Integrand size = 21, antiderivative size = 112 \[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{3/2} \, dx=\frac {24 e^{\arcsin (a x)}}{85 a}+\frac {24}{85} e^{\arcsin (a x)} x \sqrt {1-a^2 x^2}+\frac {12 e^{\arcsin (a x)} \left (1-a^2 x^2\right )}{85 a}+\frac {4}{17} e^{\arcsin (a x)} x \left (1-a^2 x^2\right )^{3/2}+\frac {e^{\arcsin (a x)} \left (1-a^2 x^2\right )^2}{17 a} \] Output:
24/85*exp(arcsin(a*x))/a+24/85*exp(arcsin(a*x))*x*(-a^2*x^2+1)^(1/2)+12/85 *exp(arcsin(a*x))*(-a^2*x^2+1)/a+4/17*exp(arcsin(a*x))*x*(-a^2*x^2+1)^(3/2 )+1/17*exp(arcsin(a*x))*(-a^2*x^2+1)^2/a
Time = 0.17 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.46 \[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{3/2} \, dx=\frac {e^{\arcsin (a x)} (255+68 \cos (2 \arcsin (a x))+5 \cos (4 \arcsin (a x))+136 \sin (2 \arcsin (a x))+20 \sin (4 \arcsin (a x)))}{680 a} \] Input:
Integrate[E^ArcSin[a*x]*(1 - a^2*x^2)^(3/2),x]
Output:
(E^ArcSin[a*x]*(255 + 68*Cos[2*ArcSin[a*x]] + 5*Cos[4*ArcSin[a*x]] + 136*S in[2*ArcSin[a*x]] + 20*Sin[4*ArcSin[a*x]]))/(680*a)
Time = 0.79 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5335, 7292, 7271, 4935, 4935, 2624}
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.
\(\displaystyle \int \left (1-a^2 x^2\right )^{3/2} e^{\arcsin (a x)} \, dx\) |
\(\Big \downarrow \) 5335 |
\(\displaystyle \frac {\int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^2d\arcsin (a x)}{a}\) |
\(\Big \downarrow \) 4935 |
\(\displaystyle \frac {\frac {12}{17} \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )d\arcsin (a x)+\frac {1}{17} \left (1-a^2 x^2\right )^2 e^{\arcsin (a x)}+\frac {4}{17} a x \left (1-a^2 x^2\right )^{3/2} e^{\arcsin (a x)}}{a}\) |
\(\Big \downarrow \) 4935 |
\(\displaystyle \frac {\frac {12}{17} \left (\frac {2}{5} \int e^{\arcsin (a x)}d\arcsin (a x)+\frac {2}{5} a x \sqrt {1-a^2 x^2} e^{\arcsin (a x)}+\frac {1}{5} \left (1-a^2 x^2\right ) e^{\arcsin (a x)}\right )+\frac {1}{17} \left (1-a^2 x^2\right )^2 e^{\arcsin (a x)}+\frac {4}{17} a x \left (1-a^2 x^2\right )^{3/2} e^{\arcsin (a x)}}{a}\) |
\(\Big \downarrow \) 2624 |
\(\displaystyle \frac {\frac {1}{17} \left (1-a^2 x^2\right )^2 e^{\arcsin (a x)}+\frac {4}{17} a x \left (1-a^2 x^2\right )^{3/2} e^{\arcsin (a x)}+\frac {12}{17} \left (\frac {2}{5} a x \sqrt {1-a^2 x^2} e^{\arcsin (a x)}+\frac {1}{5} \left (1-a^2 x^2\right ) e^{\arcsin (a x)}+\frac {2}{5} e^{\arcsin (a x)}\right )}{a}\) |
Input:
Int[E^ArcSin[a*x]*(1 - a^2*x^2)^(3/2),x]
Output:
((4*a*E^ArcSin[a*x]*x*(1 - a^2*x^2)^(3/2))/17 + (E^ArcSin[a*x]*(1 - a^2*x^ 2)^2)/17 + (12*((2*E^ArcSin[a*x])/5 + (2*a*E^ArcSin[a*x]*x*Sqrt[1 - a^2*x^ 2])/5 + (E^ArcSin[a*x]*(1 - a^2*x^2))/5))/17)/a
Int[((F_)^(v_))^(n_.), x_Symbol] :> Simp[(F^v)^n/(n*Log[F]*D[v, x]), x] /; FreeQ[{F, n}, x] && LinearQ[v, x]
Int[Cos[(d_.) + (e_.)*(x_)]^(m_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbo l] :> Simp[b*c*Log[F]*F^(c*(a + b*x))*(Cos[d + e*x]^m/(e^2*m^2 + b^2*c^2*Lo g[F]^2)), x] + (Simp[e*m*F^(c*(a + b*x))*Sin[d + e*x]*(Cos[d + e*x]^(m - 1) /(e^2*m^2 + b^2*c^2*Log[F]^2)), x] + Simp[(m*(m - 1)*e^2)/(e^2*m^2 + b^2*c^ 2*Log[F]^2) Int[F^(c*(a + b*x))*Cos[d + e*x]^(m - 2), x], x]) /; FreeQ[{F , a, b, c, d, e}, x] && NeQ[e^2*m^2 + b^2*c^2*Log[F]^2, 0] && GtQ[m, 1]
Int[(u_.)*(f_)^(ArcSin[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> Simp[ 1/b Subst[Int[(u /. x -> -a/b + Sin[x]/b)*f^(c*x^n)*Cos[x], x], x, ArcSin [a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ FracPart[p]/v^(m*FracPart[p])) Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) && !(Eq Q[v, x] && EqQ[m, 1])
\[\int {\mathrm e}^{\arcsin \left (a x \right )} \left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}d x\]
Input:
int(exp(arcsin(a*x))*(-a^2*x^2+1)^(3/2),x)
Output:
int(exp(arcsin(a*x))*(-a^2*x^2+1)^(3/2),x)
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.49 \[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{3/2} \, dx=\frac {{\left (5 \, a^{4} x^{4} - 22 \, a^{2} x^{2} - 4 \, {\left (5 \, a^{3} x^{3} - 11 \, a x\right )} \sqrt {-a^{2} x^{2} + 1} + 41\right )} e^{\left (\arcsin \left (a x\right )\right )}}{85 \, a} \] Input:
integrate(exp(arcsin(a*x))*(-a^2*x^2+1)^(3/2),x, algorithm="fricas")
Output:
1/85*(5*a^4*x^4 - 22*a^2*x^2 - 4*(5*a^3*x^3 - 11*a*x)*sqrt(-a^2*x^2 + 1) + 41)*e^(arcsin(a*x))/a
Time = 1.13 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.85 \[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{3/2} \, dx=\begin {cases} \frac {a^{3} x^{4} e^{\operatorname {asin}{\left (a x \right )}}}{17} - \frac {4 a^{2} x^{3} \sqrt {- a^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a x \right )}}}{17} - \frac {22 a x^{2} e^{\operatorname {asin}{\left (a x \right )}}}{85} + \frac {44 x \sqrt {- a^{2} x^{2} + 1} e^{\operatorname {asin}{\left (a x \right )}}}{85} + \frac {41 e^{\operatorname {asin}{\left (a x \right )}}}{85 a} & \text {for}\: a \neq 0 \\x & \text {otherwise} \end {cases} \] Input:
integrate(exp(asin(a*x))*(-a**2*x**2+1)**(3/2),x)
Output:
Piecewise((a**3*x**4*exp(asin(a*x))/17 - 4*a**2*x**3*sqrt(-a**2*x**2 + 1)* exp(asin(a*x))/17 - 22*a*x**2*exp(asin(a*x))/85 + 44*x*sqrt(-a**2*x**2 + 1 )*exp(asin(a*x))/85 + 41*exp(asin(a*x))/(85*a), Ne(a, 0)), (x, True))
\[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{3/2} \, dx=\int { {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} e^{\left (\arcsin \left (a x\right )\right )} \,d x } \] Input:
integrate(exp(arcsin(a*x))*(-a^2*x^2+1)^(3/2),x, algorithm="maxima")
Output:
integrate((-a^2*x^2 + 1)^(3/2)*e^(arcsin(a*x)), x)
Exception generated. \[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{3/2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(exp(arcsin(a*x))*(-a^2*x^2+1)^(3/2),x, algorithm="giac")
Output:
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. \[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{3/2} \, dx=\int {\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}\,{\left (1-a^2\,x^2\right )}^{3/2} \,d x \] Input:
int(exp(asin(a*x))*(1 - a^2*x^2)^(3/2),x)
Output:
int(exp(asin(a*x))*(1 - a^2*x^2)^(3/2), x)
\[ \int e^{\arcsin (a x)} \left (1-a^2 x^2\right )^{3/2} \, dx=-\left (\int e^{\mathit {asin} \left (a x \right )} \sqrt {-a^{2} x^{2}+1}\, x^{2}d x \right ) a^{2}+\int e^{\mathit {asin} \left (a x \right )} \sqrt {-a^{2} x^{2}+1}d x \] Input:
int(exp(asin(a*x))*(-a^2*x^2+1)^(3/2),x)
Output:
- int(e**asin(a*x)*sqrt( - a**2*x**2 + 1)*x**2,x)*a**2 + int(e**asin(a*x) *sqrt( - a**2*x**2 + 1),x)