### 3.196 $$\int \cosh (2 x) \sinh (x) \, dx$$

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

[Out]

-Cosh[x]/2 + Cosh[3*x]/6

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Rubi [A]  time = 0.009962, 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 = {4284} $\frac{1}{6} \cosh (3 x)-\frac{\cosh (x)}{2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Cosh[x]/2 + Cosh[3*x]/6

Rule 4284

Int[cos[(c_.) + (d_.)*(x_)]*sin[(a_.) + (b_.)*(x_)], x_Symbol] :> -Simp[Cos[a - c + (b - d)*x]/(2*(b - d)), x]
- Simp[Cos[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 (2 x) \sinh (x) \, dx &=-\frac{\cosh (x)}{2}+\frac{1}{6} \cosh (3 x)\\ \end{align*}

Mathematica [A]  time = 0.0050589, size = 15, normalized size = 1. $\frac{1}{6} \cosh (3 x)-\frac{\cosh (x)}{2}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Cosh[x]/2 + Cosh[3*x]/6

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Maple [A]  time = 0.01, size = 12, normalized size = 0.8 \begin{align*} -{\frac{\cosh \left ( x \right ) }{2}}+{\frac{\cosh \left ( 3\,x \right ) }{6}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*cosh(x)+1/6*cosh(3*x)

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Maxima [B]  time = 1.09112, 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(2*x)*sinh(x),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.07606, size = 72, normalized size = 4.8 \begin{align*} \frac{1}{6} \, \cosh \left (x\right )^{3} + \frac{1}{2} \, \cosh \left (x\right ) \sinh \left (x\right )^{2} - \frac{1}{2} \, \cosh \left (x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.608357, size = 20, normalized size = 1.33 \begin{align*} \frac{2 \sinh{\left (x \right )} \sinh{\left (2 x \right )}}{3} - \frac{\cosh{\left (x \right )} \cosh{\left (2 x \right )}}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.1749, 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(2*x)*sinh(x),x, algorithm="giac")

[Out]

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