### 3.31 $$\int e^{\text{csch}^{-1}(a x)} \, dx$$

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

[Out]

E^ArcCsch[a*x]*x - ArcCsch[a*x]/a + Log[x]/a

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Rubi [A]  time = 0.015288, antiderivative size = 31, normalized size of antiderivative = 1.29, number of steps used = 5, number of rules used = 5, integrand size = 6, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.833, Rules used = {6331, 29, 242, 277, 215} $x \sqrt{\frac{1}{a^2 x^2}+1}+\frac{\log (x)}{a}-\frac{\text{csch}^{-1}(a x)}{a}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

Rule 6331

Int[E^ArcCsch[(a_.)*(x_)^(p_.)], x_Symbol] :> Dist[1/a, Int[1/x^p, x], x] + Int[Sqrt[1 + 1/(a^2*x^(2*p))], x]
/; FreeQ[{a, p}, x]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 242

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^2, x], x, 1/x] /; FreeQ[{a, b, p},
x] && ILtQ[n, 0]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 215

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

Rubi steps

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

Mathematica [A]  time = 0.0147928, size = 35, normalized size = 1.46 $\frac{a x \sqrt{\frac{1}{a^2 x^2}+1}+\log (a x)-\sinh ^{-1}\left (\frac{1}{a x}\right )}{a}$

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [B]  time = 0.169, size = 113, normalized size = 4.7 \begin{align*} -{\frac{x}{{a}^{2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}} \left ( -\sqrt{{a}^{-2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}{a}^{2}+\ln \left ( 2\,{\frac{1}{{a}^{2}x} \left ( \sqrt{{a}^{-2}}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}{a}^{2}+1 \right ) } \right ) \right ){\frac{1}{\sqrt{{a}^{-2}}}}{\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}}}}+{\frac{\ln \left ( x \right ) }{a}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.00209, size = 86, normalized size = 3.58 \begin{align*} x \sqrt{\frac{1}{a^{2} x^{2}} + 1} - \frac{\log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right )}{2 \, a} + \frac{\log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right )}{2 \, a} + \frac{\log \left (x\right )}{a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.77596, size = 203, normalized size = 8.46 \begin{align*} \frac{a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) + \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) + \log \left (x\right )}{a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.01817, size = 48, normalized size = 2. \begin{align*} \frac{x}{\sqrt{1 + \frac{1}{a^{2} x^{2}}}} + \frac{\log{\left (x \right )}}{a} - \frac{\operatorname{asinh}{\left (\frac{1}{a x} \right )}}{a} + \frac{1}{a^{2} x \sqrt{1 + \frac{1}{a^{2} x^{2}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16093, size = 82, normalized size = 3.42 \begin{align*} \frac{{\left (2 \, \sqrt{a^{2} x^{2} + 1} - \log \left (\sqrt{a^{2} x^{2} + 1} + 1\right ) + \log \left (\sqrt{a^{2} x^{2} + 1} - 1\right )\right )}{\left | a \right |} \mathrm{sgn}\left (x\right )}{2 \, a^{2}} + \frac{\log \left ({\left | x \right |}\right )}{a} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*sqrt(a^2*x^2 + 1) - log(sqrt(a^2*x^2 + 1) + 1) + log(sqrt(a^2*x^2 + 1) - 1))*abs(a)*sgn(x)/a^2 + log(ab
s(x))/a