3.576 \(\int \cosh ^2(x) \, dx\)

Optimal. Leaf size=14 \[ \frac{x}{2}+\frac{1}{2} \sinh (x) \cosh (x) \]

[Out]

x/2 + (Cosh[x]*Sinh[x])/2

________________________________________________________________________________________

Rubi [A]  time = 0.0071077, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 4, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {2635, 8} \[ \frac{x}{2}+\frac{1}{2} \sinh (x) \cosh (x) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^2,x]

[Out]

x/2 + (Cosh[x]*Sinh[x])/2

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cosh ^2(x) \, dx &=\frac{1}{2} \cosh (x) \sinh (x)+\frac{\int 1 \, dx}{2}\\ &=\frac{x}{2}+\frac{1}{2} \cosh (x) \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.0018328, size = 14, normalized size = 1. \[ \frac{x}{2}+\frac{1}{4} \sinh (2 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^2,x]

[Out]

x/2 + Sinh[2*x]/4

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 11, normalized size = 0.8 \begin{align*}{\frac{x}{2}}+{\frac{\cosh \left ( x \right ) \sinh \left ( x \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^2,x)

[Out]

1/2*x+1/2*cosh(x)*sinh(x)

________________________________________________________________________________________

Maxima [A]  time = 0.926173, size = 22, normalized size = 1.57 \begin{align*} \frac{1}{2} \, x + \frac{1}{8} \, e^{\left (2 \, x\right )} - \frac{1}{8} \, e^{\left (-2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^2,x, algorithm="maxima")

[Out]

1/2*x + 1/8*e^(2*x) - 1/8*e^(-2*x)

________________________________________________________________________________________

Fricas [A]  time = 2.03191, size = 39, normalized size = 2.79 \begin{align*} \frac{1}{2} \, \cosh \left (x\right ) \sinh \left (x\right ) + \frac{1}{2} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^2,x, algorithm="fricas")

[Out]

1/2*cosh(x)*sinh(x) + 1/2*x

________________________________________________________________________________________

Sympy [B]  time = 0.184481, size = 24, normalized size = 1.71 \begin{align*} - \frac{x \sinh ^{2}{\left (x \right )}}{2} + \frac{x \cosh ^{2}{\left (x \right )}}{2} + \frac{\sinh{\left (x \right )} \cosh{\left (x \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**2,x)

[Out]

-x*sinh(x)**2/2 + x*cosh(x)**2/2 + sinh(x)*cosh(x)/2

________________________________________________________________________________________

Giac [B]  time = 1.09885, size = 32, normalized size = 2.29 \begin{align*} -\frac{1}{8} \,{\left (2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )} + \frac{1}{2} \, x + \frac{1}{8} \, e^{\left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^2,x, algorithm="giac")

[Out]

-1/8*(2*e^(2*x) + 1)*e^(-2*x) + 1/2*x + 1/8*e^(2*x)