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

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 = {4284} \[ \frac {1}{4} \cosh (2 x)+\frac {1}{8} \cosh (4 x) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Sinh[3*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 (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 \[ \frac {\cosh ^2(x)}{2}+\frac {1}{8} \cosh (4 x) \]

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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fricas [B]  time = 0.47, size = 33, normalized size = 1.94 \[ \frac {1}{8} \, \cosh \relax (x)^{4} + \frac {1}{8} \, \sinh \relax (x)^{4} + \frac {1}{4} \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + \frac {1}{4} \, \cosh \relax (x)^{2} \]

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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giac [A]  time = 0.11, size = 26, normalized size = 1.53 \[ \frac {1}{16} \, {\left (e^{\left (2 \, x\right )} + e^{\left (-2 \, x\right )}\right )}^{2} + \frac {1}{8} \, e^{\left (2 \, x\right )} + \frac {1}{8} \, e^{\left (-2 \, x\right )} \]

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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maple [A]  time = 0.11, size = 12, normalized size = 0.71 \[ \cosh ^{4}\relax (x )-\frac {\left (\cosh ^{2}\relax (x )\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.33, size = 27, normalized size = 1.59 \[ \frac {1}{16} \, {\left (2 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (4 \, x\right )} + \frac {1}{8} \, e^{\left (-2 \, x\right )} + \frac {1}{16} \, e^{\left (-4 \, x\right )} \]

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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mupad [B]  time = 1.45, size = 11, normalized size = 0.65 \[ {\mathrm {cosh}\relax (x)}^4-\frac {{\mathrm {cosh}\relax (x)}^2}{2} \]

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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sympy [A]  time = 0.41, size = 20, normalized size = 1.18 \[ - \frac {\sinh {\relax (x )} \sinh {\left (3 x \right )}}{8} + \frac {3 \cosh {\relax (x )} \cosh {\left (3 x \right )}}{8} \]

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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