\(\int \text {csch}(x) \, dx\) [575]

Optimal result

Integrand size = 2, antiderivative size = 5 \[ \int \text {csch}(x) \, dx=-\text {arctanh}(\cosh (x)) \]

[Out]

-arctanh(cosh(x))

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3855} \[ \int \text {csch}(x) \, dx=-\text {arctanh}(\cosh (x)) \]

[In]

Int[Csch[x],x]

[Out]

-ArcTanh[Cosh[x]]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = -\text {arctanh}(\cosh (x)) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(17\) vs. \(2(5)=10\).

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 3.40 \[ \int \text {csch}(x) \, dx=-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right ) \]

[In]

Integrate[Csch[x],x]

[Out]

-Log[Cosh[x/2]] + Log[Sinh[x/2]]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.20

method	result	size
lookup	\(\ln \left (\tanh \left (\frac {x}{2}\right )\right )\)	\(6\)
default	\(\ln \left (\tanh \left (\frac {x}{2}\right )\right )\)	\(6\)
parallelrisch	\(\ln \left (\tanh \left (\frac {x}{2}\right )\right )\)	\(6\)
risch	\(\ln \left (-1+{\mathrm e}^{x}\right )-\ln \left (1+{\mathrm e}^{x}\right )\)	\(14\)

[In]

int(csch(x),x,method=_RETURNVERBOSE)

[Out]

ln(tanh(1/2*x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (5) = 10\).

Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 3.40 \[ \int \text {csch}(x) \, dx=-\log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \]

[In]

integrate(csch(x),x, algorithm="fricas")

[Out]

-log(cosh(x) + sinh(x) + 1) + log(cosh(x) + sinh(x) - 1)

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \text {csch}(x) \, dx=\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} \]

[In]

integrate(csch(x),x)

[Out]

log(tanh(x/2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \text {csch}(x) \, dx=\log \left (\tanh \left (\frac {1}{2} \, x\right )\right ) \]

[In]

integrate(csch(x),x, algorithm="maxima")

[Out]

log(tanh(1/2*x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (5) = 10\).

Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 2.80 \[ \int \text {csch}(x) \, dx=-\log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \]

[In]

integrate(csch(x),x, algorithm="giac")

[Out]

-log(e^x + 1) + log(abs(e^x - 1))

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.00 \[ \int \text {csch}(x) \, dx=\ln \left (\mathrm {tanh}\left (\frac {x}{2}\right )\right ) \]

[In]

int(1/sinh(x),x)

[Out]

log(tanh(x/2))