3.197 \(\int \cosh (3 x) \sinh (x) \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{8} \cosh (4 x)-\frac{1}{4} \cosh (2 x) \]

[Out]

-Cosh[2*x]/4 + Cosh[4*x]/8

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Rubi [A]  time = 0.0105323, antiderivative size = 17, 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} \[ \frac{1}{8} \cosh (4 x)-\frac{1}{4} \cosh (2 x) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[3*x]*Sinh[x],x]

[Out]

-Cosh[2*x]/4 + Cosh[4*x]/8

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 (3 x) \sinh (x) \, dx &=-\frac{1}{4} \cosh (2 x)+\frac{1}{8} \cosh (4 x)\\ \end{align*}

Mathematica [A]  time = 0.0065284, size = 17, normalized size = 1. \[ \frac{1}{8} \cosh (4 x)-\frac{\cosh ^2(x)}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[3*x]*Sinh[x],x]

[Out]

-Cosh[x]^2/2 + Cosh[4*x]/8

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Maple [A]  time = 0.019, size = 14, normalized size = 0.8 \begin{align*} -{\frac{\cosh \left ( 2\,x \right ) }{4}}+{\frac{\cosh \left ( 4\,x \right ) }{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(3*x)*sinh(x),x)

[Out]

-1/4*cosh(2*x)+1/8*cosh(4*x)

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Maxima [A]  time = 1.02542, size = 36, normalized size = 2.12 \begin{align*} -\frac{1}{16} \,{\left (2 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (4 \, x\right )} - \frac{1}{8} \, e^{\left (-2 \, x\right )} + \frac{1}{16} \, e^{\left (-4 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(3*x)*sinh(x),x, algorithm="maxima")

[Out]

-1/16*(2*e^(-2*x) - 1)*e^(4*x) - 1/8*e^(-2*x) + 1/16*e^(-4*x)

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Fricas [B]  time = 1.98447, size = 109, normalized size = 6.41 \begin{align*} \frac{1}{8} \, \cosh \left (x\right )^{4} + \frac{1}{8} \, \sinh \left (x\right )^{4} + \frac{1}{4} \,{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \frac{1}{4} \, \cosh \left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(3*x)*sinh(x),x, algorithm="fricas")

[Out]

1/8*cosh(x)^4 + 1/8*sinh(x)^4 + 1/4*(3*cosh(x)^2 - 1)*sinh(x)^2 - 1/4*cosh(x)^2

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Sympy [A]  time = 0.586712, size = 20, normalized size = 1.18 \begin{align*} \frac{3 \sinh{\left (x \right )} \sinh{\left (3 x \right )}}{8} - \frac{\cosh{\left (x \right )} \cosh{\left (3 x \right )}}{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(3*x)*sinh(x),x)

[Out]

3*sinh(x)*sinh(3*x)/8 - cosh(x)*cosh(3*x)/8

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Giac [A]  time = 1.17471, size = 35, normalized size = 2.06 \begin{align*} \frac{1}{16} \,{\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} - \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(3*x)*sinh(x),x, algorithm="giac")

[Out]

1/16*(e^(2*x) + e^(-2*x))^2 - 1/8*e^(2*x) - 1/8*e^(-2*x)