d^3+1/2*b*c*e*arctan(x*(c^2*d+e)^(1/ 2)/d^(1/2)/(-c^2*x^2+1)^(1/2))/d^(5/2)/(c^2*d+e)^(1/2)-1/2*b*c*(-c^2*x^2+1 )^(1/2)/d^2/x
Time = 1.73 (sec) , antiderivative size = 687, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=-\frac {\frac {2 a d}{x^2}+\frac {2 a d e}{d+e x^2}+8 a e \log (x)-4 a e \log \left (d+e x^2\right )+b \left (\frac {2 d \left (c x \sqrt {1-c^2 x^2}+\arcsin (c x)\right )}{x^2}+\sqrt {d} e \left (\frac {\arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \arctan \left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )-i \sqrt {d} e \left (-\frac {\arcsin (c x)}{i \sqrt {d}+\sqrt {e} x}-\frac {c \text {arctanh}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )+2 i e \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )+2 i e \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )-4 i e \left (\arcsin (c x) \left (\arcsin (c x)+2 i \log \left (1-e^{2 i \arcsin (c x)}\right )\right )+\operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )\right )}{4 d^3} \]
-1/4*((2*a*d)/x^2 + (2*a*d*e)/(d + e*x^2) + 8*a*e*Log[x] - 4*a*e*Log[d + e *x^2] + b*((2*d*(c*x*Sqrt[1 - c^2*x^2] + ArcSin[c*x]))/x^2 + Sqrt[d]*e*(Ar cSin[c*x]/(Sqrt[d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqrt[e] + c^2*Sqrt[d]*x)/ (Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) - I*Sqrt[d]*e*(-(Ar cSin[c*x]/(I*Sqrt[d] + Sqrt[e]*x)) - (c*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]*x )/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]) + (2*I)*e*(ArcSin [c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sq rt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[ d] + Sqrt[c^2*d + e]))]) + (2*I)*e*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[ 1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog [2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + 2*PolyLog [2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])]) - (4*I)*e* (ArcSin[c*x]*(ArcSin[c*x] + (2*I)*Log[1 - E^((2*I)*ArcSin[c*x])]) + PolyLo g[2, E^((2*I)*ArcSin[c*x])])))/d^3
Time = 1.43 (sec) , antiderivative size = 632, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5232, 2009}
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.
| \(\displaystyle \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 5232 |
| \(\displaystyle \int \left (\frac {2 e^2 x (a+b \arcsin (c x))}{d^3 \left (d+e x^2\right )}-\frac {2 e (a+b \arcsin (c x))}{d^3 x}+\frac {e^2 x (a+b \arcsin (c x))}{d^2 \left (d+e x^2\right )^2}+\frac {a+b \arcsin (c x)}{d^2 x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{d^3}+\frac {e (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{d^3}-\frac {2 e \log \left (1-e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^3}-\frac {e (a+b \arcsin (c x))}{2 d^2 \left (d+e x^2\right )}-\frac {a+b \arcsin (c x)}{2 d^2 x^2}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{d^3}-\frac {i b e \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{d^3}+\frac {i b e \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{d^3}+\frac {b c e \arctan \left (\frac {x \sqrt {c^2 d+e}}{\sqrt {d} \sqrt {1-c^2 x^2}}\right )}{2 d^{5/2} \sqrt {c^2 d+e}}-\frac {b c \sqrt {1-c^2 x^2}}{2 d^2 x}\) |
-1/2*(b*c*Sqrt[1 - c^2*x^2])/(d^2*x) - (a + b*ArcSin[c*x])/(2*d^2*x^2) - ( e*(a + b*ArcSin[c*x]))/(2*d^2*(d + e*x^2)) + (b*c*e*ArcTan[(Sqrt[c^2*d + e ]*x)/(Sqrt[d]*Sqrt[1 - c^2*x^2])])/(2*d^(5/2)*Sqrt[c^2*d + e]) + (e*(a + b *ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2 *d + e])])/d^3 + (e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]) )/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^3 + (e*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^3 + (e*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt [c^2*d + e])])/d^3 - (2*e*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x] )])/d^3 - (I*b*e*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/d^3 - (I*b*e*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I *c*Sqrt[-d] - Sqrt[c^2*d + e])])/d^3 - (I*b*e*PolyLog[2, -((Sqrt[e]*E^(I*A rcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/d^3 - (I*b*e*PolyLog[2, ( Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/d^3 + (I*b*e *PolyLog[2, E^((2*I)*ArcSin[c*x])])/d^3
3.7.36.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, ( f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 1.96 (sec) , antiderivative size = 589, normalized size of antiderivative = 0.93
| method | result | size |
| parts | \(-\frac {a}{2 d^{2} x^{2}}-\frac {2 a e \ln \left (x \right )}{d^{3}}+\frac {a e \ln \left (e \,x^{2}+d \right )}{d^{3}}-\frac {a e}{2 d^{2} \left (e \,x^{2}+d \right )}+b \,c^{2} \left (-\frac {-i c^{4} d \,x^{2}-i e \,c^{4} x^{4}+\sqrt {-c^{2} x^{2}+1}\, c^{3} d x +\sqrt {-c^{2} x^{2}+1}\, e \,c^{3} x^{3}+\arcsin \left (c x \right ) c^{2} d +2 \arcsin \left (c x \right ) c^{2} e \,x^{2}}{2 c^{2} x^{2} d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {2 i e \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3} c^{2}}+\frac {2 i e \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3} c^{2}}-\frac {i \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {2 e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-4 c^{2} d -2 e}{4 \sqrt {d^{2} c^{4}+c^{2} e d}}\right ) e}{2 d^{3} c^{2} \left (c^{2} d +e \right )}-\frac {i e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e -4 c^{2} d -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e -2 c^{2} d -e}\right )}{2 d^{3} c^{2}}-\frac {i e^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}-1\right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1}^{2} e -2 c^{2} d -e}\right )}{2 d^{3} c^{2}}-\frac {2 e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{d^{3} c^{2}}\right )\) | \(589\) |
| derivativedivides | \(c^{2} \left (-\frac {a}{2 d^{2} c^{2} x^{2}}-\frac {2 a e \ln \left (c x \right )}{c^{2} d^{3}}-\frac {a e}{2 d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{c^{2} d^{3}}+b \,c^{4} \left (-\frac {-i c^{4} d \,x^{2}-i e \,c^{4} x^{4}+\sqrt {-c^{2} x^{2}+1}\, c^{3} d x +\sqrt {-c^{2} x^{2}+1}\, e \,c^{3} x^{3}+\arcsin \left (c x \right ) c^{2} d +2 \arcsin \left (c x \right ) c^{2} e \,x^{2}}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) d^{2} c^{6} x^{2}}-\frac {2 i e \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}+\frac {2 i e \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}+\frac {i e^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}-1\right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}-\frac {i e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (-\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}+\frac {i \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {4 c^{2} d -2 e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}+2 e}{4 \sqrt {d^{2} c^{4}+c^{2} e d}}\right ) e}{2 c^{6} d^{3} \left (c^{2} d +e \right )}-\frac {2 e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}\right )\right )\) | \(615\) |
| default | \(c^{2} \left (-\frac {a}{2 d^{2} c^{2} x^{2}}-\frac {2 a e \ln \left (c x \right )}{c^{2} d^{3}}-\frac {a e}{2 d^{2} \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a e \ln \left (c^{2} e \,x^{2}+c^{2} d \right )}{c^{2} d^{3}}+b \,c^{4} \left (-\frac {-i c^{4} d \,x^{2}-i e \,c^{4} x^{4}+\sqrt {-c^{2} x^{2}+1}\, c^{3} d x +\sqrt {-c^{2} x^{2}+1}\, e \,c^{3} x^{3}+\arcsin \left (c x \right ) c^{2} d +2 \arcsin \left (c x \right ) c^{2} e \,x^{2}}{2 \left (c^{2} e \,x^{2}+c^{2} d \right ) d^{2} c^{6} x^{2}}-\frac {2 i e \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}+\frac {2 i e \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}+\frac {i e^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2}-1\right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}-\frac {i e \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (-\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{-\textit {\_R1}^{2} e +2 c^{2} d +e}\right )}{2 c^{6} d^{3}}+\frac {i \sqrt {d \,c^{2} \left (c^{2} d +e \right )}\, \operatorname {arctanh}\left (\frac {4 c^{2} d -2 e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}+2 e}{4 \sqrt {d^{2} c^{4}+c^{2} e d}}\right ) e}{2 c^{6} d^{3} \left (c^{2} d +e \right )}-\frac {2 e \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{c^{6} d^{3}}\right )\right )\) | \(615\) |
-1/2*a/d^2/x^2-2*a/d^3*e*ln(x)+a*e/d^3*ln(e*x^2+d)-1/2*a*e/d^2/(e*x^2+d)+b *c^2*(-1/2*(-I*c^4*d*x^2-I*e*c^4*x^4+(-c^2*x^2+1)^(1/2)*c^3*d*x+(-c^2*x^2+ 1)^(1/2)*e*c^3*x^3+arcsin(c*x)*c^2*d+2*arcsin(c*x)*c^2*e*x^2)/c^2/x^2/d^2/ (c^2*e*x^2+c^2*d)-2*I*e/d^3/c^2*dilog(I*c*x+(-c^2*x^2+1)^(1/2))+2*I*e/d^3/ c^2*dilog(1+I*c*x+(-c^2*x^2+1)^(1/2))-1/2*I*(d*c^2*(c^2*d+e))^(1/2)/d^3/c^ 2/(c^2*d+e)*arctanh(1/4*(2*e*(I*c*x+(-c^2*x^2+1)^(1/2))^2-4*c^2*d-2*e)/(c^ 4*d^2+c^2*d*e)^(1/2))*e-1/2*I*e/d^3/c^2*sum((_R1^2*e-4*c^2*d-e)/(_R1^2*e-2 *c^2*d-e)*(I*arcsin(c*x)*ln((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)+dilog((_R1 -I*c*x-(-c^2*x^2+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(-4*c^2*d-2*e)*_Z^2+e)) -1/2*I*e^2/d^3/c^2*sum((_R1^2-1)/(_R1^2*e-2*c^2*d-e)*(I*arcsin(c*x)*ln((_R 1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1) ),_R1=RootOf(e*_Z^4+(-4*c^2*d-2*e)*_Z^2+e))-2*e/d^3/c^2*arcsin(c*x)*ln(1+I *c*x+(-c^2*x^2+1)^(1/2)))
\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{x^{3} \left (d + e x^{2}\right )^{2}}\, dx \]
\[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{2} x^{3}} \,d x } \]
-1/2*a*((2*e*x^2 + d)/(d^2*e*x^4 + d^3*x^2) - 2*e*log(e*x^2 + d)/d^3 + 4*e *log(x)/d^3) + b*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/(e^2 *x^7 + 2*d*e*x^5 + d^2*x^3), x)
Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^3 \left (d+e x^2\right )^2} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^3\,{\left (e\,x^2+d\right )}^2} \,d x \]