3.369 \(\int \frac {x}{\sinh ^{-1}(\sinh (x))} \, dx\)

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

[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 = 0.04, 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 = {} \[ \int \frac {x}{\sinh ^{-1}(\sinh (x))} \, dx \]

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 = 0.56, size = 28, normalized size = 1.04 \[ x \sqrt {\cosh ^2(x)} \text {sech}(x) \log \left (\sinh ^{-1}(\sinh (x))\right )-\sinh ^{-1}(\sinh (x)) \left (\log \left (\sinh ^{-1}(\sinh (x))\right )-1\right ) \]

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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fricas [A]  time = 0.48, size = 1, normalized size = 0.04 \[ x \]

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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giac [A]  time = 0.28, size = 1, normalized size = 0.04 \[ x \]

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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maple [F]  time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {x}{\arcsinh \left (\sinh \relax (x )\right )}\, dx \]

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 = 0.78, size = 1, normalized size = 0.04 \[ x \]

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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mupad [B]  time = 0.22, size = 1, normalized size = 0.04 \[ x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\operatorname {asinh}{\left (\sinh {\relax (x )} \right )}}\, dx \]

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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