Integrand size = 21, antiderivative size = 649 \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{6 d x^2}-\frac {a+b \arcsin (c x)}{3 d x^3}+\frac {e (a+b \arcsin (c x))}{d^2 x}-\frac {b c^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 d}+\frac {b c e \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}+\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}+\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{2 (-d)^{5/2}} \]
1/3*(-a-b*arcsin(c*x))/d/x^3+e*(a+b*arcsin(c*x))/d^2/x-1/6*b*c^3*arctanh(( -c^2*x^2+1)^(1/2))/d+b*c*e*arctanh((-c^2*x^2+1)^(1/2))/d^2+1/2*e^(3/2)*(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)^(5/2)-1/2*e^(3/2)*(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)^(5/2)+1/2*e^( 3/2)*(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)^(5/2)-1/2*e^(3/2)*(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)^(5/2) +1/2*I*b*e^(3/2)*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)^(5/2)-1/2*I*b*e^(3/2)*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)^(5/2)+1/2*I*b* e^(3/2)*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)^(5/2)-1/2*I*b*e^(3/2)*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)^(5/2)-1/6*b*c*(-c^2*x^2 +1)^(1/2)/d/x^2
Time = 0.30 (sec) , antiderivative size = 531, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=-\frac {a}{3 d x^3}+\frac {a e}{d^2 x}+\frac {a e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{5/2}}+b \left (-\frac {e \left (-\frac {\arcsin (c x)}{x}-c \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )\right )}{d^2}-\frac {c x \sqrt {1-c^2 x^2}+2 \arcsin (c x)+c^3 x^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 d x^3}-\frac {e^{3/2} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{4 d^{5/2}}+\frac {e^{3/2} \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{4 d^{5/2}}\right ) \]
-1/3*a/(d*x^3) + (a*e)/(d^2*x) + (a*e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/d ^(5/2) + b*(-((e*(-(ArcSin[c*x]/x) - c*ArcTanh[Sqrt[1 - c^2*x^2]]))/d^2) - (c*x*Sqrt[1 - c^2*x^2] + 2*ArcSin[c*x] + c^3*x^3*ArcTanh[Sqrt[1 - c^2*x^2 ]])/(6*d*x^3) - (e^(3/2)*(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, -((Sqr t[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]))/(4*d^(5/2)) + (e ^(3/2)*(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*d^(5/2)))
Time = 1.31 (sec) , antiderivative size = 649, 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.
\(\displaystyle \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx\) |
\(\Big \downarrow \) 5232 |
\(\displaystyle \int \left (\frac {e^2 (a+b \arcsin (c x))}{d^2 \left (d+e x^2\right )}-\frac {e (a+b \arcsin (c x))}{d^2 x^2}+\frac {a+b \arcsin (c x)}{d x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 (-d)^{5/2}}+\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 (-d)^{5/2}}-\frac {e^{3/2} (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{2 (-d)^{5/2}}+\frac {e (a+b \arcsin (c x))}{d^2 x}-\frac {a+b \arcsin (c x)}{3 d x^3}+\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}+\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}-\frac {i b e^{3/2} \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{2 (-d)^{5/2}}+\frac {b c e \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{d^2}-\frac {b c^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )}{6 d}-\frac {b c \sqrt {1-c^2 x^2}}{6 d x^2}\) |
-1/6*(b*c*Sqrt[1 - c^2*x^2])/(d*x^2) - (a + b*ArcSin[c*x])/(3*d*x^3) + (e* (a + b*ArcSin[c*x]))/(d^2*x) - (b*c^3*ArcTanh[Sqrt[1 - c^2*x^2]])/(6*d) + (b*c*e*ArcTanh[Sqrt[1 - c^2*x^2]])/d^2 + (e^(3/2)*(a + b*ArcSin[c*x])*Log[ 1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d) ^(5/2)) - (e^(3/2)*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x])) /(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + (e^(3/2)*(a + b*ArcSi n[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e ])])/(2*(-d)^(5/2)) - (e^(3/2)*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*A rcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(2*(-d)^(5/2)) + ((I/2)*b* e^(3/2)*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2* d + e]))])/(-d)^(5/2) - ((I/2)*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c *x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(-d)^(5/2) + ((I/2)*b*e^(3/2)*Pol yLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/ (-d)^(5/2) - ((I/2)*b*e^(3/2)*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c* Sqrt[-d] + Sqrt[c^2*d + e])])/(-d)^(5/2)
3.7.32.3.1 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 = 6.28 (sec) , antiderivative size = 491, normalized size of antiderivative = 0.76
method | result | size |
parts | \(a \left (-\frac {1}{3 d \,x^{3}}+\frac {e}{d^{2} x}+\frac {e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{d^{2} \sqrt {d e}}\right )-\frac {b \left (-4 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{7} d^{2} x^{3}+4 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{7} d^{2} x^{3}+4 \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} x +24 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{5} d e \,x^{3}-24 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{5} d e \,x^{3}-24 \arcsin \left (c x \right ) c^{4} d e \,x^{2}+8 c^{4} d^{2} \arcsin \left (c x \right )+3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (\textit {\_R1}^{2} e -4 c^{2} d -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e -2 c^{2} d -e \right )}\right ) c^{3} x^{3} e^{2}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (\textit {\_R1}^{2} e -2 c^{2} d -e \right )}\right ) c^{3} x^{3} e^{2}\right )}{24 c^{4} x^{3} d^{3}}\) | \(491\) |
derivativedivides | \(c^{3} \left (-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}+\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}+\frac {b \left (4 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{7} d^{2} x^{3}-4 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{7} d^{2} x^{3}-4 \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} x -24 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{5} d e \,x^{3}+24 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{5} d e \,x^{3}-8 c^{4} d^{2} \arcsin \left (c x \right )+24 \arcsin \left (c x \right ) c^{4} d e \,x^{2}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (-\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{7} x^{3} d^{3}}\right )\) | \(501\) |
default | \(c^{3} \left (-\frac {a}{3 d \,c^{3} x^{3}}+\frac {a e}{c^{3} d^{2} x}+\frac {a \,e^{2} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{c^{3} d^{2} \sqrt {d e}}+\frac {b \left (4 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{7} d^{2} x^{3}-4 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{7} d^{2} x^{3}-4 \sqrt {-c^{2} x^{2}+1}\, c^{5} d^{2} x -24 \ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right ) c^{5} d e \,x^{3}+24 \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right ) c^{5} d e \,x^{3}-8 c^{4} d^{2} \arcsin \left (c x \right )+24 \arcsin \left (c x \right ) c^{4} d e \,x^{2}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (4 \textit {\_R1}^{2} c^{2} d +\textit {\_R1}^{2} e -e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}-3 \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (e \,\textit {\_Z}^{4}+\left (-4 c^{2} d -2 e \right ) \textit {\_Z}^{2}+e \right )}{\sum }\frac {\left (-\textit {\_R1}^{2} e +4 c^{2} d +e \right ) \left (i \arcsin \left (c x \right ) \ln \left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {\textit {\_R1} -i c x -\sqrt {-c^{2} x^{2}+1}}{\textit {\_R1}}\right )\right )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}\right ) e^{2} c^{3} x^{3}\right )}{24 c^{7} x^{3} d^{3}}\right )\) | \(501\) |
a*(-1/3/d/x^3+e/d^2/x+e^2/d^2/(d*e)^(1/2)*arctan(e*x/(d*e)^(1/2)))-1/24*b/ c^4*(-4*ln(I*c*x+(-c^2*x^2+1)^(1/2)-1)*c^7*d^2*x^3+4*ln(1+I*c*x+(-c^2*x^2+ 1)^(1/2))*c^7*d^2*x^3+4*(-c^2*x^2+1)^(1/2)*c^5*d^2*x+24*ln(I*c*x+(-c^2*x^2 +1)^(1/2)-1)*c^5*d*e*x^3-24*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))*c^5*d*e*x^3-24* arcsin(c*x)*c^4*d*e*x^2+8*c^4*d^2*arcsin(c*x)+3*sum((_R1^2*e-4*c^2*d-e)/_R 1/(_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))*c^3*x^3*e^2-3*sum((4*_R1^2*c^2*d+_R1^2*e-e)/_R1/(_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))*c^ 3*x^3*e^2)/x^3/d^3
\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{4}} \,d x } \]
\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{x^{4} \left (d + e x^{2}\right )}\, dx \]
Exception generated. \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )} x^{4}} \,d x } \]
Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^4 \left (d+e x^2\right )} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^4\,\left (e\,x^2+d\right )} \,d x \]