Integrand size = 21, antiderivative size = 745 \[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\frac {a+b \arcsin (c x)}{4 e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {a+b \arcsin (c x)}{4 e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{3/2} \sqrt {c^2 d+e}}-\frac {b c \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{4 e^{3/2} \sqrt {c^2 d+e}}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{4 \sqrt {-d} e^{3/2}} \] Output:
1/4*(a+b*arcsin(c*x))/e^(3/2)/((-d)^(1/2)-e^(1/2)*x)-1/4*(a+b*arcsin(c*x)) /e^(3/2)/((-d)^(1/2)+e^(1/2)*x)-1/4*b*c*arctanh((e^(1/2)-c^2*(-d)^(1/2)*x) /(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/e^(3/2)/(c^2*d+e)^(1/2)-1/4*b*c*arcta nh((e^(1/2)+c^2*(-d)^(1/2)*x)/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/e^(3/2)/ (c^2*d+e)^(1/2)+1/4*(a+b*arcsin(c*x))*ln(1-e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/ 2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(3/2)-1/4*(a+b*arcsin(c *x))*ln(1+e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/ 2)))/(-d)^(1/2)/e^(3/2)+1/4*(a+b*arcsin(c*x))*ln(1-e^(1/2)*(I*c*x+(-c^2*x^ 2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(3/2)-1/4*(a+b* arcsin(c*x))*ln(1+e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2* d+e)^(1/2)))/(-d)^(1/2)/e^(3/2)+1/4*I*b*polylog(2,-e^(1/2)*(I*c*x+(-c^2*x^ 2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(3/2)-1/4*I*b*p olylog(2,e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2 )))/(-d)^(1/2)/e^(3/2)+1/4*I*b*polylog(2,-e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2 ))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(1/2)/e^(3/2)-1/4*I*b*polylog(2, e^(1/2)*(I*c*x+(-c^2*x^2+1)^(1/2))/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^ (1/2)/e^(3/2)
Time = 0.81 (sec) , antiderivative size = 603, normalized size of antiderivative = 0.81 \[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\frac {-\frac {4 a \sqrt {e} x}{d+e x^2}+\frac {4 a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d}}+b \left (-\frac {2 \arcsin (c x)}{i \sqrt {d}+\sqrt {e} x}-2 i \left (\frac {\arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \arctan \left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )-\frac {2 c \text {arctanh}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}-\frac {\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )}{\sqrt {d}}+\frac {\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )}{\sqrt {d}}\right )}{8 e^{3/2}} \] Input:
Integrate[(x^2*(a + b*ArcSin[c*x]))/(d + e*x^2)^2,x]
Output:
((-4*a*Sqrt[e]*x)/(d + e*x^2) + (4*a*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[d] + b*((-2*ArcSin[c*x])/(I*Sqrt[d] + Sqrt[e]*x) - (2*I)*(ArcSin[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]) - (2*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] - (ArcSin[c*x ]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - S qrt[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]) + S qrt[c^2*d + e])] + 2*PolyLog[2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))])/Sqrt[d] + (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 - (S qrt[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])])/Sqrt[d]))/(8*e ^(3/2))
Time = 2.86 (sec) , antiderivative size = 745, 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.
| \(\displaystyle \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 5232 |
| \(\displaystyle \int \left (\frac {a+b \arcsin (c x)}{e \left (d+e x^2\right )}-\frac {d (a+b \arcsin (c x))}{e \left (d+e x^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {(a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {(a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{\sqrt {c^2 d+e}+i c \sqrt {-d}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {a+b \arcsin (c x)}{4 e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {a+b \arcsin (c x)}{4 e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{4 \sqrt {-d} e^{3/2}}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{4 \sqrt {-d} e^{3/2}}-\frac {b c \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{4 e^{3/2} \sqrt {c^2 d+e}}-\frac {b c \text {arctanh}\left (\frac {c^2 \sqrt {-d} x+\sqrt {e}}{\sqrt {1-c^2 x^2} \sqrt {c^2 d+e}}\right )}{4 e^{3/2} \sqrt {c^2 d+e}}\) |
Input:
Int[(x^2*(a + b*ArcSin[c*x]))/(d + e*x^2)^2,x]
Output:
(a + b*ArcSin[c*x])/(4*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) - (a + b*ArcSin[c*x ])/(4*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) - (b*c*ArcTanh[(Sqrt[e] - c^2*Sqrt[- d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(4*e^(3/2)*Sqrt[c^2*d + e]) - (b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2] )])/(4*e^(3/2)*Sqrt[c^2*d + e]) + ((a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^ (I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(4*Sqrt[-d]*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])])/(4*Sqrt[-d]*e^(3/2)) + ((a + b*ArcSin[c*x])*Log[1 - (Sq rt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^ (3/2)) - ((a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqr t[-d] + Sqrt[c^2*d + e])])/(4*Sqrt[-d]*e^(3/2)) + ((I/4)*b*PolyLog[2, -((S qrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/(Sqrt[-d]*e^ (3/2)) - ((I/4)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - S qrt[c^2*d + e])])/(Sqrt[-d]*e^(3/2)) + ((I/4)*b*PolyLog[2, -((Sqrt[e]*E^(I *ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e]))])/(Sqrt[-d]*e^(3/2)) - (( I/4)*b*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(Sqrt[-d]*e^(3/2))
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 = 417.55 (sec) , antiderivative size = 811, normalized size of antiderivative = 1.09
| method | result | size |
| derivativedivides | \(\frac {-\frac {a \,c^{5} x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a \,c^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+b \,c^{4} \left (-\frac {\arcsin \left (c x \right ) c x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\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 {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 )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}}{4 e}-\frac {\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 {\textit {\_R1} \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}}{4 e}+\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right )}\, \left (2 c^{4} d^{2}+2 \sqrt {c^{2} d \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e +\sqrt {c^{2} d \left (c^{2} d +e \right )}\, e \right ) \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 e^{4} \left (c^{2} d +e \right )}-\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right )}\, \left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 e^{4}}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}\, \left (-2 \sqrt {c^{2} d \left (c^{2} d +e \right )}\, c^{2} d +2 c^{4} d^{2}+2 c^{2} d e -\sqrt {c^{2} d \left (c^{2} d +e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 e^{4} \left (c^{2} d +e \right )}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 e^{4}}\right )}{c^{3}}\) | \(811\) |
| default | \(\frac {-\frac {a \,c^{5} x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {a \,c^{3} \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+b \,c^{4} \left (-\frac {\arcsin \left (c x \right ) c x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}-\frac {\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 {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 )}{\textit {\_R1} \left (-\textit {\_R1}^{2} e +2 c^{2} d +e \right )}}{4 e}-\frac {\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 {\textit {\_R1} \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}}{4 e}+\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right )}\, \left (2 c^{4} d^{2}+2 \sqrt {c^{2} d \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e +\sqrt {c^{2} d \left (c^{2} d +e \right )}\, e \right ) \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 e^{4} \left (c^{2} d +e \right )}-\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right )}\, \left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 e^{4}}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}\, \left (-2 \sqrt {c^{2} d \left (c^{2} d +e \right )}\, c^{2} d +2 c^{4} d^{2}+2 c^{2} d e -\sqrt {c^{2} d \left (c^{2} d +e \right )}\, e \right ) \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 e^{4} \left (c^{2} d +e \right )}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 e^{4}}\right )}{c^{3}}\) | \(811\) |
| parts | \(-\frac {a x}{2 e \left (e \,x^{2}+d \right )}+\frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{2 e \sqrt {d e}}+\frac {b \left (-\frac {c^{5} \arcsin \left (c x \right ) x}{2 e \left (c^{2} e \,x^{2}+c^{2} d \right )}+\frac {c^{4} \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 {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 )}{\textit {\_R1} \left (\textit {\_R1}^{2} e -2 c^{2} d -e \right )}\right )}{4 e}+\frac {c^{4} \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 {\textit {\_R1} \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 )}{4 e}+\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}\, \left (-2 \sqrt {c^{2} d \left (c^{2} d +e \right )}\, c^{2} d +2 c^{4} d^{2}+2 c^{2} d e -\sqrt {c^{2} d \left (c^{2} d +e \right )}\, e \right ) c^{4} \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}}\right )}{2 e^{4} \left (c^{2} d +e \right )}-\frac {\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}\, \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) \operatorname {arctanh}\left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) e}}\right ) c^{4}}{2 e^{4}}+\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right )}\, \left (2 c^{4} d^{2}+2 \sqrt {c^{2} d \left (c^{2} d +e \right )}\, c^{2} d +2 c^{2} d e +\sqrt {c^{2} d \left (c^{2} d +e \right )}\, e \right ) c^{4} \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}-e \right ) e}}\right )}{2 e^{4} \left (c^{2} d +e \right )}-\frac {\sqrt {-e \left (2 c^{2} d -2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right )}\, \left (2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}+e \right ) \arctan \left (\frac {e \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )}{\sqrt {\left (-2 c^{2} d +2 \sqrt {c^{2} d \left (c^{2} d +e \right )}-e \right ) e}}\right ) c^{4}}{2 e^{4}}\right )}{c^{3}}\) | \(816\) |
Input:
int(x^2*(a+b*arcsin(c*x))/(e*x^2+d)^2,x,method=_RETURNVERBOSE)
Output:
1/c^3*(-1/2*a*c^5/e*x/(c^2*e*x^2+c^2*d)+1/2*a*c^3/e/(d*e)^(1/2)*arctan(e*x /(d*e)^(1/2))+b*c^4*(-1/2*arcsin(c*x)*c*x/e/(c^2*e*x^2+c^2*d)-1/4/e*sum(1/ _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))-1/4/e*sum(_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))+1/2*(-e*(2*c^2*d-2*(c^2*d*(c^2*d +e))^(1/2)+e))^(1/2)*(2*c^4*d^2+2*(c^2*d*(c^2*d+e))^(1/2)*c^2*d+2*c^2*d*e+ (c^2*d*(c^2*d+e))^(1/2)*e)*arctan(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+ 2*(c^2*d*(c^2*d+e))^(1/2)-e)*e)^(1/2))/e^4/(c^2*d+e)-1/2*(-e*(2*c^2*d-2*(c ^2*d*(c^2*d+e))^(1/2)+e))^(1/2)*(2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*arct an(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)-e)*e) ^(1/2))/e^4+1/2*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(-2*(c^2*d *(c^2*d+e))^(1/2)*c^2*d+2*c^4*d^2+2*c^2*d*e-(c^2*d*(c^2*d+e))^(1/2)*e)*arc tanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e )^(1/2))/e^4/(c^2*d+e)-1/2*((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2) *(2*c^2*d-2*(c^2*d*(c^2*d+e))^(1/2)+e)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2) )/((2*c^2*d+2*(c^2*d*(c^2*d+e))^(1/2)+e)*e)^(1/2))/e^4))
\[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate(x^2*(a+b*arcsin(c*x))/(e*x^2+d)^2,x, algorithm="fricas")
Output:
integral((b*x^2*arcsin(c*x) + a*x^2)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)
\[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{2}}\, dx \] Input:
integrate(x**2*(a+b*asin(c*x))/(e*x**2+d)**2,x)
Output:
Integral(x**2*(a + b*asin(c*x))/(d + e*x**2)**2, x)
Exception generated. \[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(x^2*(a+b*arcsin(c*x))/(e*x^2+d)^2,x, algorithm="maxima")
Output:
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
\[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{2}} \,d x } \] Input:
integrate(x^2*(a+b*arcsin(c*x))/(e*x^2+d)^2,x, algorithm="giac")
Output:
integrate((b*arcsin(c*x) + a)*x^2/(e*x^2 + d)^2, x)
Timed out. \[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^2} \,d x \] Input:
int((x^2*(a + b*asin(c*x)))/(d + e*x^2)^2,x)
Output:
int((x^2*(a + b*asin(c*x)))/(d + e*x^2)^2, x)
\[ \int \frac {x^2 (a+b \arcsin (c x))}{\left (d+e x^2\right )^2} \, dx=\frac {\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a d +\sqrt {e}\, \sqrt {d}\, \mathit {atan} \left (\frac {e x}{\sqrt {e}\, \sqrt {d}}\right ) a e \,x^{2}+2 \left (\int \frac {\mathit {asin} \left (c x \right ) x^{2}}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b \,d^{2} e^{2}+2 \left (\int \frac {\mathit {asin} \left (c x \right ) x^{2}}{e^{2} x^{4}+2 d e \,x^{2}+d^{2}}d x \right ) b d \,e^{3} x^{2}-a d e x}{2 d \,e^{2} \left (e \,x^{2}+d \right )} \] Input:
int(x^2*(a+b*asin(c*x))/(e*x^2+d)^2,x)
Output:
(sqrt(e)*sqrt(d)*atan((e*x)/(sqrt(e)*sqrt(d)))*a*d + sqrt(e)*sqrt(d)*atan( (e*x)/(sqrt(e)*sqrt(d)))*a*e*x**2 + 2*int((asin(c*x)*x**2)/(d**2 + 2*d*e*x **2 + e**2*x**4),x)*b*d**2*e**2 + 2*int((asin(c*x)*x**2)/(d**2 + 2*d*e*x** 2 + e**2*x**4),x)*b*d*e**3*x**2 - a*d*e*x)/(2*d*e**2*(d + e*x**2))