3.11 \(\int \sinh (x) \, dx\)

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

[Out]

Cosh[x]

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Rubi [A]  time = 0.003047, 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 = {2638} \[ \cosh (x) \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x],x]

[Out]

Cosh[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 \sinh (x) \, dx &=\cosh (x)\\ \end{align*}

Mathematica [A]  time = 0.0010163, size = 2, normalized size = 1. \[ \cosh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x],x]

[Out]

Cosh[x]

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Maple [A]  time = 0., size = 3, normalized size = 1.5 \begin{align*} \cosh \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x),x)

[Out]

cosh(x)

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Maxima [A]  time = 0.918757, size = 3, normalized size = 1.5 \begin{align*} \cosh \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosh(x)

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Fricas [A]  time = 1.90352, size = 12, normalized size = 6. \begin{align*} \cosh \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosh(x)

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Sympy [A]  time = 0.109566, size = 2, normalized size = 1. \begin{align*} \cosh{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x),x)

[Out]

cosh(x)

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Giac [B]  time = 1.05852, 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(sinh(x),x, algorithm="giac")

[Out]

1/2*e^(-x) + 1/2*e^x