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

Optimal. Leaf size=87 \[ \frac{(2-2 i) e^{(1+i) \csc ^{-1}(a x)} \, _2F_1\left (\frac{1}{2}-\frac{i}{2},2;\frac{3}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a}-\frac{(1-i) e^{(1+i) \csc ^{-1}(a x)} \, _2F_1\left (\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a} \]

[Out]

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

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Rubi [A]  time = 0.0941809, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5267, 4471, 2251} \[ \frac{(2-2 i) e^{(1+i) \csc ^{-1}(a x)} \, _2F_1\left (\frac{1}{2}-\frac{i}{2},2;\frac{3}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a}-\frac{(1-i) e^{(1+i) \csc ^{-1}(a x)} \, _2F_1\left (\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.103721, size = 54, normalized size = 0.62 \[ \frac{e^{\csc ^{-1}(a x)} \left (a x+(1+i) e^{i \csc ^{-1}(a x)} \, _2F_1\left (\frac{1}{2}-\frac{i}{2},1;\frac{3}{2}-\frac{i}{2};e^{2 i \csc ^{-1}(a x)}\right )\right )}{a} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(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

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Maple [F]  time = 0.18, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{{\rm arccsc} \left (ax\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(arccsc(a*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(e^(arccsc(a*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(exp(acsc(a*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(arccsc(a*x)), x)