2.10 \(\int \frac {x^3 e^{\sin ^{-1}(x)}}{\sqrt {1-x^2}} \,dx\)

2.10.1 Mathematica

ClearAll[x]; 
integrand = (x^3 Exp[ArcSin[x]])/Sqrt[1 - x^2]; 
res = Integrate[integrand, x] 
TeXForm[res]
 

\[ -\frac {1}{40} e^{\sin ^{-1}(x)} \left (15 \left (\sqrt {1-x^2}-x\right )+\sin \left (3 \sin ^{-1}(x)\right )-3 \cos \left (3 \sin ^{-1}(x)\right )\right ) \]

2.10.2 Rubi

<< Rubi` 
ClearAll[x] 
integrand = (x^3 Exp[ArcSin[x]])/Sqrt[1 - x^2]; 
res = Int[integrand, x]; 
TeXForm[res]
 

\[ \frac {1}{10} x^3 e^{\sin ^{-1}(x)}-\frac {3}{10} \sqrt {1-x^2} x^2 e^{\sin ^{-1}(x)}-\frac {3}{10} \sqrt {1-x^2} e^{\sin ^{-1}(x)}+\frac {3}{10} x e^{\sin ^{-1}(x)} \]

2.10.3 Maple

restart; 
integrand := (x^3*exp(1)^arcsin(x))/sqrt(1 - x^2); 
res:=int(integrand,x); 
latex(res)
 

\[ {\frac {{{\rm e}^{\arcsin \left ( x \right ) }}{x}^{2}}{10} \left ( x-3\, \sqrt {-{x}^{2}+1} \right ) }+{\frac {3\,{{\rm e}^{\arcsin \left ( x \right ) }}}{10} \left ( x-\sqrt {-{x}^{2}+1} \right ) } \]

2.10.4 Fricas

set output tex off 
setSimplifyDenomsFlag(true) 
integrand := (x^3*exp(asin(x)))/sqrt(1 - x^2); 
res:=integrate(integrand,x); 
latex(res)
 

\[ {{\left ( {{\left ( -{3 \ {{x} \sp {2}}} -3 \right )} \ {\sqrt {{-{{x} \sp {2 }}+1}}}}+{{x} \sp {3}}+{3 \ x} \right )} \ {{e} \sp {{\arcsin \left ( {x} \right )}}}} \over {10} \]

2.10.5 Maxima

integrand : (x^3*exp(asin(x)))/sqrt(1 - x^2); 
res : integrate(integrand,x); 
tex(res);
 

\[ \text {did not solve} \]

2.10.6 XCAS

integrand := (x^3*exp(asin(x)))/sqrt(1 - x^2); 
res := integrate(integrand,x); 
latex(res)
 

\[ \frac {2}{5} x \mathrm {e}^{\arcsin x}-\frac {3}{5} \sqrt {-x^{2}+1} \mathrm {e}^{\arcsin x}+\frac {3}{10} \sqrt {-x^{2}+1} \left (-x^{2}+1\right ) \mathrm {e}^{\arcsin x}-\frac {1}{10} x \left (-x^{2}+1\right ) \mathrm {e}^{\arcsin x} \]

2.10.7 Sympy

>python 
Python 3.7.3 (default, Mar 27 2019, 22:11:17) 
[GCC 7.3.0] :: Anaconda, Inc. on linux 
 
from sympy import * 
x = symbols('x') 
integrand = (x**3*exp(asin(x)))/sqrt(1 - x**2); 
res = integrate(integrand,x); 
latex(res)
 

\[ \frac {x^{3} e^{\operatorname {asin}{\left (x \right )}}}{10} - \frac {3 x^{2} \sqrt {1 - x^{2}} e^{\operatorname {asin}{\left (x \right )}}}{10} + \frac {3 x e^{\operatorname {asin}{\left (x \right )}}}{10} - \frac {3 \sqrt {1 - x^{2}} e^{\operatorname {asin}{\left (x \right )}}}{10} \]

2.10.8 MuPad

evalin(symengine,'int( (x^3*exp(arcsin(x)))/sqrt(1 - x^2),x)')
 

\[ \text {did not solve} \]