### 3.674 $$\int (\text{csch}(x)+\sinh (x)) \, dx$$

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

[Out]

-ArcTanh[Cosh[x]] + Cosh[x]

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Rubi [A]  time = 0.0075202, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 5, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, Rules used = {3770, 2638} $\cosh (x)-\tanh ^{-1}(\cosh (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Csch[x] + Sinh[x],x]

[Out]

-ArcTanh[Cosh[x]] + Cosh[x]

Rule 3770

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

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[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*}

Mathematica [A]  time = 0.0035631, size = 10, normalized size = 1.25 $\cosh (x)+\log \left (\tanh \left (\frac{x}{2}\right )\right )$

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 = 0.001, size = 9, normalized size = 1.1 \begin{align*} \ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) +\cosh \left ( x \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)+sinh(x),x)

[Out]

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

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Maxima [A]  time = 1.18091, size = 11, normalized size = 1.38 \begin{align*} \cosh \left (x\right ) + \log \left (\tanh \left (\frac{1}{2} \, x\right )\right ) \end{align*}

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]  time = 1.6715, size = 236, normalized size = 29.5 \begin{align*} \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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \left (\sinh{\left (x \right )} + \operatorname{csch}{\left (x \right )}\right )\, dx \end{align*}

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]  time = 1.13579, size = 32, normalized size = 4. \begin{align*} \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{align*}

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))