3.680 \(\int (-\cosh (x)+\text{sech}(x)) \, dx\)

Optimal. Leaf size=8 \[ \tan ^{-1}(\sinh (x))-\sinh (x) \]

[Out]

ArcTan[Sinh[x]] - Sinh[x]

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Rubi [A]  time = 0.0064484, antiderivative size = 8, normalized size of antiderivative = 1., 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 = {2637, 3770} \[ \tan ^{-1}(\sinh (x))-\sinh (x) \]

Antiderivative was successfully verified.

[In]

Int[-Cosh[x] + Sech[x],x]

[Out]

ArcTan[Sinh[x]] - Sinh[x]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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{sech}(x)) \, dx &=-\int \cosh (x) \, dx+\int \text{sech}(x) \, dx\\ &=\tan ^{-1}(\sinh (x))-\sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.0042033, size = 14, normalized size = 1.75 \[ 2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )-\sinh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[-Cosh[x] + Sech[x],x]

[Out]

2*ArcTan[Tanh[x/2]] - Sinh[x]

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Maple [A]  time = 0.002, size = 9, normalized size = 1.1 \begin{align*} \arctan \left ( \sinh \left ( x \right ) \right ) -\sinh \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-cosh(x)+sech(x),x)

[Out]

arctan(sinh(x))-sinh(x)

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Maxima [A]  time = 1.18032, size = 11, normalized size = 1.38 \begin{align*} \arctan \left (\sinh \left (x\right )\right ) - \sinh \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cosh(x)+sech(x),x, algorithm="maxima")

[Out]

arctan(sinh(x)) - sinh(x)

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Fricas [B]  time = 1.78094, size = 166, normalized size = 20.75 \begin{align*} \frac{4 \,{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - \cosh \left (x\right )^{2} - 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(-cosh(x)+sech(x),x, algorithm="fricas")

[Out]

1/2*(4*(cosh(x) + sinh(x))*arctan(cosh(x) + sinh(x)) - cosh(x)^2 - 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 (- \cosh{\left (x \right )} + \operatorname{sech}{\left (x \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cosh(x)+sech(x),x)

[Out]

Integral(-cosh(x) + sech(x), x)

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Giac [A]  time = 1.16438, size = 22, normalized size = 2.75 \begin{align*} 2 \, \arctan \left (e^{x}\right ) + \frac{1}{2} \, e^{\left (-x\right )} - \frac{1}{2} \, e^{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-cosh(x)+sech(x),x, algorithm="giac")

[Out]

2*arctan(e^x) + 1/2*e^(-x) - 1/2*e^x