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

Optimal. Leaf size=48 \[ 2 \tan ^{-1}\left (\sqrt{\frac{1-a x}{a x+1}}\right )-\frac{2}{1-\sqrt{\frac{1-a x}{a x+1}}} \]

[Out]

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

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Rubi [A]  time = 0.0308311, antiderivative size = 64, normalized size of antiderivative = 1.33, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6334, 97, 12, 41, 216} \[ -\frac{\sqrt{1-a x}}{a x \sqrt{\frac{1}{a x+1}}}-\frac{1}{a x}-\sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \sin ^{-1}(a x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(1/(a*x)) - Sqrt[1 - a*x]/(a*x*Sqrt[(1 + a*x)^(-1)]) - Sqrt[(1 + a*x)^(-1)]*Sqrt[1 + a*x]*ArcSin[a*x]

Rule 6334

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

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])

Rule 12

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

Rule 41

Int[((a_) + (b_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(a*c + b*d*x^2)^m, x] /; FreeQ[{a, b
, c, d, m}, x] && EqQ[b*c + a*d, 0] && (IntegerQ[m] || (GtQ[a, 0] && GtQ[c, 0]))

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

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

Mathematica [C]  time = 0.0497251, size = 75, normalized size = 1.56 \[ \sqrt{\frac{1-a x}{a x+1}} \left (-\frac{1}{a x}-1\right )-\frac{1}{a x}-i \log \left (2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)-2 i a x\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [C]  time = 0.189, size = 92, normalized size = 1.9 \begin{align*} -{{\it csgn} \left ( a \right ) \sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( \arctan \left ({{\it csgn} \left ( a \right ) ax{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) xa+{\it csgn} \left ( a \right ) \sqrt{-{a}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}-{\frac{1}{ax}} \end{align*}

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,x)

[Out]

-(-(a*x-1)/a/x)^(1/2)*((a*x+1)/a/x)^(1/2)*(arctan(csgn(a)*a*x/(-a^2*x^2+1)^(1/2))*x*a+csgn(a)*(-a^2*x^2+1)^(1/
2))*csgn(a)/(-a^2*x^2+1)^(1/2)-1/a/x

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{-\frac{a^{2} \arcsin \left (\frac{a^{2} x}{\sqrt{a^{2}}}\right )}{\sqrt{a^{2}}} - \frac{\sqrt{-a^{2} x^{2} + 1}}{x}}{a} - \frac{1}{a x} \end{align*}

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,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.02397, size = 165, normalized size = 3.44 \begin{align*} -\frac{a x \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} - a x \arctan \left (\sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}\right ) + 1}{a x} \end{align*}

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,x, algorithm="fricas")

[Out]

-(a*x*sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)) - a*x*arctan(sqrt((a*x + 1)/(a*x))*sqrt(-(a*x - 1)/(a*x)))
+ 1)/(a*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{x^{2}}\, dx + \int \frac{a \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a x}}}{x}\, dx}{a} \end{align*}

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,x)

[Out]

(Integral(x**(-2), x) + Integral(a*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x))/x, x))/a

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}}{x}\,{d x} \end{align*}

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,x, algorithm="giac")

[Out]

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