### 3.54 $$\int \frac{e^{2 \text{csch}^{-1}(a x)}}{x} \, dx$$

Optimal. Leaf size=38 $-\frac{\sqrt{\frac{1}{a^2 x^2}+1}}{a x}-\frac{1}{a^2 x^2}-\text{csch}^{-1}(a x)+\log (x)$

[Out]

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

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Rubi [A]  time = 0.214397, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.417, Rules used = {6338, 6742, 335, 195, 215} $-\frac{\sqrt{\frac{1}{a^2 x^2}+1}}{a x}-\frac{1}{a^2 x^2}-\text{csch}^{-1}(a x)+\log (x)$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(2*ArcCsch[a*x])/x,x]

[Out]

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

Rule 6338

Int[E^(ArcCsch[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(1/u + Sqrt[1 + 1/u^2])^n, x] /; FreeQ[m, x] && Int
egerQ[n]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 335

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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 \frac{e^{2 \text{csch}^{-1}(a x)}}{x} \, dx &=\int \frac{\left (\sqrt{1+\frac{1}{a^2 x^2}}+\frac{1}{a x}\right )^2}{x} \, dx\\ &=\int \left (\frac{2}{a^2 x^3}+\frac{2 \sqrt{1+\frac{1}{a^2 x^2}}}{a x^2}+\frac{1}{x}\right ) \, dx\\ &=-\frac{1}{a^2 x^2}+\log (x)+\frac{2 \int \frac{\sqrt{1+\frac{1}{a^2 x^2}}}{x^2} \, dx}{a}\\ &=-\frac{1}{a^2 x^2}+\log (x)-\frac{2 \operatorname{Subst}\left (\int \sqrt{1+\frac{x^2}{a^2}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{1}{a^2 x^2}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{a x}+\log (x)-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{a^2}}} \, dx,x,\frac{1}{x}\right )}{a}\\ &=-\frac{1}{a^2 x^2}-\frac{\sqrt{1+\frac{1}{a^2 x^2}}}{a x}-\text{csch}^{-1}(a x)+\log (x)\\ \end{align*}

Mathematica [A]  time = 0.0571885, size = 39, normalized size = 1.03 $-\frac{a x \sqrt{\frac{1}{a^2 x^2}+1}+1}{a^2 x^2}-\sinh ^{-1}\left (\frac{1}{a x}\right )+\log (x)$

Warning: Unable to verify antiderivative.

[In]

Integrate[E^(2*ArcCsch[a*x])/x,x]

[Out]

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

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Maple [B]  time = 0.177, size = 150, normalized size = 4. \begin{align*} \ln \left ( x \right ) -{\frac{1}{{a}^{2}{x}^{2}}}-{\frac{1}{ax}\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}{x}^{2}}}} \left ({a}^{2} \left ({\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}} \right ) ^{{\frac{3}{2}}}\sqrt{{a}^{-2}}-\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}\sqrt{{a}^{-2}}{x}^{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 ){x}^{2} \right ){\frac{1}{\sqrt{{a}^{-2}}}}{\frac{1}{\sqrt{{\frac{{a}^{2}{x}^{2}+1}{{a}^{2}}}}}}} \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))^2/x,x)

[Out]

ln(x)-1/a^2/x^2-1/a*((a^2*x^2+1)/a^2/x^2)^(1/2)/x*(a^2*((a^2*x^2+1)/a^2)^(3/2)*(1/a^2)^(1/2)-((a^2*x^2+1)/a^2)
^(1/2)*(1/a^2)^(1/2)*x^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)*x^2)/(1/a^2)^(1/2)/((a^
2*x^2+1)/a^2)^(1/2)

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Maxima [B]  time = 1.01631, size = 126, normalized size = 3.32 \begin{align*} -\frac{\frac{2 \, a^{2} x \sqrt{\frac{1}{a^{2} x^{2}} + 1}}{a^{2} x^{2}{\left (\frac{1}{a^{2} x^{2}} + 1\right )} - 1} + a \log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} + 1\right ) - a \log \left (a x \sqrt{\frac{1}{a^{2} x^{2}} + 1} - 1\right )}{2 \, a} - \frac{1}{a^{2} x^{2}} + \log \left (x\right ) \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))^2/x,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.59509, size = 252, normalized size = 6.63 \begin{align*} -\frac{a^{2} x^{2} \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) - a^{2} x^{2} \log \left (x\right ) + a x \sqrt{\frac{a^{2} x^{2} + 1}{a^{2} x^{2}}} + 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))^2/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.73874, size = 34, normalized size = 0.89 \begin{align*} \log{\left (x \right )} - \operatorname{asinh}{\left (\frac{1}{a x} \right )} - \frac{\sqrt{1 + \frac{1}{a^{2} x^{2}}}}{a x} - \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))**2/x,x)

[Out]

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

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \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))^2/x,x, algorithm="giac")

[Out]

Exception raised: TypeError