\(\int \frac {x}{\text {arcsinh}(\sinh (x))} \, dx\) [369]

Optimal result

Integrand size = 7, antiderivative size = 27 \[ \int \frac {x}{\text {arcsinh}(\sinh (x))} \, dx=\text {arcsinh}(\sinh (x))+\log (\text {arcsinh}(\sinh (x))) \left (-\text {arcsinh}(\sinh (x))+x \sqrt {\cosh ^2(x)} \text {sech}(x)\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {x}{\text {arcsinh}(\sinh (x))} \, dx=\int \frac {x}{\text {arcsinh}(\sinh (x))} \, dx \]

[In]

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

[Out]

Defer[Int][x/ArcSinh[Sinh[x]], x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x}{\text {arcsinh}(\sinh (x))} \, dx \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(148\) vs. \(2(27)=54\).

Time = 0.22 (sec) , antiderivative size = 148, normalized size of antiderivative = 5.48 \[ \int \frac {x}{\text {arcsinh}(\sinh (x))} \, dx=-\log \left (\frac {1}{2} e^{-x} \left (-1+e^{2 x}+e^x \sqrt {2+e^{-2 x}+e^{2 x}}\right )\right ) \left (-1+\log \left (\log \left (\frac {1}{2} e^{-x} \left (-1+e^{2 x}+e^x \sqrt {2+e^{-2 x}+e^{2 x}}\right )\right )\right )\right )+\frac {e^x \sqrt {2+e^{-2 x}+e^{2 x}} x \log \left (\log \left (\frac {1}{2} e^{-x} \left (-1+e^{2 x}+e^x \sqrt {2+e^{-2 x}+e^{2 x}}\right )\right )\right )}{1+e^{2 x}} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {x}{\operatorname {arcsinh}\left (\sinh \left (x \right )\right )}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.04 \[ \int \frac {x}{\text {arcsinh}(\sinh (x))} \, dx=x \]

[In]

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

[Out]

x

Sympy [F]

\[ \int \frac {x}{\text {arcsinh}(\sinh (x))} \, dx=\int \frac {x}{\operatorname {asinh}{\left (\sinh {\left (x \right )} \right )}}\, dx \]

[In]

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

[Out]

Integral(x/asinh(sinh(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.04 \[ \int \frac {x}{\text {arcsinh}(\sinh (x))} \, dx=x \]

[In]

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

[Out]

x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.04 \[ \int \frac {x}{\text {arcsinh}(\sinh (x))} \, dx=x \]

[In]

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

[Out]

x

Mupad [B] (verification not implemented)

Time = 2.53 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.04 \[ \int \frac {x}{\text {arcsinh}(\sinh (x))} \, dx=x \]

[In]

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

[Out]

x