3.4.2 \(\int \sinh ^{-1}(\frac {a}{x}) \, dx\) [302]

Optimal. Leaf size=25 \[ x \text {csch}^{-1}\left (\frac {x}{a}\right )+a \tanh ^{-1}\left (\sqrt {1+\frac {a^2}{x^2}}\right ) \]

[Out]

x*arccsch(x/a)+a*arctanh((1+a^2/x^2)^(1/2))

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Rubi [A]
time = 0.01, antiderivative size = 25, normalized size of antiderivative = 1.00, 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 = {5870, 6413, 272, 65, 214} \begin {gather*} a \tanh ^{-1}\left (\sqrt {\frac {a^2}{x^2}+1}\right )+x \text {csch}^{-1}\left (\frac {x}{a}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcSinh[a/x],x]

[Out]

x*ArcCsch[x/a] + a*ArcTanh[Sqrt[1 + a^2/x^2]]

Rule 65

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 214

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

Rule 272

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 5870

Int[ArcSinh[(c_.)/((a_.) + (b_.)*(x_)^(n_.))]^(m_.)*(u_.), x_Symbol] :> Int[u*ArcCsch[a/c + b*(x^n/c)]^m, x] /
; FreeQ[{a, b, c, n, m}, x]

Rule 6413

Int[ArcCsch[(c_.)*(x_)], x_Symbol] :> Simp[x*ArcCsch[c*x], x] + Dist[1/c, Int[1/(x*Sqrt[1 + 1/(c^2*x^2)]), x],
x] /; FreeQ[c, x]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(77\) vs. \(2(25)=50\).
time = 0.06, size = 77, normalized size = 3.08 \begin {gather*} x \sinh ^{-1}\left (\frac {a}{x}\right )+\frac {a \sqrt {a^2+x^2} \left (-\log \left (1-\frac {x}{\sqrt {a^2+x^2}}\right )+\log \left (1+\frac {x}{\sqrt {a^2+x^2}}\right )\right )}{2 \sqrt {1+\frac {a^2}{x^2}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcSinh[a/x],x]

[Out]

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

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Maple [A]
time = 0.88, size = 31, normalized size = 1.24

method	result	size
derivativedivides	\(-a \left (-\frac {x \arcsinh \left (\frac {a}{x}\right )}{a}-\arctanh \left (\frac {1}{\sqrt {1+\frac {a^{2}}{x^{2}}}}\right )\right )\)	\(31\)
default	\(-a \left (-\frac {x \arcsinh \left (\frac {a}{x}\right )}{a}-\arctanh \left (\frac {1}{\sqrt {1+\frac {a^{2}}{x^{2}}}}\right )\right )\)	\(31\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arcsinh(a/x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.27, size = 43, normalized size = 1.72 \begin {gather*} \frac {1}{2} \, a {\left (\log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} - 1\right )\right )} + x \operatorname {arsinh}\left (\frac {a}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a/x),x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (23) = 46\).
time = 0.40, size = 96, normalized size = 3.84 \begin {gather*} -a \log \left (x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} - x\right ) + {\left (x - 1\right )} \log \left (\frac {x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} + a}{x}\right ) + \log \left (x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} + a - x\right ) - \log \left (x \sqrt {\frac {a^{2} + x^{2}}{x^{2}}} - a - x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a/x),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {asinh}{\left (\frac {a}{x} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(asinh(a/x),x)

[Out]

Integral(asinh(a/x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (23) = 46\).
time = 0.40, size = 49, normalized size = 1.96 \begin {gather*} a \log \left ({\left | a \right |}\right ) \mathrm {sgn}\left (x\right ) + x \log \left (\sqrt {\frac {a^{2}}{x^{2}} + 1} + \frac {a}{x}\right ) - \frac {a \log \left (-x + \sqrt {a^{2} + x^{2}}\right )}{\mathrm {sgn}\left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(a/x),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.55, size = 25, normalized size = 1.00 \begin {gather*} x\,\mathrm {asinh}\left (\frac {a}{x}\right )+a\,\ln \left (x+\sqrt {a^2+x^2}\right )\,\mathrm {sign}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(a/x),x)

[Out]

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

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