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

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

[Out]

-ArcTanh[Cosh[x]]/2

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Rubi [A]  time = 0.0150622, antiderivative size = 7, normalized size of antiderivative = 1., 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 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]

Rule 3770

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

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*}

Mathematica [A]  time = 0.0022791, 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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Maple [A]  time = 0.013, size = 6, normalized size = 0.9 \begin{align*} -{\it Artanh} \left ({{\rm e}^{x}} \right ) \end{align*}

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 = 1.07637, size = 26, normalized size = 3.71 \begin{align*} -\frac{1}{2} \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac{1}{2} \, \log \left (e^{\left (-x\right )} - 1\right ) \end{align*}

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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Fricas [B]  time = 1.98568, size = 89, normalized size = 12.71 \begin{align*} -\frac{1}{2} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) + \frac{1}{2} \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) - 1\right ) \end{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh{\left (x \right )} \operatorname{csch}{\left (2 x \right )}\, dx \end{align*}

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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Giac [B]  time = 1.1409, size = 22, normalized size = 3.14 \begin{align*} -\frac{1}{2} \, \log \left (e^{x} + 1\right ) + \frac{1}{2} \, \log \left ({\left | e^{x} - 1 \right |}\right ) \end{align*}

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))