\(\int \frac {e^{\sec ^{-1}(a x)}}{x} \, dx\) [46]

Optimal result

Integrand size = 10, antiderivative size = 45 \[ \int \frac {e^{\sec ^{-1}(a x)}}{x} \, dx=-i e^{\sec ^{-1}(a x)}+2 i e^{\sec ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (-\frac {i}{2},1,1-\frac {i}{2},-e^{2 i \sec ^{-1}(a x)}\right ) \]

[Out]

-I*exp(arcsec(a*x))+2*I*exp(arcsec(a*x))*hypergeom([1, -1/2*I],[1-1/2*I],-(1/a/x+I*(1-1/a^2/x^2)^(1/2))^2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5374, 12, 4527, 2225, 2283} \[ \int \frac {e^{\sec ^{-1}(a x)}}{x} \, dx=2 i e^{\sec ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (-\frac {i}{2},1,1-\frac {i}{2},-e^{2 i \sec ^{-1}(a x)}\right )-i e^{\sec ^{-1}(a x)} \]

[In]

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

[Out]

(-I)*E^ArcSec[a*x] + (2*I)*E^ArcSec[a*x]*Hypergeometric2F1[-1/2*I, 1, 1 - I/2, -E^((2*I)*ArcSec[a*x])]

Rule 12

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

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2283

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[a^p*(G^(h*(f + g*x))/(g*h*Log[G]))*Hypergeometric2F1[-p, g*h*(Log[G]/(d*e*Log[F])), g*h*(Log[G]/(d*e*Log[F]))
+ 1, Simplify[(-b/a)*F^(e*(c + d*x))]], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] || G
tQ[a, 0])

Rule 4527

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*Tan[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol] :> Dist[I^n, Int[ExpandIntegran
d[F^(c*(a + b*x))*((1 - E^(2*I*(d + e*x)))^n/(1 + E^(2*I*(d + e*x)))^n), x], x], x] /; FreeQ[{F, a, b, c, d, e
}, x] && IntegerQ[n]

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 e^x \tan (x) \, dx,x,\sec ^{-1}(a x)\right )}{a} \\ & = \text {Subst}\left (\int e^x \tan (x) \, dx,x,\sec ^{-1}(a x)\right ) \\ & = i \text {Subst}\left (\int \left (-e^x+\frac {2 e^x}{1+e^{2 i x}}\right ) \, dx,x,\sec ^{-1}(a x)\right ) \\ & = -\left (i \text {Subst}\left (\int e^x \, dx,x,\sec ^{-1}(a x)\right )\right )+2 i \text {Subst}\left (\int \frac {e^x}{1+e^{2 i x}} \, dx,x,\sec ^{-1}(a x)\right ) \\ & = -i e^{\sec ^{-1}(a x)}+2 i e^{\sec ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (-\frac {i}{2},1,1-\frac {i}{2},-e^{2 i \sec ^{-1}(a x)}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.76 \[ \int \frac {e^{\sec ^{-1}(a x)}}{x} \, dx=-i \left (-e^{\sec ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (-\frac {i}{2},1,1-\frac {i}{2},-e^{2 i \sec ^{-1}(a x)}\right )+\left (\frac {1}{5}-\frac {2 i}{5}\right ) e^{(1+2 i) \sec ^{-1}(a x)} \operatorname {Hypergeometric2F1}\left (1,1-\frac {i}{2},2-\frac {i}{2},-e^{2 i \sec ^{-1}(a x)}\right )\right ) \]

[In]

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

[Out]

(-I)*(-(E^ArcSec[a*x]*Hypergeometric2F1[-1/2*I, 1, 1 - I/2, -E^((2*I)*ArcSec[a*x])]) + (1/5 - (2*I)/5)*E^((1 +
2*I)*ArcSec[a*x])*Hypergeometric2F1[1, 1 - I/2, 2 - I/2, -E^((2*I)*ArcSec[a*x])])

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(e^(arcsec(a*x))/x, x)

Sympy [F]

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

[In]

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

[Out]

Integral(exp(asec(a*x))/x, x)

Maxima [F]

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

[In]

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

[Out]

integrate(e^(arcsec(a*x))/x, x)

Giac [F]

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

[In]

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

[Out]

integrate(e^(arcsec(a*x))/x, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(exp(acos(1/(a*x)))/x, x)