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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \[ \frac {1}{6} \cosh (3 x)-\frac {\cosh (x)}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Cosh[x] + Cosh[3*x]/6

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fricas [A]  time = 0.40, size = 19, normalized size = 1.27 \[ \frac {1}{6} \, \cosh \relax (x)^{3} + \frac {1}{2} \, \cosh \relax (x) \sinh \relax (x)^{2} - \frac {1}{2} \, \cosh \relax (x) \]

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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giac [B]  time = 0.11, size = 25, normalized size = 1.67 \[ -\frac {1}{12} \, {\left (3 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-3 \, x\right )} + \frac {1}{12} \, e^{\left (3 \, x\right )} - \frac {1}{4} \, e^{x} \]

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

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maple [A]  time = 0.13, size = 12, normalized size = 0.80 \[ -\frac {\cosh \relax (x )}{2}+\frac {\cosh \left (3 x \right )}{6} \]

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 = 0.30, size = 27, normalized size = 1.80 \[ -\frac {1}{12} \, {\left (3 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (3 \, x\right )} - \frac {1}{4} \, e^{\left (-x\right )} + \frac {1}{12} \, e^{\left (-3 \, x\right )} \]

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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mupad [B]  time = 1.42, size = 11, normalized size = 0.73 \[ \frac {2\,{\mathrm {cosh}\relax (x)}^3}{3}-\mathrm {cosh}\relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*cosh(x)^3)/3 - cosh(x)

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sympy [A]  time = 0.45, size = 20, normalized size = 1.33 \[ \frac {2 \sinh {\relax (x )} \sinh {\left (2 x \right )}}{3} - \frac {\cosh {\relax (x )} \cosh {\left (2 x \right )}}{3} \]

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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