\(\int \frac {e^{\sec ^{-1}(a x)}}{x^3} \, dx\) [48]

Optimal result

Integrand size = 10, antiderivative size = 41 \[ \int \frac {e^{\sec ^{-1}(a x)}}{x^3} \, dx=-\frac {1}{5} a^2 e^{\sec ^{-1}(a x)} \cos \left (2 \sec ^{-1}(a x)\right )+\frac {1}{10} a^2 e^{\sec ^{-1}(a x)} \sin \left (2 \sec ^{-1}(a x)\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5374, 12, 4557, 4517} \[ \int \frac {e^{\sec ^{-1}(a x)}}{x^3} \, dx=\frac {1}{10} a^2 e^{\sec ^{-1}(a x)} \sin \left (2 \sec ^{-1}(a x)\right )-\frac {1}{5} a^2 e^{\sec ^{-1}(a x)} \cos \left (2 \sec ^{-1}(a x)\right ) \]

[In]

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

[Out]

-1/5*(a^2*E^ArcSec[a*x]*Cos[2*ArcSec[a*x]]) + (a^2*E^ArcSec[a*x]*Sin[2*ArcSec[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 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 5374

Int[(u_.)*(f_)^(ArcSec[(a_.) + (b_.)*(x_)]^(n_.)*(c_.)), x_Symbol] :> Dist[1/b, Subst[Int[(u /. x -> -a/b + Se
c[x]/b)*f^(c*x^n)*Sec[x]*Tan[x], x], x, ArcSec[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^3 e^x \cos (x) \sin (x) \, dx,x,\sec ^{-1}(a x)\right )}{a} \\ & = a^2 \text {Subst}\left (\int e^x \cos (x) \sin (x) \, dx,x,\sec ^{-1}(a x)\right ) \\ & = a^2 \text {Subst}\left (\int \frac {1}{2} e^x \sin (2 x) \, dx,x,\sec ^{-1}(a x)\right ) \\ & = \frac {1}{2} a^2 \text {Subst}\left (\int e^x \sin (2 x) \, dx,x,\sec ^{-1}(a x)\right ) \\ & = -\frac {1}{5} a^2 e^{\sec ^{-1}(a x)} \cos \left (2 \sec ^{-1}(a x)\right )+\frac {1}{10} a^2 e^{\sec ^{-1}(a x)} \sin \left (2 \sec ^{-1}(a x)\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.73 \[ \int \frac {e^{\sec ^{-1}(a x)}}{x^3} \, dx=\frac {1}{10} a^2 e^{\sec ^{-1}(a x)} \left (-2 \cos \left (2 \sec ^{-1}(a x)\right )+\sin \left (2 \sec ^{-1}(a x)\right )\right ) \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {{\mathrm e}^{\operatorname {arcsec}\left (a x \right )}}{x^{3}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.73 \[ \int \frac {e^{\sec ^{-1}(a x)}}{x^3} \, dx=\frac {{\left (a^{2} x^{2} + \sqrt {a^{2} x^{2} - 1} - 2\right )} e^{\left (\operatorname {arcsec}\left (a x\right )\right )}}{5 \, x^{2}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {e^{\sec ^{-1}(a x)}}{x^3} \, dx=\int \frac {e^{\operatorname {asec}{\left (a x \right )}}}{x^{3}}\, dx \]

[In]

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

[Out]

Integral(exp(asec(a*x))/x**3, x)

Maxima [F]

\[ \int \frac {e^{\sec ^{-1}(a x)}}{x^3} \, dx=\int { \frac {e^{\left (\operatorname {arcsec}\left (a x\right )\right )}}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate(e^(arcsec(a*x))/x^3, x)

Giac [F]

\[ \int \frac {e^{\sec ^{-1}(a x)}}{x^3} \, dx=\int { \frac {e^{\left (\operatorname {arcsec}\left (a x\right )\right )}}{x^{3}} \,d x } \]

[In]

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

[Out]

integrate(e^(arcsec(a*x))/x^3, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {e^{\sec ^{-1}(a x)}}{x^3} \, dx=\int \frac {{\mathrm {e}}^{\mathrm {acos}\left (\frac {1}{a\,x}\right )}}{x^3} \,d x \]

[In]

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

[Out]

int(exp(acos(1/(a*x)))/x^3, x)