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

Optimal. Leaf size=38 \[ -\sqrt {\frac {1}{a^2 x^2}+1}+\tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )-\frac {1}{a x} \]

[Out]

-1/a/x+arctanh((1+1/a^2/x^2)^(1/2))-(1+1/a^2/x^2)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6336, 30, 266, 50, 63, 208} \[ -\sqrt {\frac {1}{a^2 x^2}+1}+\tanh ^{-1}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )-\frac {1}{a x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rule 266

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

Rule 6336

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

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 42, normalized size = 1.11 \[ -\sqrt {\frac {1}{a^2 x^2}+1}+\log \left (x \left (\sqrt {\frac {1}{a^2 x^2}+1}+1\right )\right )-\frac {1}{a x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [A]  time = 0.59, size = 64, normalized size = 1.68 \[ -\frac {a x \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right ) + a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + a x + 1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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))/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(x)]Error: Bad Argument Type

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maple [B]  time = 0.04, size = 107, normalized size = 2.82 \[ \frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (-a^{2} \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}}+\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, x^{2} a^{2}+\ln \left (x +\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\right ) x \right )}{\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}}-\frac {1}{a x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 54, normalized size = 1.42 \[ -\sqrt {\frac {1}{a^{2} x^{2}} + 1} - \frac {1}{a x} + \frac {1}{2} \, \log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.46, size = 34, normalized size = 0.89 \[ \mathrm {atanh}\left (\sqrt {\frac {1}{a^2\,x^2}+1}\right )-\sqrt {\frac {1}{a^2\,x^2}+1}-\frac {1}{a\,x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 4.81, size = 41, normalized size = 1.08 \[ - \frac {a x}{\sqrt {a^{2} x^{2} + 1}} + \operatorname {asinh}{\left (a x \right )} - \frac {1}{a x} - \frac {1}{a x \sqrt {a^{2} x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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