3.48 \(\int \frac{e^{\csc ^{-1}(a x)}}{x^5} \, dx\)

Optimal. Leaf size=81 \[ \frac{1}{10} a^4 e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )-\frac{1}{34} a^4 e^{\csc ^{-1}(a x)} \cos \left (4 \csc ^{-1}(a x)\right )-\frac{1}{20} a^4 e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right )+\frac{1}{136} a^4 e^{\csc ^{-1}(a x)} \sin \left (4 \csc ^{-1}(a x)\right ) \]

[Out]

(a^4*E^ArcCsc[a*x]*Cos[2*ArcCsc[a*x]])/10 - (a^4*E^ArcCsc[a*x]*Cos[4*ArcCsc[a*x]])/34 - (a^4*E^ArcCsc[a*x]*Sin
[2*ArcCsc[a*x]])/20 + (a^4*E^ArcCsc[a*x]*Sin[4*ArcCsc[a*x]])/136

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Rubi [A]  time = 0.0679689, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5267, 12, 4469, 4432} \[ \frac{1}{10} a^4 e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )-\frac{1}{34} a^4 e^{\csc ^{-1}(a x)} \cos \left (4 \csc ^{-1}(a x)\right )-\frac{1}{20} a^4 e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right )+\frac{1}{136} a^4 e^{\csc ^{-1}(a x)} \sin \left (4 \csc ^{-1}(a x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCsc[a*x]/x^5,x]

[Out]

(a^4*E^ArcCsc[a*x]*Cos[2*ArcCsc[a*x]])/10 - (a^4*E^ArcCsc[a*x]*Cos[4*ArcCsc[a*x]])/34 - (a^4*E^ArcCsc[a*x]*Sin
[2*ArcCsc[a*x]])/20 + (a^4*E^ArcCsc[a*x]*Sin[4*ArcCsc[a*x]])/136

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 4469

Int[Cos[(f_.) + (g_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)]^(m_.), x_Symbol] :
> Int[ExpandTrigReduce[F^(c*(a + b*x)), Sin[d + e*x]^m*Cos[f + g*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e, f, g
}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 4432

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Sin[(d_.) + (e_.)*(x_)], x_Symbol] :> Simp[(b*c*Log[F]*F^(c*(a + b*x))*S
in[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x] - Simp[(e*F^(c*(a + b*x))*Cos[d + e*x])/(e^2 + b^2*c^2*Log[F]^2), x]
/; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2 + b^2*c^2*Log[F]^2, 0]

Rubi steps

\begin{align*} \int \frac{e^{\csc ^{-1}(a x)}}{x^5} \, dx &=-\frac{\operatorname{Subst}\left (\int a^5 e^x \cos (x) \sin ^3(x) \, dx,x,\csc ^{-1}(a x)\right )}{a}\\ &=-\left (a^4 \operatorname{Subst}\left (\int e^x \cos (x) \sin ^3(x) \, dx,x,\csc ^{-1}(a x)\right )\right )\\ &=-\left (a^4 \operatorname{Subst}\left (\int \left (\frac{1}{4} e^x \sin (2 x)-\frac{1}{8} e^x \sin (4 x)\right ) \, dx,x,\csc ^{-1}(a x)\right )\right )\\ &=\frac{1}{8} a^4 \operatorname{Subst}\left (\int e^x \sin (4 x) \, dx,x,\csc ^{-1}(a x)\right )-\frac{1}{4} a^4 \operatorname{Subst}\left (\int e^x \sin (2 x) \, dx,x,\csc ^{-1}(a x)\right )\\ &=\frac{1}{10} a^4 e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )-\frac{1}{34} a^4 e^{\csc ^{-1}(a x)} \cos \left (4 \csc ^{-1}(a x)\right )-\frac{1}{20} a^4 e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right )+\frac{1}{136} a^4 e^{\csc ^{-1}(a x)} \sin \left (4 \csc ^{-1}(a x)\right )\\ \end{align*}

Mathematica [A]  time = 0.128638, size = 50, normalized size = 0.62 \[ -\frac{1}{680} a^4 e^{\csc ^{-1}(a x)} \left (-68 \cos \left (2 \csc ^{-1}(a x)\right )+20 \cos \left (4 \csc ^{-1}(a x)\right )+34 \sin \left (2 \csc ^{-1}(a x)\right )-5 \sin \left (4 \csc ^{-1}(a x)\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcCsc[a*x]/x^5,x]

[Out]

-(a^4*E^ArcCsc[a*x]*(-68*Cos[2*ArcCsc[a*x]] + 20*Cos[4*ArcCsc[a*x]] + 34*Sin[2*ArcCsc[a*x]] - 5*Sin[4*ArcCsc[a
*x]]))/680

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Maple [F]  time = 0.179, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{\rm arccsc} \left (ax\right )}}}{{x}^{5}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arccsc(a*x))/x^5,x)

[Out]

int(exp(arccsc(a*x))/x^5,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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccsc(a*x))/x^5,x, algorithm="maxima")

[Out]

integrate(e^(arccsc(a*x))/x^5, x)

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Fricas [A]  time = 3.53715, size = 123, normalized size = 1.52 \begin{align*} \frac{{\left (6 \, a^{4} x^{4} + 3 \, a^{2} x^{2} -{\left (6 \, a^{2} x^{2} + 5\right )} \sqrt{a^{2} x^{2} - 1} - 20\right )} e^{\left (\operatorname{arccsc}\left (a x\right )\right )}}{85 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccsc(a*x))/x^5,x, algorithm="fricas")

[Out]

1/85*(6*a^4*x^4 + 3*a^2*x^2 - (6*a^2*x^2 + 5)*sqrt(a^2*x^2 - 1) - 20)*e^(arccsc(a*x))/x^4

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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^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(acsc(a*x))/x**5,x)

[Out]

Integral(exp(acsc(a*x))/x**5, 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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccsc(a*x))/x^5,x, algorithm="giac")

[Out]

integrate(e^(arccsc(a*x))/x^5, x)