### 3.221 $$\int \cosh (x) \sinh (2 x) \, dx$$

Optimal. Leaf size=8 $\frac{2 \cosh ^3(x)}{3}$

[Out]

(2*Cosh[x]^3)/3

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Rubi [A]  time = 0.0105073, antiderivative size = 15, normalized size of antiderivative = 1.88, 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} $\frac{\cosh (x)}{2}+\frac{1}{6} \cosh (3 x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[x]*Sinh[2*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 (x) \sinh (2 x) \, dx &=\frac{\cosh (x)}{2}+\frac{1}{6} \cosh (3 x)\\ \end{align*}

Mathematica [A]  time = 0.0052221, size = 15, normalized size = 1.88 $\frac{\cosh (x)}{2}+\frac{1}{6} \cosh (3 x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

Cosh[x]/2 + Cosh[3*x]/6

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Maple [A]  time = 0.009, size = 7, normalized size = 0.9 \begin{align*}{\frac{2\, \left ( \cosh \left ( x \right ) \right ) ^{3}}{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*cosh(x)^3

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Maxima [B]  time = 1.00849, size = 36, normalized size = 4.5 \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)*sinh(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)

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Fricas [B]  time = 1.98672, size = 72, normalized size = 9. \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(x)*sinh(2*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 [B]  time = 0.647082, size = 20, normalized size = 2.5 \begin{align*} - \frac{\sinh{\left (x \right )} \sinh{\left (2 x \right )}}{3} + \frac{2 \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(x)*sinh(2*x),x)

[Out]

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

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Giac [B]  time = 1.22339, size = 34, normalized size = 4.25 \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)*sinh(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