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

Optimal result

Integrand size = 2, antiderivative size = 2 \[ \int \sinh (x) \, dx=\cosh (x) \]

[Out]

cosh(x)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2718} \[ \int \sinh (x) \, dx=\cosh (x) \]

[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*} \text {integral}& = \cosh (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \sinh (x) \, dx=\cosh (x) \]

[In]

Integrate[Sinh[x],x]

[Out]

Cosh[x]

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.50

method	result	size
lookup	\(\cosh \left (x \right )\)	\(3\)
default	\(\cosh \left (x \right )\)	\(3\)
risch	\(\frac {{\mathrm e}^{x}}{2}+\frac {{\mathrm e}^{-x}}{2}\)	\(12\)
parallelrisch	\(-\frac {2}{\tanh ^{2}\left (\frac {x}{2}\right )-1}\)	\(13\)
meijerg	\(-\sqrt {\pi }\, \left (\frac {1}{\sqrt {\pi }}-\frac {\cosh \left (x \right )}{\sqrt {\pi }}\right )\)	\(17\)

[In]

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

[Out]

cosh(x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \sinh (x) \, dx=\cosh \left (x\right ) \]

[In]

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

[Out]

cosh(x)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \sinh (x) \, dx=\cosh {\left (x \right )} \]

[In]

integrate(sinh(x),x)

[Out]

cosh(x)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \sinh (x) \, dx=\cosh \left (x\right ) \]

[In]

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

[Out]

cosh(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11 vs. \(2 (2) = 4\).

Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 5.50 \[ \int \sinh (x) \, dx=\frac {1}{2} \, e^{\left (-x\right )} + \frac {1}{2} \, e^{x} \]

[In]

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

[Out]

1/2*e^(-x) + 1/2*e^x

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 2, normalized size of antiderivative = 1.00 \[ \int \sinh (x) \, dx=\mathrm {cosh}\left (x\right ) \]

[In]

int(sinh(x),x)

[Out]

cosh(x)