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

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

[Out]

2/3*cosh(x)^3

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Rubi [A]  time = 0.01, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [B]  time = 0.45, size = 19, normalized size = 2.38 \[ \frac {1}{6} \, \cosh \relax (x)^{3} + \frac {1}{2} \, \cosh \relax (x) \sinh \relax (x)^{2} + \frac {1}{2} \, \cosh \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(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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giac [B]  time = 0.13, size = 25, normalized size = 3.12 \[ \frac {1}{12} \, {\left (3 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} + \frac {1}{12} \, e^{\left (3 \, x\right )} + \frac {1}{4} \, e^{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(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

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maple [A]  time = 0.10, size = 7, normalized size = 0.88 \[ \frac {2 \left (\cosh ^{3}\relax (x )\right )}{3} \]

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 = 0.34, size = 27, normalized size = 3.38 \[ \frac {1}{12} \, {\left (3 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (3 \, x\right )} + \frac {1}{4} \, e^{\left (-x\right )} + \frac {1}{12} \, e^{\left (-3 \, x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(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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mupad [B]  time = 0.05, size = 6, normalized size = 0.75 \[ \frac {2\,{\mathrm {cosh}\relax (x)}^3}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*cosh(x)^3)/3

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sympy [B]  time = 0.45, size = 20, normalized size = 2.50 \[ - \frac {\sinh {\relax (x )} \sinh {\left (2 x \right )}}{3} + \frac {2 \cosh {\relax (x )} \cosh {\left (2 x \right )}}{3} \]

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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