Integrand size = 10, antiderivative size = 83 \[ \int \frac {e^{\arcsin (a x)}}{x^2} \, dx=(1-i) a e^{(1+i) \arcsin (a x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},1,\frac {3}{2}-\frac {i}{2},e^{2 i \arcsin (a x)}\right )-(2-2 i) a e^{(1+i) \arcsin (a x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},2,\frac {3}{2}-\frac {i}{2},e^{2 i \arcsin (a x)}\right ) \] Output:
(1-I)*a*exp((1+I)*arcsin(a*x))*hypergeom([1, 1/2-1/2*I],[3/2-1/2*I],(I*a*x +(-a^2*x^2+1)^(1/2))^2)+(-2+2*I)*a*exp((1+I)*arcsin(a*x))*hypergeom([2, 1/ 2-1/2*I],[3/2-1/2*I],(I*a*x+(-a^2*x^2+1)^(1/2))^2)
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.65 \[ \int \frac {e^{\arcsin (a x)}}{x^2} \, dx=-\frac {e^{\arcsin (a x)}+(1+i) a e^{(1+i) \arcsin (a x)} x \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},1,\frac {3}{2}-\frac {i}{2},e^{2 i \arcsin (a x)}\right )}{x} \] Input:
Integrate[E^ArcSin[a*x]/x^2,x]
Output:
-((E^ArcSin[a*x] + (1 + I)*a*E^((1 + I)*ArcSin[a*x])*x*Hypergeometric2F1[1 /2 - I/2, 1, 3/2 - I/2, E^((2*I)*ArcSin[a*x])])/x)
Time = 0.30 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5335, 27, 4974, 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.
\(\displaystyle \int \frac {e^{\arcsin (a x)}}{x^2} \, dx\) |
\(\Big \downarrow \) 5335 |
\(\displaystyle \frac {\int \frac {e^{\arcsin (a x)} \sqrt {1-a^2 x^2}}{x^2}d\arcsin (a x)}{a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a \int \frac {e^{\arcsin (a x)} \sqrt {1-a^2 x^2}}{a^2 x^2}d\arcsin (a x)\) |
\(\Big \downarrow \) 4974 |
\(\displaystyle a \int \left (\frac {2 e^{(1+i) \arcsin (a x)}}{1-e^{2 i \arcsin (a x)}}-\frac {4 e^{(1+i) \arcsin (a x)}}{\left (-1+e^{2 i \arcsin (a x)}\right )^2}\right )d\arcsin (a x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle a \left ((1-i) e^{(1+i) \arcsin (a x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},1,\frac {3}{2}-\frac {i}{2},e^{2 i \arcsin (a x)}\right )-(2-2 i) e^{(1+i) \arcsin (a x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},2,\frac {3}{2}-\frac {i}{2},e^{2 i \arcsin (a x)}\right )\right )\) |
Input:
Int[E^ArcSin[a*x]/x^2,x]
Output:
a*((1 - I)*E^((1 + I)*ArcSin[a*x])*Hypergeometric2F1[1/2 - I/2, 1, 3/2 - I /2, E^((2*I)*ArcSin[a*x])] - (2 - 2*I)*E^((1 + I)*ArcSin[a*x])*Hypergeomet ric2F1[1/2 - I/2, 2, 3/2 - I/2, E^((2*I)*ArcSin[a*x])])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*(G_)[(d_.) + (e_.)*(x_)]^(m_.)*(H_)[( d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigToExp[F^(c*(a + b*x)), G[d + e*x]^m*H[d + e*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] && IGtQ[ m, 0] && IGtQ[n, 0] && TrigQ[G] && TrigQ[H]
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]
\[\int \frac {{\mathrm e}^{\arcsin \left (a x \right )}}{x^{2}}d x\]
Input:
int(exp(arcsin(a*x))/x^2,x)
Output:
int(exp(arcsin(a*x))/x^2,x)
\[ \int \frac {e^{\arcsin (a x)}}{x^2} \, dx=\int { \frac {e^{\left (\arcsin \left (a x\right )\right )}}{x^{2}} \,d x } \] Input:
integrate(exp(arcsin(a*x))/x^2,x, algorithm="fricas")
Output:
integral(e^(arcsin(a*x))/x^2, x)
\[ \int \frac {e^{\arcsin (a x)}}{x^2} \, dx=\int \frac {e^{\operatorname {asin}{\left (a x \right )}}}{x^{2}}\, dx \] Input:
integrate(exp(asin(a*x))/x**2,x)
Output:
Integral(exp(asin(a*x))/x**2, x)
\[ \int \frac {e^{\arcsin (a x)}}{x^2} \, dx=\int { \frac {e^{\left (\arcsin \left (a x\right )\right )}}{x^{2}} \,d x } \] Input:
integrate(exp(arcsin(a*x))/x^2,x, algorithm="maxima")
Output:
integrate(e^(arcsin(a*x))/x^2, x)
\[ \int \frac {e^{\arcsin (a x)}}{x^2} \, dx=\int { \frac {e^{\left (\arcsin \left (a x\right )\right )}}{x^{2}} \,d x } \] Input:
integrate(exp(arcsin(a*x))/x^2,x, algorithm="giac")
Output:
integrate(e^(arcsin(a*x))/x^2, x)
Timed out. \[ \int \frac {e^{\arcsin (a x)}}{x^2} \, dx=\int \frac {{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}}{x^2} \,d x \] Input:
int(exp(asin(a*x))/x^2,x)
Output:
int(exp(asin(a*x))/x^2, x)
\[ \int \frac {e^{\arcsin (a x)}}{x^2} \, dx=\int \frac {e^{\mathit {asin} \left (a x \right )}}{x^{2}}d x \] Input:
int(exp(asin(a*x))/x^2,x)
Output:
int(e**asin(a*x)/x**2,x)