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

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Rubi [F]  time = 0.0393858, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 0.519165, 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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Maple [F]  time = 0.041, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{{\it Arcsinh} \left ( \sinh \left ( x \right ) \right ) }}\, dx \end{align*}

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 = 1.81796, size = 1, normalized size = 0.04 \begin{align*} x \end{align*}

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 = 2.20826, size = 4, normalized size = 0.15 \begin{align*} x \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\operatorname{asinh}{\left (\sinh{\left (x \right )} \right )}}\, dx \end{align*}

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 = 1.35227, size = 1, normalized size = 0.04 \begin{align*} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x