3.2.96 \(\int \cosh (2 x) \sinh (x) \, dx\) [196]

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

[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 = {4369} \begin {gather*} \frac {1}{6} \cosh (3 x)-\frac {\cosh (x)}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*Cosh[x] + Cosh[3*x]/6

Rule 4369

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 \begin {gather*} -\frac {\cosh (x)}{2}+\frac {1}{6} \cosh (3 x) \end {gather*}

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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Maple [A]
time = 0.66, size = 12, normalized size = 0.80

method	result	size
default	\(-\frac {\cosh \left (x \right )}{2}+\frac {\cosh \left (3 x \right )}{6}\)	\(12\)
risch	\(\frac {{\mathrm e}^{3 x}}{12}-\frac {{\mathrm e}^{x}}{4}-\frac {{\mathrm e}^{-x}}{4}+\frac {{\mathrm e}^{-3 x}}{12}\)	\(24\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)*cosh(2*x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (11) = 22\).
time = 0.27, size = 27, normalized size = 1.80 \begin {gather*} -\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 {gather*}

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 = 0.36, size = 19, normalized size = 1.27 \begin {gather*} \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 {gather*}

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 = 0.12, size = 20, normalized size = 1.33 \begin {gather*} \frac {2 \sinh {\left (x \right )} \sinh {\left (2 x \right )}}{3} - \frac {\cosh {\left (x \right )} \cosh {\left (2 x \right )}}{3} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 25 vs. \(2 (11) = 22\).
time = 0.39, size = 25, normalized size = 1.67 \begin {gather*} -\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 {gather*}

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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Mupad [B]
time = 1.42, size = 11, normalized size = 0.73 \begin {gather*} \frac {2\,{\mathrm {cosh}\left (x\right )}^3}{3}-\mathrm {cosh}\left (x\right ) \end {gather*}

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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