3.6.71 \(\int \sinh (x) \, dx\) [571]

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

[Out]

cosh(x)

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Rubi [A]
time = 0.00, antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2718} \begin {gather*} \cosh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[x],x]

[Out]

Cosh[x]

Rule 2718

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*}

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Mathematica [A]
time = 0.00, size = 2, normalized size = 1.00 \begin {gather*} \cosh (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x],x]

[Out]

Cosh[x]

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Maple [A]
time = 0.01, size = 3, normalized size = 1.50

method	result	size
lookup	\(\cosh \left (x \right )\)	\(3\)
default	\(\cosh \left (x \right )\)	\(3\)
risch	\(\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}\)	\(12\)
meijerg	\(-\sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (x \right )}{\sqrt {\pi }}\right )\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x),x,method=_RETURNVERBOSE)

[Out]

cosh(x)

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Maxima [A]
time = 4.62, size = 2, normalized size = 1.00 \begin {gather*} \cosh \left (x\right ) \end {gather*}

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 = 0.50, size = 2, normalized size = 1.00 \begin {gather*} \cosh \left (x\right ) \end {gather*}

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 = 0.05, size = 2, normalized size = 1.00 \begin {gather*} \cosh {\left (x \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x),x)

[Out]

cosh(x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 11 vs. \(2 (2) = 4\).
time = 1.85, size = 11, normalized size = 5.50 \begin {gather*} \frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \end {gather*}

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

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Mupad [B]
time = 0.02, size = 2, normalized size = 1.00 \begin {gather*} \mathrm {cosh}\left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x),x)

[Out]

cosh(x)

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