Integrand size = 10, antiderivative size = 81 \[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=\frac {1}{10} a^4 e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )-\frac {1}{34} a^4 e^{\csc ^{-1}(a x)} \cos \left (4 \csc ^{-1}(a x)\right )-\frac {1}{20} a^4 e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right )+\frac {1}{136} a^4 e^{\csc ^{-1}(a x)} \sin \left (4 \csc ^{-1}(a x)\right ) \] Output:
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))
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.62 \[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=-\frac {1}{680} a^4 e^{\csc ^{-1}(a x)} \left (-68 \cos \left (2 \csc ^{-1}(a x)\right )+20 \cos \left (4 \csc ^{-1}(a x)\right )+34 \sin \left (2 \csc ^{-1}(a x)\right )-5 \sin \left (4 \csc ^{-1}(a x)\right )\right ) \] Input:
Integrate[E^ArcCsc[a*x]/x^5,x]
Output:
-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*ArcCsc[a*x]]))
Time = 0.31 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5790, 27, 4972, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx\) |
\(\Big \downarrow \) 5790 |
\(\displaystyle -\frac {\int \frac {a^2 e^{\csc ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}}{x^3}d\csc ^{-1}(a x)}{a}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -a^4 \int \frac {e^{\csc ^{-1}(a x)} \sqrt {1-\frac {1}{a^2 x^2}}}{a^3 x^3}d\csc ^{-1}(a x)\) |
\(\Big \downarrow \) 4972 |
\(\displaystyle -a^4 \int \left (\frac {1}{4} e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right )-\frac {1}{8} e^{\csc ^{-1}(a x)} \sin \left (4 \csc ^{-1}(a x)\right )\right )d\csc ^{-1}(a x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -a^4 \left (-\frac {1}{10} e^{\csc ^{-1}(a x)} \cos \left (2 \csc ^{-1}(a x)\right )+\frac {1}{34} e^{\csc ^{-1}(a x)} \cos \left (4 \csc ^{-1}(a x)\right )+\frac {1}{20} e^{\csc ^{-1}(a x)} \sin \left (2 \csc ^{-1}(a x)\right )-\frac {1}{136} e^{\csc ^{-1}(a x)} \sin \left (4 \csc ^{-1}(a x)\right )\right )\) |
Input:
Int[E^ArcCsc[a*x]/x^5,x]
Output:
-(a^4*(-1/10*(E^ArcCsc[a*x]*Cos[2*ArcCsc[a*x]]) + (E^ArcCsc[a*x]*Cos[4*Arc Csc[a*x]])/34 + (E^ArcCsc[a*x]*Sin[2*ArcCsc[a*x]])/20 - (E^ArcCsc[a*x]*Sin [4*ArcCsc[a*x]])/136))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[(u_.)*(f_)^(ArcCsc[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> Simp[ -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]
\[\int \frac {{\mathrm e}^{\operatorname {arccsc}\left (a x \right )}}{x^{5}}d x\]
Input:
int(exp(arccsc(a*x))/x^5,x)
Output:
int(exp(arccsc(a*x))/x^5,x)
Time = 0.15 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.63 \[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=\frac {{\left (6 \, a^{4} x^{4} + 3 \, a^{2} x^{2} - {\left (6 \, a^{2} x^{2} + 5\right )} \sqrt {a^{2} x^{2} - 1} - 20\right )} e^{\left (\operatorname {arccsc}\left (a x\right )\right )}}{85 \, x^{4}} \] Input:
integrate(exp(arccsc(a*x))/x^5,x, algorithm="fricas")
Output:
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^(a rccsc(a*x))/x^4
\[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=\int \frac {e^{\operatorname {acsc}{\left (a x \right )}}}{x^{5}}\, dx \] Input:
integrate(exp(acsc(a*x))/x**5,x)
Output:
Integral(exp(acsc(a*x))/x**5, x)
\[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=\int { \frac {e^{\left (\operatorname {arccsc}\left (a x\right )\right )}}{x^{5}} \,d x } \] Input:
integrate(exp(arccsc(a*x))/x^5,x, algorithm="maxima")
Output:
integrate(e^(arccsc(a*x))/x^5, x)
\[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=\int { \frac {e^{\left (\operatorname {arccsc}\left (a x\right )\right )}}{x^{5}} \,d x } \] Input:
integrate(exp(arccsc(a*x))/x^5,x, algorithm="giac")
Output:
integrate(e^(arccsc(a*x))/x^5, x)
Timed out. \[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=\int \frac {{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{a\,x}\right )}}{x^5} \,d x \] Input:
int(exp(asin(1/(a*x)))/x^5,x)
Output:
int(exp(asin(1/(a*x)))/x^5, x)
\[ \int \frac {e^{\csc ^{-1}(a x)}}{x^5} \, dx=\int \frac {e^{\mathit {acsc} \left (a x \right )}}{x^{5}}d x \] Input:
int(exp(acsc(a*x))/x^5,x)
Output:
int(e**acsc(a*x)/x**5,x)