3.4.69 \(\int \frac {x}{\sinh ^{-1}(\sinh (x))} \, dx\) [369]

Optimal. Leaf size=27 \[ \sinh ^{-1}(\sinh (x))+\log \left (\sinh ^{-1}(\sinh (x))\right ) \left (-\sinh ^{-1}(\sinh (x))+x \sqrt {\cosh ^2(x)} \text {sech}(x)\right ) \]

[Out]

arcsinh(sinh(x))+ln(arcsinh(sinh(x)))*(-arcsinh(sinh(x))+x*sech(x)*(cosh(x)^2)^(1/2))

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Rubi [F]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {x}{\sinh ^{-1}(\sinh (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x/ArcSinh[Sinh[x]],x]

[Out]

Defer[Int][x/ArcSinh[Sinh[x]], x]

Rubi steps

\begin {align*} \int \frac {x}{\sinh ^{-1}(\sinh (x))} \, dx &=\int \frac {x}{\sinh ^{-1}(\sinh (x))} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.36, size = 28, normalized size = 1.04 \begin {gather*} -\sinh ^{-1}(\sinh (x)) \left (-1+\log \left (\sinh ^{-1}(\sinh (x))\right )\right )+x \sqrt {\cosh ^2(x)} \log \left (\sinh ^{-1}(\sinh (x))\right ) \text {sech}(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/ArcSinh[Sinh[x]],x]

[Out]

-(ArcSinh[Sinh[x]]*(-1 + Log[ArcSinh[Sinh[x]]])) + x*Sqrt[Cosh[x]^2]*Log[ArcSinh[Sinh[x]]]*Sech[x]

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Maple [F]
time = 0.82, size = 0, normalized size = 0.00 \[\int \frac {x}{\arcsinh \left (\sinh \left (x \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/arcsinh(sinh(x)),x)

[Out]

int(x/arcsinh(sinh(x)),x)

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Maxima [A]
time = 0.49, size = 1, normalized size = 0.04 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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Fricas [A]
time = 0.32, size = 1, normalized size = 0.04 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/asinh(sinh(x)),x)

[Out]

Integral(x/asinh(sinh(x)), x)

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Giac [A]
time = 0.39, size = 1, normalized size = 0.04 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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Mupad [B]
time = 0.22, size = 1, normalized size = 0.04 \begin {gather*} x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/asinh(sinh(x)),x)

[Out]

x

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