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

Optimal. Leaf size=53 \[ \frac{\sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \sin ^{-1}(a x)}{2 a^2}+\frac{1}{2} x^2 e^{\text{sech}^{-1}(a x)}+\frac{x}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0181567, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6335, 8, 41, 216} \[ \frac{\sqrt{\frac{1}{a x+1}} \sqrt{a x+1} \sin ^{-1}(a x)}{2 a^2}+\frac{1}{2} x^2 e^{\text{sech}^{-1}(a x)}+\frac{x}{2 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6335

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 e^{\text{sech}^{-1}(a x)} x \, dx &=\frac{1}{2} e^{\text{sech}^{-1}(a x)} x^2+\frac{\int 1 \, dx}{2 a}+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{1-a x} \sqrt{1+a x}} \, dx}{2 a}\\ &=\frac{x}{2 a}+\frac{1}{2} e^{\text{sech}^{-1}(a x)} x^2+\frac{\left (\sqrt{\frac{1}{1+a x}} \sqrt{1+a x}\right ) \int \frac{1}{\sqrt{1-a^2 x^2}} \, dx}{2 a}\\ &=\frac{x}{2 a}+\frac{1}{2} e^{\text{sech}^{-1}(a x)} x^2+\frac{\sqrt{\frac{1}{1+a x}} \sqrt{1+a x} \sin ^{-1}(a x)}{2 a^2}\\ \end{align*}

Mathematica [C]  time = 0.0759349, size = 75, normalized size = 1.42 \[ \frac{2 a x+a x \sqrt{\frac{1-a x}{a x+1}} (a x+1)+i \log \left (2 \sqrt{\frac{1-a x}{a x+1}} (a x+1)-2 i a x\right )}{2 a^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [C]  time = 0.184, size = 92, normalized size = 1.7 \begin{align*}{\frac{x{\it csgn} \left ( a \right ) }{2\,a}\sqrt{-{\frac{ax-1}{ax}}}\sqrt{{\frac{ax+1}{ax}}} \left ( x\sqrt{-{a}^{2}{x}^{2}+1}{\it csgn} \left ( a \right ) a+\arctan \left ({x{\it csgn} \left ( a \right ) a{\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}} \right ) \right ){\frac{1}{\sqrt{-{a}^{2}{x}^{2}+1}}}}+{\frac{x}{a}} \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]

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

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

[Out]

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

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Fricas [A]  time = 1.80814, size = 171, normalized size = 3.23 \begin{align*} \frac{a^{2} x^{2} \sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}} + 2 \, a x - \arctan \left (\sqrt{\frac{a x + 1}{a x}} \sqrt{-\frac{a x - 1}{a x}}\right )}{2 \, a^{2}} \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]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int 1\, dx + \int a x \sqrt{-1 + \frac{1}{a x}} \sqrt{1 + \frac{1}{a 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(1, x) + Integral(a*x*sqrt(-1 + 1/(a*x))*sqrt(1 + 1/(a*x)), x))/a

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x{\left (\sqrt{\frac{1}{a x} + 1} \sqrt{\frac{1}{a x} - 1} + \frac{1}{a x}\right )}\,{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(x*(sqrt(1/(a*x) + 1)*sqrt(1/(a*x) - 1) + 1/(a*x)), x)