3.12 \(\int \cosh (x) \, dx\)

Optimal. Leaf size=2 \[ \sinh (x) \]

[Out]

Sinh[x]

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Rubi [A]  time = 0.0020568, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2637} \[ \sinh (x) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x],x]

[Out]

Sinh[x]

Rule 2637

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

Rubi steps

\begin{align*} \int \cosh (x) \, dx &=\sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.0011039, size = 2, normalized size = 1. \[ \sinh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x],x]

[Out]

Sinh[x]

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Maple [A]  time = 0.002, size = 3, normalized size = 1.5 \begin{align*} \sinh \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x),x)

[Out]

sinh(x)

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Maxima [A]  time = 0.92125, size = 3, normalized size = 1.5 \begin{align*} \sinh \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x),x, algorithm="maxima")

[Out]

sinh(x)

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Fricas [A]  time = 1.89097, size = 12, normalized size = 6. \begin{align*} \sinh \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x),x, algorithm="fricas")

[Out]

sinh(x)

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Sympy [A]  time = 0.109905, size = 2, normalized size = 1. \begin{align*} \sinh{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x),x)

[Out]

sinh(x)

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Giac [B]  time = 1.05604, size = 15, normalized size = 7.5 \begin{align*} -\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),x, algorithm="giac")

[Out]

-1/2*e^(-x) + 1/2*e^x