3.17 \(\int \text {csch}^{-1}(\sqrt {x}) \, dx\)

Optimal. Leaf size=31 \[ \frac {\sqrt {-x-1} \sqrt {x}}{\sqrt {-x}}+x \text {csch}^{-1}\left (\sqrt {x}\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6344, 12, 32} \[ \frac {\sqrt {-x-1} \sqrt {x}}{\sqrt {-x}}+x \text {csch}^{-1}\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCsch[Sqrt[x]],x]

[Out]

(Sqrt[-1 - x]*Sqrt[x])/Sqrt[-x] + x*ArcCsch[Sqrt[x]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 6344

Int[ArcCsch[u_], x_Symbol] :> Simp[x*ArcCsch[u], x] - Dist[u/Sqrt[-u^2], Int[SimplifyIntegrand[(x*D[u, x])/(u*
Sqrt[-1 - u^2]), x], x], x] /; InverseFunctionFreeQ[u, x] &&  !FunctionOfExponentialQ[u, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.77 \[ \sqrt {\frac {1}{x}+1} \sqrt {x}+x \text {csch}^{-1}\left (\sqrt {x}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCsch[Sqrt[x]],x]

[Out]

Sqrt[1 + x^(-1)]*Sqrt[x] + x*ArcCsch[Sqrt[x]]

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fricas [A]  time = 0.52, size = 36, normalized size = 1.16 \[ x \log \left (\frac {x \sqrt {\frac {x + 1}{x}} + \sqrt {x}}{x}\right ) + \sqrt {x} \sqrt {\frac {x + 1}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*log((x*sqrt((x + 1)/x) + sqrt(x))/x) + sqrt(x)*sqrt((x + 1)/x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {arcsch}\left (\sqrt {x}\right )\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccsch(sqrt(x)), x)

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maple [A]  time = 0.05, size = 24, normalized size = 0.77 \[ x \,\mathrm {arccsch}\left (\sqrt {x}\right )+\frac {1+x}{\sqrt {\frac {1+x}{x}}\, \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccsch(x^(1/2)),x)

[Out]

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

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maxima [A]  time = 0.32, size = 18, normalized size = 0.58 \[ x \operatorname {arcsch}\left (\sqrt {x}\right ) + \sqrt {x} \sqrt {\frac {1}{x} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*arccsch(sqrt(x)) + sqrt(x)*sqrt(1/x + 1)

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mupad [B]  time = 2.57, size = 18, normalized size = 0.58 \[ x\,\mathrm {asinh}\left (\frac {1}{\sqrt {x}}\right )+\sqrt {x}\,\sqrt {\frac {1}{x}+1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(asinh(1/x^(1/2)),x)

[Out]

x*asinh(1/x^(1/2)) + x^(1/2)*(1/x + 1)^(1/2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {acsch}{\left (\sqrt {x} \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acsch(x**(1/2)),x)

[Out]

Integral(acsch(sqrt(x)), x)

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