3.3.22 \(\int \cosh (x) \sinh (3 x) \, dx\) [222]

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

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 17, 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}{4} \cosh (2 x)+\frac {1}{8} \cosh (4 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Cosh[2*x]/4 + Cosh[4*x]/8

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

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Mathematica [A]
time = 0.01, size = 17, normalized size = 1.00 \begin {gather*} \frac {\cosh ^2(x)}{2}+\frac {1}{8} \cosh (4 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Cosh[x]^2/2 + Cosh[4*x]/8

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Maple [A]
time = 1.06, size = 13, normalized size = 0.76

method	result	size
derivativedivides	\(\frac {\left (4 \left (\cosh ^{2}\left (x \right )\right )-1\right )^{2}}{16}\)	\(13\)
default	\(\frac {\left (4 \left (\cosh ^{2}\left (x \right )\right )-1\right )^{2}}{16}\)	\(13\)
risch	\(\frac {{\mathrm e}^{4 x}}{16}+\frac {{\mathrm e}^{2 x}}{8}+\frac {{\mathrm e}^{-2 x}}{8}+\frac {{\mathrm e}^{-4 x}}{16}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*sinh(3*x),x,method=_RETURNVERBOSE)

[Out]

1/16*(4*cosh(x)^2-1)^2

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (13) = 26\).
time = 0.26, size = 27, normalized size = 1.59 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(3*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] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (13) = 26\).
time = 0.36, size = 33, normalized size = 1.94 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(3*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 = 0.12, size = 20, normalized size = 1.18 \begin {gather*} - \frac {\sinh {\left (x \right )} \sinh {\left (3 x \right )}}{8} + \frac {3 \cosh {\left (x \right )} \cosh {\left (3 x \right )}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.39, size = 26, normalized size = 1.53 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.45, size = 11, normalized size = 0.65 \begin {gather*} {\mathrm {cosh}\left (x\right )}^4-\frac {{\mathrm {cosh}\left (x\right )}^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

cosh(x)^4 - cosh(x)^2/2

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