\(\int \text {csch}(x) \, dx\) [26]

Optimal result

Integrand size = 2, antiderivative size = 7 \[ \int \text {csch}(x) \, dx=\log \left (\tanh \left (\frac {x}{2}\right )\right ) \]

[Out]

ln(tanh(1/2*x))

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.71, 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. \(2(7)=14\).

Time = 0.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 2.43 \[ \int \text {csch}(x) \, dx=-\log \left (\cosh \left (\frac {x}{2}\right )\right )+\log \left (\sinh \left (\frac {x}{2}\right )\right ) \]

[In]

Integrate[Csch[x],x]

[Out]

-Log[Cosh[x/2]] + Log[Sinh[x/2]]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.86

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

[In]

int(1/sinh(x),x,method=_RETURNVERBOSE)

[Out]

-2*arctanh(exp(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 = 0.23 (sec) , antiderivative size = 17, normalized size of antiderivative = 2.43 \[ \int \text {csch}(x) \, dx=-\log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \]

[In]

integrate(1/sinh(x),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.71 \[ \int \text {csch}(x) \, dx=\log {\left (\tanh {\left (\frac {x}{2} \right )} \right )} \]

[In]

integrate(1/sinh(x),x)

[Out]

log(tanh(x/2))

Maxima [B] (verification not implemented)

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

Time = 0.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 2.43 \[ \int \text {csch}(x) \, dx=-\log \left (e^{\left (-x\right )} + 1\right ) + \log \left (e^{\left (-x\right )} - 1\right ) \]

[In]

integrate(1/sinh(x),x, algorithm="maxima")

[Out]

-log(e^(-x) + 1) + log(e^(-x) - 1)

Giac [B] (verification not implemented)

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

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 2.00 \[ \int \text {csch}(x) \, dx=-\log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \]

[In]

integrate(1/sinh(x),x, algorithm="giac")

[Out]

-log(e^x + 1) + log(abs(e^x - 1))

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.71 \[ \int \text {csch}(x) \, dx=\ln \left (\mathrm {tanh}\left (\frac {x}{2}\right )\right ) \]

[In]

int(1/sinh(x),x)

[Out]

log(tanh(x/2))