3.7.68 \(\int (-\coth (x)+\text {csch}(x)) \, dx\) [668]

Optimal. Leaf size=11 \[ -\tanh ^{-1}(\cosh (x))-\log (\sinh (x)) \]

[Out]

-arctanh(cosh(x))-ln(sinh(x))

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Rubi [A]
time = 0.01, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3556, 3855} \begin {gather*} -\log (\sinh (x))-\tanh ^{-1}(\cosh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-Coth[x] + Csch[x],x]

[Out]

-ArcTanh[Cosh[x]] - Log[Sinh[x]]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, 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 (-\coth (x)+\text {csch}(x)) \, dx &=-\int \coth (x) \, dx+\int \text {csch}(x) \, dx\\ &=-\tanh ^{-1}(\cosh (x))-\log (\sinh (x))\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.18 \begin {gather*} -\log (\sinh (x))+\log \left (\tanh \left (\frac {x}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-Coth[x] + Csch[x],x]

[Out]

-Log[Sinh[x]] + Log[Tanh[x/2]]

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Maple [A]
time = 0.60, size = 12, normalized size = 1.09

method	result	size
default	\(-\ln \left (\sinh \left (x \right )\right )+\ln \left (\tanh \left (\frac {x}{2}\right )\right )\)	\(12\)
risch	\(x -\ln \left ({\mathrm e}^{2 x}-1\right )-\ln \left ({\mathrm e}^{x}+1\right )+\ln \left ({\mathrm e}^{x}-1\right )\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-coth(x)+csch(x),x,method=_RETURNVERBOSE)

[Out]

-ln(sinh(x))+ln(tanh(1/2*x))

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Maxima [A]
time = 0.27, size = 11, normalized size = 1.00 \begin {gather*} -\log \left (\sinh \left (x\right )\right ) + \log \left (\tanh \left (\frac {1}{2} \, x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-coth(x)+csch(x),x, algorithm="maxima")

[Out]

-log(sinh(x)) + log(tanh(1/2*x))

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Fricas [A]
time = 0.37, size = 11, normalized size = 1.00 \begin {gather*} x - 2 \, \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(-coth(x)+csch(x),x, algorithm="fricas")

[Out]

x - 2*log(cosh(x) + sinh(x) + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-coth(x)+csch(x),x)

[Out]

Integral(-coth(x) + csch(x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (11) = 22\).
time = 0.40, size = 25, normalized size = 2.27 \begin {gather*} x - \log \left (e^{x} + 1\right ) - \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) + \log \left ({\left | e^{x} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-coth(x)+csch(x),x, algorithm="giac")

[Out]

x - log(e^x + 1) - log(abs(e^(2*x) - 1)) + log(abs(e^x - 1))

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Mupad [B]
time = 1.54, size = 9, normalized size = 0.82 \begin {gather*} x-2\,\ln \left ({\mathrm {e}}^x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sinh(x) - coth(x),x)

[Out]

x - 2*log(exp(x) + 1)

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