3.444 \(\int \frac {e^{\sin ^{-1}(a x)}}{x^2} \, dx\)

Optimal. Leaf size=83 \[ (1-i) a e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};e^{2 i \sin ^{-1}(a x)}\right )-(2-2 i) a e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},2;\frac {3}{2}-\frac {i}{2};e^{2 i \sin ^{-1}(a x)}\right ) \]

[Out]

(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*e
xp((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)

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4836, 12, 4471, 2251} \[ (1-i) a e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};e^{2 i \sin ^{-1}(a x)}\right )-(2-2 i) a e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},2;\frac {3}{2}-\frac {i}{2};e^{2 i \sin ^{-1}(a x)}\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^ArcSin[a*x]/x^2,x]

[Out]

(1 - I)*a*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
)*a*E^((1 + I)*ArcSin[a*x])*Hypergeometric2F1[1/2 - I/2, 2, 3/2 - I/2, E^((2*I)*ArcSin[a*x])]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2251

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[(a^p*G^(h*(f + g*x))*Hypergeometric2F1[-p, (g*h*Log[G])/(d*e*Log[F]), (g*h*Log[G])/(d*e*Log[F]) + 1, Simplify
[-((b*F^(e*(c + d*x)))/a)]])/(g*h*Log[G]), x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] ||
GtQ[a, 0])

Rule 4471

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]

Rule 4836

Int[(u_.)*(f_)^(ArcSin[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> Dist[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]

Rubi steps

\begin {align*} \int \frac {e^{\sin ^{-1}(a x)}}{x^2} \, dx &=\frac {\operatorname {Subst}\left (\int a^2 e^x \cot (x) \csc (x) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=a \operatorname {Subst}\left (\int e^x \cot (x) \csc (x) \, dx,x,\sin ^{-1}(a x)\right )\\ &=a \operatorname {Subst}\left (\int \left (\frac {2 e^{(1+i) x}}{1-e^{2 i x}}-\frac {4 e^{(1+i) x}}{\left (-1+e^{2 i x}\right )^2}\right ) \, dx,x,\sin ^{-1}(a x)\right )\\ &=(2 a) \operatorname {Subst}\left (\int \frac {e^{(1+i) x}}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )-(4 a) \operatorname {Subst}\left (\int \frac {e^{(1+i) x}}{\left (-1+e^{2 i x}\right )^2} \, dx,x,\sin ^{-1}(a x)\right )\\ &=(1-i) a e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};e^{2 i \sin ^{-1}(a x)}\right )-(2-2 i) a e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},2;\frac {3}{2}-\frac {i}{2};e^{2 i \sin ^{-1}(a x)}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.11, size = 54, normalized size = 0.65 \[ -\frac {e^{\sin ^{-1}(a x)}+(1+i) a x e^{(1+i) \sin ^{-1}(a x)} \, _2F_1\left (\frac {1}{2}-\frac {i}{2},1;\frac {3}{2}-\frac {i}{2};e^{2 i \sin ^{-1}(a x)}\right )}{x} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSin[a*x]/x^2,x]

[Out]

-((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)*Arc
Sin[a*x])])/x)

________________________________________________________________________________________

fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {e^{\left (\arcsin \left (a x\right )\right )}}{x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(a*x))/x^2,x, algorithm="fricas")

[Out]

integral(e^(arcsin(a*x))/x^2, x)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (\arcsin \left (a x\right )\right )}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(a*x))/x^2,x, algorithm="giac")

[Out]

integrate(e^(arcsin(a*x))/x^2, x)

________________________________________________________________________________________

maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{\arcsin \left (a x \right )}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arcsin(a*x))/x^2,x)

[Out]

int(exp(arcsin(a*x))/x^2,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (\arcsin \left (a x\right )\right )}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arcsin(a*x))/x^2,x, algorithm="maxima")

[Out]

integrate(e^(arcsin(a*x))/x^2, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{\mathrm {asin}\left (a\,x\right )}}{x^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(asin(a*x))/x^2,x)

[Out]

int(exp(asin(a*x))/x^2, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {asin}{\left (a x \right )}}}{x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(asin(a*x))/x**2,x)

[Out]

Integral(exp(asin(a*x))/x**2, x)

________________________________________________________________________________________