Integrand size = 21, antiderivative size = 45 \[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {\left (\frac {4}{5}-\frac {8 i}{5}\right ) e^{(1+2 i) \arcsin (a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},-e^{2 i \arcsin (a x)}\right )}{a} \] Output:
(4/5-8/5*I)*exp((1+2*I)*arcsin(a*x))*hypergeom([2, 1-1/2*I],[2-1/2*I],-(I* a*x+(-a^2*x^2+1)^(1/2))^2)/a
Time = 0.08 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\frac {\left (\frac {4}{5}-\frac {8 i}{5}\right ) e^{(1+2 i) \arcsin (a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},-e^{2 i \arcsin (a x)}\right )}{a} \] Input:
Integrate[E^ArcSin[a*x]/(1 - a^2*x^2)^(3/2),x]
Output:
((4/5 - (8*I)/5)*E^((1 + 2*I)*ArcSin[a*x])*Hypergeometric2F1[1 - I/2, 2, 2 - I/2, -E^((2*I)*ArcSin[a*x])])/a
Time = 0.74 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5335, 7292, 7271, 4951}
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.
\(\displaystyle \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 5335 |
\(\displaystyle \frac {\int \frac {e^{\arcsin (a x)}}{1-a^2 x^2}d\arcsin (a x)}{a}\) |
\(\Big \downarrow \) 4951 |
\(\displaystyle \frac {\left (\frac {4}{5}-\frac {8 i}{5}\right ) e^{(1+2 i) \arcsin (a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},-e^{2 i \arcsin (a x)}\right )}{a}\) |
Input:
Int[E^ArcSin[a*x]/(1 - a^2*x^2)^(3/2),x]
Output:
((4/5 - (8*I)/5)*E^((1 + 2*I)*ArcSin[a*x])*Hypergeometric2F1[1 - I/2, 2, 2 - I/2, -E^((2*I)*ArcSin[a*x])])/a
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sec[(d_.) + (e_.)*(x_)]^(n_.), x_Symb ol] :> Simp[2^n*E^(I*n*(d + e*x))*(F^(c*(a + b*x))/(I*e*n + b*c*Log[F]))*Hy pergeometric2F1[n, n/2 - I*b*c*(Log[F]/(2*e)), 1 + n/2 - I*b*c*(Log[F]/(2*e )), -E^(2*I*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]
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])
\[\int \frac {{\mathrm e}^{\arcsin \left (a x \right )}}{\left (-a^{2} x^{2}+1\right )^{\frac {3}{2}}}d x\]
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)
\[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {e^{\left (\arcsin \left (a x\right )\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(exp(arcsin(a*x))/(-a^2*x^2+1)^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(-a^2*x^2 + 1)*e^(arcsin(a*x))/(a^4*x^4 - 2*a^2*x^2 + 1), x)
\[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {e^{\operatorname {asin}{\left (a x \right )}}}{\left (- \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(exp(asin(a*x))/(-a**2*x**2+1)**(3/2),x)
Output:
Integral(exp(asin(a*x))/(-(a*x - 1)*(a*x + 1))**(3/2), x)
\[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {e^{\left (\arcsin \left (a x\right )\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(exp(arcsin(a*x))/(-a^2*x^2+1)^(3/2),x, algorithm="maxima")
Output:
integrate(e^(arcsin(a*x))/(-a^2*x^2 + 1)^(3/2), x)
\[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int { \frac {e^{\left (\arcsin \left (a x\right )\right )}}{{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(exp(arcsin(a*x))/(-a^2*x^2+1)^(3/2),x, algorithm="giac")
Output:
integrate(e^(arcsin(a*x))/(-a^2*x^2 + 1)^(3/2), x)
Timed out. \[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=\int \frac {{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}}{{\left (1-a^2\,x^2\right )}^{3/2}} \,d x \] 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)
\[ \int \frac {e^{\arcsin (a x)}}{\left (1-a^2 x^2\right )^{3/2}} \, dx=-\left (\int \frac {e^{\mathit {asin} \left (a x \right )}}{\sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}-\sqrt {-a^{2} x^{2}+1}}d x \right ) \] 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)*a**2*x**2 - sqrt( - a**2*x**2 + 1)),x)