3.245 \(\int \cosh (x) \text {csch}(2 x) \, dx\)

Optimal. Leaf size=7 \[ -\frac {1}{2} \tanh ^{-1}(\cosh (x)) \]

[Out]

-1/2*arctanh(cosh(x))

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Rubi [A]  time = 0.02, antiderivative size = 7, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4287, 3770} \[ -\frac {1}{2} \tanh ^{-1}(\cosh (x)) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Csch[2*x],x]

[Out]

-ArcTanh[Cosh[x]]/2

Rule 3770

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

Rule 4287

Int[(cos[(a_.) + (b_.)*(x_)]*(e_.))^(m_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Dist[2^p/e^p, Int[(e*Cos
[a + b*x])^(m + p)*Sin[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
&& IntegerQ[p]

Rubi steps

\begin {align*} \int \cosh (x) \text {csch}(2 x) \, dx &=\frac {1}{2} \int \text {csch}(x) \, dx\\ &=-\frac {1}{2} \tanh ^{-1}(\cosh (x))\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 11, normalized size = 1.57 \[ \frac {1}{2} \log \left (\tanh \left (\frac {x}{2}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Csch[2*x],x]

[Out]

Log[Tanh[x/2]]/2

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fricas [B]  time = 0.45, size = 19, normalized size = 2.71 \[ -\frac {1}{2} \, \log \left (\cosh \relax (x) + \sinh \relax (x) + 1\right ) + \frac {1}{2} \, \log \left (\cosh \relax (x) + \sinh \relax (x) - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*csch(2*x),x, algorithm="fricas")

[Out]

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

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giac [B]  time = 0.13, size = 16, normalized size = 2.29 \[ -\frac {1}{2} \, \log \left (e^{x} + 1\right ) + \frac {1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*csch(2*x),x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.15, size = 6, normalized size = 0.86 \[ -\arctanh \left ({\mathrm e}^{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*csch(2*x),x)

[Out]

-arctanh(exp(x))

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maxima [B]  time = 0.34, size = 19, normalized size = 2.71 \[ -\frac {1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*csch(2*x),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.06, size = 19, normalized size = 2.71 \[ \frac {\ln \left (1-{\mathrm {e}}^x\right )}{2}-\frac {\ln \left (-{\mathrm {e}}^x-1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)/sinh(2*x),x)

[Out]

log(1 - exp(x))/2 - log(- exp(x) - 1)/2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\relax (x )} \operatorname {csch}{\left (2 x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*csch(2*x),x)

[Out]

Integral(cosh(x)*csch(2*x), x)

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