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

Optimal. Leaf size=43 \[ 2 i e^{\csc ^{-1}(a x)} \, _2F_1\left (-\frac{i}{2},1;1-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )-i e^{\csc ^{-1}(a x)} \]

[Out]

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

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Rubi [A]  time = 0.0585483, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5267, 12, 4443, 2194, 2251} \[ 2 i e^{\csc ^{-1}(a x)} \, _2F_1\left (-\frac{i}{2},1;1-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )-i e^{\csc ^{-1}(a x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5267

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

Rule 12

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

Rule 4443

Int[Cot[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Dist[(-I)^n, Int[ExpandInteg
rand[(F^(c*(a + b*x))*(1 + E^(2*I*(d + e*x)))^n)/(1 - E^(2*I*(d + e*x)))^n, x], x], x] /; FreeQ[{F, a, b, c, d
, e}, x] && IntegerQ[n]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, 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])

Rubi steps

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

Mathematica [A]  time = 0.0541905, size = 75, normalized size = 1.74 \[ -i \left (-e^{\csc ^{-1}(a x)} \, _2F_1\left (-\frac{i}{2},1;1-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )-\left (\frac{1}{5}-\frac{2 i}{5}\right ) e^{(1+2 i) \csc ^{-1}(a x)} \, _2F_1\left (1,1-\frac{i}{2};2-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.191, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{\rm arccsc} \left (ax\right )}}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\operatorname{arccsc}\left (a x\right )\right )}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(arccsc(a*x))/x, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{e^{\left (\operatorname{arccsc}\left (a x\right )\right )}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(e^(arccsc(a*x))/x, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\operatorname{acsc}{\left (a x \right )}}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(acsc(a*x))/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{\left (\operatorname{arccsc}\left (a x\right )\right )}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(arccsc(a*x))/x, x)