### 3.225 $$\int \cosh (x) \cosh (2 x) \, dx$$

Optimal. Leaf size=15 $\frac{\sinh (x)}{2}+\frac{1}{6} \sinh (3 x)$

[Out]

Sinh[x]/2 + Sinh[3*x]/6

________________________________________________________________________________________

Rubi [A]  time = 0.0088413, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.143, Rules used = {4283} $\frac{\sinh (x)}{2}+\frac{1}{6} \sinh (3 x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[x]*Cosh[2*x],x]

[Out]

Sinh[x]/2 + Sinh[3*x]/6

Rule 4283

Int[cos[(a_.) + (b_.)*(x_)]*cos[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[a - c + (b - d)*x]/(2*(b - d)), x]
+ Simp[Sin[a + c + (b + d)*x]/(2*(b + d)), x] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - d^2, 0]

Rubi steps

\begin{align*} \int \cosh (x) \cosh (2 x) \, dx &=\frac{\sinh (x)}{2}+\frac{1}{6} \sinh (3 x)\\ \end{align*}

Mathematica [A]  time = 0.005055, size = 15, normalized size = 1. $\frac{\sinh (x)}{2}+\frac{1}{6} \sinh (3 x)$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cosh[x]*Cosh[2*x],x]

[Out]

Sinh[x]/2 + Sinh[3*x]/6

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 12, normalized size = 0.8 \begin{align*}{\frac{\sinh \left ( x \right ) }{2}}+{\frac{\sinh \left ( 3\,x \right ) }{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*cosh(2*x),x)

[Out]

1/2*sinh(x)+1/6*sinh(3*x)

________________________________________________________________________________________

Maxima [B]  time = 1.03474, size = 36, normalized size = 2.4 \begin{align*} \frac{1}{12} \,{\left (3 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (3 \, x\right )} - \frac{1}{4} \, e^{\left (-x\right )} - \frac{1}{12} \, e^{\left (-3 \, x\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*e^(-2*x) + 1)*e^(3*x) - 1/4*e^(-x) - 1/12*e^(-3*x)

________________________________________________________________________________________

Fricas [A]  time = 1.96405, size = 61, normalized size = 4.07 \begin{align*} \frac{1}{6} \, \sinh \left (x\right )^{3} + \frac{1}{2} \,{\left (\cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*sinh(x)^3 + 1/2*(cosh(x)^2 + 1)*sinh(x)

________________________________________________________________________________________

Sympy [A]  time = 0.618163, size = 20, normalized size = 1.33 \begin{align*} - \frac{\sinh{\left (x \right )} \cosh{\left (2 x \right )}}{3} + \frac{2 \sinh{\left (2 x \right )} \cosh{\left (x \right )}}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sinh(x)*cosh(2*x)/3 + 2*sinh(2*x)*cosh(x)/3

________________________________________________________________________________________

Giac [B]  time = 1.16954, size = 34, normalized size = 2.27 \begin{align*} -\frac{1}{12} \,{\left (3 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} + \frac{1}{12} \, e^{\left (3 \, x\right )} + \frac{1}{4} \, e^{x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(3*e^(2*x) + 1)*e^(-3*x) + 1/12*e^(3*x) + 1/4*e^x