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

Optimal. Leaf size=5 \[ -\tanh ^{-1}(\cosh (x)) \]

[Out]

-arctanh(cosh(x))

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Rubi [A]
time = 0.00, antiderivative size = 5, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3855} \begin {gather*} -\tanh ^{-1}(\cosh (x)) \end {gather*}

Antiderivative was successfully verified.

[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*} \int \text {csch}(x) \, dx &=-\tanh ^{-1}(\cosh (x))\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 7, normalized size = 1.40 \begin {gather*} \log \left (\tanh \left (\frac {x}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[x],x]

[Out]

Log[Tanh[x/2]]

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Maple [A]
time = 0.01, size = 6, normalized size = 1.20

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(tanh(1/2*x))

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Maxima [A]
time = 1.83, size = 5, normalized size = 1.00 \begin {gather*} \log \left (\tanh \left (\frac {1}{2} \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(tanh(1/2*x))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (5) = 10\).
time = 0.44, size = 17, normalized size = 3.40 \begin {gather*} -\log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \operatorname {csch}{\left (x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x),x)

[Out]

Integral(csch(x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 14 vs. \(2 (5) = 10\).
time = 1.13, size = 14, normalized size = 2.80 \begin {gather*} -\log \left (e^{x} + 1\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(e^x + 1) + log(abs(e^x - 1))

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Mupad [B]
time = 0.01, size = 5, normalized size = 1.00 \begin {gather*} \ln \left (\mathrm {tanh}\left (\frac {x}{2}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sinh(x),x)

[Out]

log(tanh(x/2))

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