Integrand size = 6, antiderivative size = 87 \[ \int e^{\csc ^{-1}(a x)} \, dx=-\frac {(1-i) e^{(1+i) \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},1,\frac {3}{2}-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )}{a}+\frac {(2-2 i) e^{(1+i) \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},2,\frac {3}{2}-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )}{a} \] Output:
(-1+I)*exp((1+I)*arccsc(a*x))*hypergeom([1, 1/2-1/2*I],[3/2-1/2*I],(I/a/x+ (1-1/a^2/x^2)^(1/2))^2)/a+(2-2*I)*exp((1+I)*arccsc(a*x))*hypergeom([2, 1/2 -1/2*I],[3/2-1/2*I],(I/a/x+(1-1/a^2/x^2)^(1/2))^2)/a
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.62 \[ \int e^{\csc ^{-1}(a x)} \, dx=\frac {e^{\csc ^{-1}(a x)} \left (a x+(1+i) e^{i \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},1,\frac {3}{2}-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )\right )}{a} \] Input:
Integrate[E^ArcCsc[a*x],x]
Output:
(E^ArcCsc[a*x]*(a*x + (1 + I)*E^(I*ArcCsc[a*x])*Hypergeometric2F1[1/2 - I/ 2, 1, 3/2 - I/2, E^((2*I)*ArcCsc[a*x])]))/a
Time = 0.32 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5790, 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.
\(\displaystyle \int e^{\csc ^{-1}(a x)} \, dx\) |
\(\Big \downarrow \) 5790 |
\(\displaystyle -\frac {\int a^2 e^{\csc ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x^2d\csc ^{-1}(a x)}{a}\) |
\(\Big \downarrow \) 4974 |
\(\displaystyle -\frac {\int \left (\frac {2 e^{(1+i) \csc ^{-1}(a x)}}{1-e^{2 i \csc ^{-1}(a x)}}-\frac {4 e^{(1+i) \csc ^{-1}(a x)}}{\left (-1+e^{2 i \csc ^{-1}(a x)}\right )^2}\right )d\csc ^{-1}(a x)}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {(1-i) e^{(1+i) \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},1,\frac {3}{2}-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )-(2-2 i) e^{(1+i) \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (\frac {1}{2}-\frac {i}{2},2,\frac {3}{2}-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )}{a}\) |
Input:
Int[E^ArcCsc[a*x],x]
Output:
-(((1 - I)*E^((1 + I)*ArcCsc[a*x])*Hypergeometric2F1[1/2 - I/2, 1, 3/2 - I /2, E^((2*I)*ArcCsc[a*x])] - (2 - 2*I)*E^((1 + I)*ArcCsc[a*x])*Hypergeomet ric2F1[1/2 - I/2, 2, 3/2 - I/2, E^((2*I)*ArcCsc[a*x])])/a)
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_)^(ArcCsc[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> Simp[ -b^(-1) Subst[Int[(u /. x -> -a/b + Csc[x]/b)*f^(c*x^n)*Csc[x]*Cot[x], x] , x, ArcCsc[a + b*x]], x] /; FreeQ[{a, b, c, f}, x] && IGtQ[n, 0]
\[\int {\mathrm e}^{\operatorname {arccsc}\left (a x \right )}d x\]
Input:
int(exp(arccsc(a*x)),x)
Output:
int(exp(arccsc(a*x)),x)
\[ \int e^{\csc ^{-1}(a x)} \, dx=\int { e^{\left (\operatorname {arccsc}\left (a x\right )\right )} \,d x } \] Input:
integrate(exp(arccsc(a*x)),x, algorithm="fricas")
Output:
integral(e^(arccsc(a*x)), x)
\[ \int e^{\csc ^{-1}(a x)} \, dx=\int e^{\operatorname {acsc}{\left (a x \right )}}\, dx \] Input:
integrate(exp(acsc(a*x)),x)
Output:
Integral(exp(acsc(a*x)), x)
\[ \int e^{\csc ^{-1}(a x)} \, dx=\int { e^{\left (\operatorname {arccsc}\left (a x\right )\right )} \,d x } \] Input:
integrate(exp(arccsc(a*x)),x, algorithm="maxima")
Output:
integrate(e^(arccsc(a*x)), x)
\[ \int e^{\csc ^{-1}(a x)} \, dx=\int { e^{\left (\operatorname {arccsc}\left (a x\right )\right )} \,d x } \] Input:
integrate(exp(arccsc(a*x)),x, algorithm="giac")
Output:
integrate(e^(arccsc(a*x)), x)
Timed out. \[ \int e^{\csc ^{-1}(a x)} \, dx=\int {\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{a\,x}\right )} \,d x \] Input:
int(exp(asin(1/(a*x))),x)
Output:
int(exp(asin(1/(a*x))), x)
\[ \int e^{\csc ^{-1}(a x)} \, dx=\int e^{\mathit {acsc} \left (a x \right )}d x \] Input:
int(exp(acsc(a*x)),x)
Output:
int(e**acsc(a*x),x)