3.36 \(\int e^{\text {sech}^{-1}(a x)} \, dx\)

Optimal. Leaf size=24 \[ \frac {\log (x)}{a}+x e^{\text {sech}^{-1}(a x)}-\frac {\text {sech}^{-1}(a x)}{a} \]

[Out]

(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2))*x-arcsech(a*x)/a+ln(x)/a

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Rubi [A]  time = 0.14, antiderivative size = 39, normalized size of antiderivative = 1.62, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6329, 1962, 208} \[ \frac {\log (x)}{a}-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{a x+1}}\right )}{a}+x e^{\text {sech}^{-1}(a x)} \]

Warning: Unable to verify antiderivative.

[In]

Int[E^ArcSech[a*x],x]

[Out]

E^ArcSech[a*x]*x - (2*ArcTanh[Sqrt[(1 - a*x)/(1 + a*x)]])/a + Log[x]/a

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 1962

Int[(u_)^(r_.)*(x_)^(m_.)*(((e_.)*((a_.) + (b_.)*(x_)^(n_.)))/((c_) + (d_.)*(x_)^(n_.)))^(p_), x_Symbol] :> Wi
th[{q = Denominator[p]}, Dist[(q*e*(b*c - a*d))/n, Subst[Int[SimplifyIntegrand[(x^(q*(p + 1) - 1)*(-(a*e) + c*
x^q)^((m + 1)/n - 1)*(u /. x -> (-(a*e) + c*x^q)^(1/n)/(b*e - d*x^q)^(1/n))^r)/(b*e - d*x^q)^((m + 1)/n + 1),
x], x], x, ((e*(a + b*x^n))/(c + d*x^n))^(1/q)], x]] /; FreeQ[{a, b, c, d, e}, x] && PolynomialQ[u, x] && Frac
tionQ[p] && IntegerQ[1/n] && IntegersQ[m, r]

Rule 6329

Int[E^ArcSech[(a_.)*(x_)], x_Symbol] :> Simp[x*E^ArcSech[a*x], x] + (Dist[1/a, Int[(1*Sqrt[(1 - a*x)/(1 + a*x)
])/(x*(1 - a*x)), x], x] + Simp[Log[x]/a, x]) /; FreeQ[a, x]

Rubi steps

\begin {align*} \int e^{\text {sech}^{-1}(a x)} \, dx &=e^{\text {sech}^{-1}(a x)} x+\frac {\log (x)}{a}+\frac {\int \frac {\sqrt {\frac {1-a x}{1+a x}}}{x (1-a x)} \, dx}{a}\\ &=e^{\text {sech}^{-1}(a x)} x+\frac {\log (x)}{a}-4 \operatorname {Subst}\left (\int \frac {1}{2 a-2 a x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\\ &=e^{\text {sech}^{-1}(a x)} x-\frac {2 \tanh ^{-1}\left (\sqrt {\frac {1-a x}{1+a x}}\right )}{a}+\frac {\log (x)}{a}\\ \end {align*}

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Mathematica [B]  time = 0.04, size = 79, normalized size = 3.29 \[ \frac {\sqrt {\frac {1-a x}{a x+1}} (a x+1)+2 \log (a x)-\log \left (a x \sqrt {\frac {1-a x}{a x+1}}+\sqrt {\frac {1-a x}{a x+1}}+1\right )}{a} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[E^ArcSech[a*x],x]

[Out]

(Sqrt[(1 - a*x)/(1 + a*x)]*(1 + a*x) + 2*Log[a*x] - Log[1 + Sqrt[(1 - a*x)/(1 + a*x)] + a*x*Sqrt[(1 - a*x)/(1
+ a*x)]])/a

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fricas [B]  time = 0.70, size = 115, normalized size = 4.79 \[ \frac {2 \, a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} + 1\right ) + \log \left (a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - 1\right ) + 2 \, \log \relax (x)}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2),x, algorithm="fricas")

[Out]

1/2*(2*a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x))
+ 1) + log(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - 1) + 2*log(x))/a

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x), x)

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maple [A]  time = 0.06, size = 80, normalized size = 3.33 \[ \frac {\ln \relax (x )}{a}-\frac {\sqrt {-\frac {a x -1}{a x}}\, x \sqrt {\frac {a x +1}{a x}}\, \left (-\sqrt {-a^{2} x^{2}+1}+\arctanh \left (\frac {1}{\sqrt {-a^{2} x^{2}+1}}\right )\right )}{\sqrt {-a^{2} x^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2),x)

[Out]

ln(x)/a-(-(a*x-1)/a/x)^(1/2)*x*((a*x+1)/a/x)^(1/2)*(-(-a^2*x^2+1)^(1/2)+arctanh(1/(-a^2*x^2+1)^(1/2)))/(-a^2*x
^2+1)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/a/x+(1/a/x-1)^(1/2)*(1+1/a/x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x), x)

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mupad [B]  time = 2.98, size = 182, normalized size = 7.58 \[ \frac {\ln \relax (x)}{a}-\frac {4\,\mathrm {atanh}\left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )}{a}+\frac {\frac {5\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+1}{\frac {4\,a\,\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}{\sqrt {\frac {1}{a\,x}+1}-1}+\frac {4\,a\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^3}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^3}}+\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{4\,a\,\left (\sqrt {\frac {1}{a\,x}+1}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/(a*x) - 1)^(1/2)*(1/(a*x) + 1)^(1/2) + 1/(a*x),x)

[Out]

log(x)/a - (4*atanh(((1/(a*x) - 1)^(1/2) - 1i)/((1/(a*x) + 1)^(1/2) - 1)))/a + ((5*((1/(a*x) - 1)^(1/2) - 1i)^
2)/((1/(a*x) + 1)^(1/2) - 1)^2 + 1)/((4*a*((1/(a*x) - 1)^(1/2) - 1i))/((1/(a*x) + 1)^(1/2) - 1) + (4*a*((1/(a*
x) - 1)^(1/2) - 1i)^3)/((1/(a*x) + 1)^(1/2) - 1)^3) + ((1/(a*x) - 1)^(1/2) - 1i)/(4*a*((1/(a*x) + 1)^(1/2) - 1
))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {1}{x}\, dx + \int a \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/a/x+(1/a/x-1)**(1/2)*(1+1/a/x)**(1/2),x)

[Out]

(Integral(1/x, x) + Integral(a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)), x))/a

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