\(\int e^{\csc ^{-1}(a x)} x \, dx\) [42]

Optimal result

Integrand size = 8, antiderivative size = 87 \[ \int e^{\csc ^{-1}(a x)} x \, dx=\frac {\left (\frac {8}{5}+\frac {4 i}{5}\right ) e^{(1+2 i) \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )}{a^2}-\frac {\left (\frac {16}{5}+\frac {8 i}{5}\right ) e^{(1+2 i) \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},3,2-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )}{a^2} \] Output:

(8/5+4/5*I)*exp((1+2*I)*arccsc(a*x))*hypergeom([2, 1-1/2*I],[2-1/2*I],(I/a 
/x+(1-1/a^2/x^2)^(1/2))^2)/a^2-(16/5+8/5*I)*exp((1+2*I)*arccsc(a*x))*hyper 
geom([3, 1-1/2*I],[2-1/2*I],(I/a/x+(1-1/a^2/x^2)^(1/2))^2)/a^2
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.16 \[ \int e^{\csc ^{-1}(a x)} x \, dx=\frac {\left (\frac {1}{5}+\frac {i}{10}\right ) e^{\csc ^{-1}(a x)} \left ((2-i) a x \left (\sqrt {1-\frac {1}{a^2 x^2}}+a x\right )+(1+2 i) \operatorname {Hypergeometric2F1}\left (-\frac {i}{2},1,1-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )+e^{2 i \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i}{2},2-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )\right )}{a^2} \] Input:

Integrate[E^ArcCsc[a*x]*x,x]
 

Output:

((1/5 + I/10)*E^ArcCsc[a*x]*((2 - I)*a*x*(Sqrt[1 - 1/(a^2*x^2)] + a*x) + ( 
1 + 2*I)*Hypergeometric2F1[-1/2*I, 1, 1 - I/2, E^((2*I)*ArcCsc[a*x])] + E^ 
((2*I)*ArcCsc[a*x])*Hypergeometric2F1[1, 1 - I/2, 2 - I/2, E^((2*I)*ArcCsc 
[a*x])]))/a^2
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5790, 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.

\(\displaystyle \int x e^{\csc ^{-1}(a x)} \, dx\)

\(\Big \downarrow \) 5790

\(\displaystyle -\frac {\int a^2 e^{\csc ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x^3d\csc ^{-1}(a x)}{a}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int a^3 e^{\csc ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}} x^3d\csc ^{-1}(a x)}{a^2}\)

\(\Big \downarrow \) 4974

\(\displaystyle -\frac {\int \left (-\frac {4 i e^{(1+2 i) \csc ^{-1}(a x)}}{\left (-1+e^{2 i \csc ^{-1}(a x)}\right )^2}-\frac {8 i e^{(1+2 i) \csc ^{-1}(a x)}}{\left (-1+e^{2 i \csc ^{-1}(a x)}\right )^3}\right )d\csc ^{-1}(a x)}{a^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\left (\frac {16}{5}+\frac {8 i}{5}\right ) e^{(1+2 i) \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},3,2-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )-\left (\frac {8}{5}+\frac {4 i}{5}\right ) e^{(1+2 i) \csc ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (1-\frac {i}{2},2,2-\frac {i}{2},e^{2 i \csc ^{-1}(a x)}\right )}{a^2}\)

Input:

Int[E^ArcCsc[a*x]*x,x]
 

Output:

-(((-8/5 - (4*I)/5)*E^((1 + 2*I)*ArcCsc[a*x])*Hypergeometric2F1[1 - I/2, 2 
, 2 - I/2, E^((2*I)*ArcCsc[a*x])] + (16/5 + (8*I)/5)*E^((1 + 2*I)*ArcCsc[a 
*x])*Hypergeometric2F1[1 - I/2, 3, 2 - I/2, E^((2*I)*ArcCsc[a*x])])/a^2)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 4974

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 5790

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]
 

Maple [F]

\[\int {\mathrm e}^{\operatorname {arccsc}\left (a x \right )} x d x\]

Input:

int(exp(arccsc(a*x))*x,x)
 

Output:

int(exp(arccsc(a*x))*x,x)
 

Fricas [F]

\[ \int e^{\csc ^{-1}(a x)} x \, dx=\int { x e^{\left (\operatorname {arccsc}\left (a x\right )\right )} \,d x } \] Input:

integrate(exp(arccsc(a*x))*x,x, algorithm="fricas")
 

Output:

integral(x*e^(arccsc(a*x)), x)
 

Sympy [F]

\[ \int e^{\csc ^{-1}(a x)} x \, dx=\int x e^{\operatorname {acsc}{\left (a x \right )}}\, dx \] Input:

integrate(exp(acsc(a*x))*x,x)
 

Output:

Integral(x*exp(acsc(a*x)), x)
 

Maxima [F]

\[ \int e^{\csc ^{-1}(a x)} x \, dx=\int { x e^{\left (\operatorname {arccsc}\left (a x\right )\right )} \,d x } \] Input:

integrate(exp(arccsc(a*x))*x,x, algorithm="maxima")
 

Output:

integrate(x*e^(arccsc(a*x)), x)
 

Giac [F]

\[ \int e^{\csc ^{-1}(a x)} x \, dx=\int { x e^{\left (\operatorname {arccsc}\left (a x\right )\right )} \,d x } \] Input:

integrate(exp(arccsc(a*x))*x,x, algorithm="giac")
 

Output:

integrate(x*e^(arccsc(a*x)), x)
 

Mupad [F(-1)]

Timed out. \[ \int e^{\csc ^{-1}(a x)} x \, dx=\int x\,{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{a\,x}\right )} \,d x \] Input:

int(x*exp(asin(1/(a*x))),x)
 

Output:

int(x*exp(asin(1/(a*x))), x)
 

Reduce [F]

\[ \int e^{\csc ^{-1}(a x)} x \, dx=\int e^{\mathit {acsc} \left (a x \right )} x d x \] Input:

int(exp(acsc(a*x))*x,x)
 

Output:

int(e**acsc(a*x)*x,x)