3.7.74 \(\int (\text {csch}(x)+\sinh (x)) \, dx\) [674]

Optimal. Leaf size=8 \[ -\tanh ^{-1}(\cosh (x))+\cosh (x) \]

[Out]

-arctanh(cosh(x))+cosh(x)

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Rubi [A]
time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3855, 2718} \begin {gather*} \cosh (x)-\tanh ^{-1}(\cosh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x] + Sinh[x],x]

[Out]

-ArcTanh[Cosh[x]] + Cosh[x]

Rule 2718

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

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Mathematica [A]
time = 0.00, size = 10, normalized size = 1.25 \begin {gather*} \cosh (x)+\log \left (\tanh \left (\frac {x}{2}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[x] + Sinh[x],x]

[Out]

Cosh[x] + Log[Tanh[x/2]]

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Maple [A]
time = 0.38, size = 9, normalized size = 1.12

method	result	size
default	\(\ln \left (\tanh \left (\frac {x}{2}\right )\right )+\cosh \left (x \right )\)	\(9\)
risch	\(-\ln \left ({\mathrm e}^{x}+1\right )+\ln \left ({\mathrm e}^{x}-1\right )+\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(tanh(1/2*x))+cosh(x)

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Maxima [A]
time = 0.25, size = 8, normalized size = 1.00 \begin {gather*} \cosh \left (x\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(csch(x)+sinh(x),x, algorithm="maxima")

[Out]

cosh(x) + log(tanh(1/2*x))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (8) = 16\).
time = 0.50, size = 53, normalized size = 6.62 \begin {gather*} \frac {\cosh \left (x\right )^{2} - 2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + 2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}{2 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(cosh(x)^2 - 2*(cosh(x) + sinh(x))*log(cosh(x) + sinh(x) + 1) + 2*(cosh(x) + sinh(x))*log(cosh(x) + sinh(x
) - 1) + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)/(cosh(x) + sinh(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)+sinh(x),x)

[Out]

Integral(sinh(x) + csch(x), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (8) = 16\).
time = 0.40, size = 24, normalized size = 3.00 \begin {gather*} \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} - \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)+sinh(x),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.04, size = 27, normalized size = 3.38 \begin {gather*} \ln \left (2-2\,{\mathrm {e}}^x\right )-\ln \left (-2\,{\mathrm {e}}^x-2\right )+\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x) + 1/sinh(x),x)

[Out]

log(2 - 2*exp(x)) - log(- 2*exp(x) - 2) + exp(-x)/2 + exp(x)/2

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