\(\int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx\) [48]

Optimal result

Integrand size = 10, antiderivative size = 81 \[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=\frac {1}{10} a^4 e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )-\frac {1}{34} a^4 e^{\csc ^{-1}(a x)} \cos \left (4 \csc ^{-1}(a x)\right )-\frac {1}{20} a^4 e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right )+\frac {1}{136} a^4 e^{\csc ^{-1}(a x)} \sin \left (4 \csc ^{-1}(a x)\right ) \]

[Out]

1/10*a^4*exp(arccsc(a*x))*cos(2*arccsc(a*x))-1/34*a^4*exp(arccsc(a*x))*cos(4*arccsc(a*x))-1/20*a^4*exp(arccsc(
a*x))*sin(2*arccsc(a*x))+1/136*a^4*exp(arccsc(a*x))*sin(4*arccsc(a*x))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5375, 12, 4557, 4517} \[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=\frac {1}{10} a^4 e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )-\frac {1}{34} a^4 e^{\csc ^{-1}(a x)} \cos \left (4 \csc ^{-1}(a x)\right )-\frac {1}{20} a^4 e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right )+\frac {1}{136} a^4 e^{\csc ^{-1}(a x)} \sin \left (4 \csc ^{-1}(a x)\right ) \]

[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 12

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

Rule 4517

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 4557

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 5375

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*} \text {integral}& = -\frac {\text {Subst}\left (\int a^5 e^x \cos (x) \sin ^3(x) \, dx,x,\csc ^{-1}(a x)\right )}{a} \\ & = -\left (a^4 \text {Subst}\left (\int e^x \cos (x) \sin ^3(x) \, dx,x,\csc ^{-1}(a x)\right )\right ) \\ & = -\left (a^4 \text {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 \text {Subst}\left (\int e^x \sin (4 x) \, dx,x,\csc ^{-1}(a x)\right )-\frac {1}{4} a^4 \text {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] (verified)

Time = 0.12 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.62 \[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=-\frac {1}{680} a^4 e^{\csc ^{-1}(a x)} \left (-68 \cos \left (2 \csc ^{-1}(a x)\right )+20 \cos \left (4 \csc ^{-1}(a x)\right )+34 \sin \left (2 \csc ^{-1}(a x)\right )-5 \sin \left (4 \csc ^{-1}(a x)\right )\right ) \]

[In]

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

[Out]

-1/680*(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*Ar
cCsc[a*x]]))

Maple [F]

\[\int \frac {{\mathrm e}^{\operatorname {arccsc}\left (a x \right )}}{x^{5}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.63 \[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=\frac {{\left (6 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - {\left (6 \, a^{2} x^{2} + 5\right )} \sqrt {a^{2} x^{2} - 1} - 20\right )} e^{\left (\operatorname {arccsc}\left (a x\right )\right )}}{85 \, x^{4}} \]

[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

Sympy [F]

\[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=\int \frac {e^{\operatorname {acsc}{\left (a x \right )}}}{x^{5}}\, dx \]

[In]

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

[Out]

Integral(exp(acsc(a*x))/x**5, x)

Maxima [F]

\[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=\int { \frac {e^{\left (\operatorname {arccsc}\left (a x\right )\right )}}{x^{5}} \,d x } \]

[In]

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

[Out]

integrate(e^(arccsc(a*x))/x^5, x)

Giac [F]

\[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=\int { \frac {e^{\left (\operatorname {arccsc}\left (a x\right )\right )}}{x^{5}} \,d x } \]

[In]

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

[Out]

integrate(e^(arccsc(a*x))/x^5, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=\int \frac {{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{a\,x}\right )}}{x^5} \,d x \]

[In]

int(exp(asin(1/(a*x)))/x^5,x)

[Out]

int(exp(asin(1/(a*x)))/x^5, x)