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

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Rubi [A]  time = 0.21, antiderivative size = 38, normalized size of antiderivative = 1.00, 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 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]

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 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]]

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*}

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Mathematica [A]  time = 0.06, 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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fricas [B]  time = 0.60, size = 112, normalized size = 2.95 \[ -\frac {a^{2} x^{2} \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x + 1\right ) - a^{2} x^{2} \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x - 1\right ) - a^{2} x^{2} \log \relax (x) + a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + 1}{a^{2} x^{2}} \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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 >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Error: Bad Argument Type

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maple [B]  time = 0.05, size = 150, normalized size = 3.95 \[ \ln \relax (x )-\frac {1}{x^{2} a^{2}}-\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (a^{2} \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}} \sqrt {\frac {1}{a^{2}}}-\sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, x^{2} a^{2}+\ln \left (\frac {2 \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2}+2}{a^{2} x}\right ) x^{2}\right )}{a x \sqrt {\frac {1}{a^{2}}}\, \sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.32, size = 93, normalized size = 2.45 \[ -\frac {\frac {2 \, a^{2} x \sqrt {\frac {1}{a^{2} x^{2}} + 1}}{a^{2} x^{2} {\left (\frac {1}{a^{2} x^{2}} + 1\right )} - 1} + a \log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right ) - a \log \left (a x \sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right )}{2 \, a} - \frac {1}{a^{2} x^{2}} + \log \relax (x) \]

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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mupad [B]  time = 2.25, size = 44, normalized size = 1.16 \[ -\ln \left (\frac {1}{x}\right )-\mathrm {asinh}\left (\frac {1}{a\,x}\right )-\frac {1}{a^2\,x^2}-\frac {\sqrt {\frac {1}{a^2\,x^2}+1}}{a\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 4.08, size = 34, normalized size = 0.89 \[ \log {\relax (x )} - \operatorname {asinh}{\left (\frac {1}{a x} \right )} - \frac {\sqrt {1 + \frac {1}{a^{2} x^{2}}}}{a x} - \frac {1}{a^{2} x^{2}} \]

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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