3.46 \(\int \frac {e^{\csc ^{-1}(a x)}}{x^3} \, dx\)

Optimal. Leaf size=41 \[ \frac {1}{5} a^2 e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )-\frac {1}{10} a^2 e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right ) \]

[Out]

1/5*a^2*exp(arccsc(a*x))*cos(2*arccsc(a*x))-1/10*a^2*exp(arccsc(a*x))*sin(2*arccsc(a*x))

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Rubi [A]  time = 0.04, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5267, 12, 4469, 4432} \[ \frac {1}{5} a^2 e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )-\frac {1}{10} a^2 e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[E^ArcCsc[a*x]/x^3,x]

[Out]

(a^2*E^ArcCsc[a*x]*Cos[2*ArcCsc[a*x]])/5 - (a^2*E^ArcCsc[a*x]*Sin[2*ArcCsc[a*x]])/10

Rule 12

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

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]

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 30, normalized size = 0.73 \[ -\frac {1}{10} a^2 e^{\csc ^{-1}(a x)} \left (\sin \left (2 \csc ^{-1}(a x)\right )-2 \cos \left (2 \csc ^{-1}(a x)\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[E^ArcCsc[a*x]/x^3,x]

[Out]

-1/10*(a^2*E^ArcCsc[a*x]*(-2*Cos[2*ArcCsc[a*x]] + Sin[2*ArcCsc[a*x]]))

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fricas [A]  time = 0.80, size = 32, normalized size = 0.78 \[ \frac {{\left (a^{2} x^{2} - \sqrt {a^{2} x^{2} - 1} - 2\right )} e^{\left (\operatorname {arccsc}\left (a x\right )\right )}}{5 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccsc(a*x))/x^3,x, algorithm="fricas")

[Out]

1/5*(a^2*x^2 - sqrt(a^2*x^2 - 1) - 2)*e^(arccsc(a*x))/x^2

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (\operatorname {arccsc}\left (a x\right )\right )}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccsc(a*x))/x^3,x, algorithm="giac")

[Out]

integrate(e^(arccsc(a*x))/x^3, x)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{\mathrm {arccsc}\left (a x \right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(arccsc(a*x))/x^3,x)

[Out]

int(exp(arccsc(a*x))/x^3,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\left (\operatorname {arccsc}\left (a x\right )\right )}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(arccsc(a*x))/x^3,x, algorithm="maxima")

[Out]

integrate(e^(arccsc(a*x))/x^3, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{a\,x}\right )}}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(asin(1/(a*x)))/x^3,x)

[Out]

int(exp(asin(1/(a*x)))/x^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {e^{\operatorname {acsc}{\left (a x \right )}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(acsc(a*x))/x**3,x)

[Out]

Integral(exp(acsc(a*x))/x**3, x)

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