3.223 \(\int \cosh (x) \sinh (4 x) \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{6} \cosh (3 x)+\frac{1}{10} \cosh (5 x) \]

[Out]

Cosh[3*x]/6 + Cosh[5*x]/10

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Rubi [A]  time = 0.0108065, antiderivative size = 17, normalized size of antiderivative = 1., 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} \[ \frac{1}{6} \cosh (3 x)+\frac{1}{10} \cosh (5 x) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]*Sinh[4*x],x]

[Out]

Cosh[3*x]/6 + Cosh[5*x]/10

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 (4 x) \, dx &=\frac{1}{6} \cosh (3 x)+\frac{1}{10} \cosh (5 x)\\ \end{align*}

Mathematica [A]  time = 0.0059692, size = 17, normalized size = 1. \[ \frac{1}{6} \cosh (3 x)+\frac{1}{10} \cosh (5 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]*Sinh[4*x],x]

[Out]

Cosh[3*x]/6 + Cosh[5*x]/10

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Maple [A]  time = 0.011, size = 14, normalized size = 0.8 \begin{align*}{\frac{8\, \left ( \cosh \left ( x \right ) \right ) ^{5}}{5}}-{\frac{4\, \left ( \cosh \left ( x \right ) \right ) ^{3}}{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)*sinh(4*x),x)

[Out]

8/5*cosh(x)^5-4/3*cosh(x)^3

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Maxima [A]  time = 1.03567, size = 36, normalized size = 2.12 \begin{align*} \frac{1}{60} \,{\left (5 \, e^{\left (-2 \, x\right )} + 3\right )} e^{\left (5 \, x\right )} + \frac{1}{12} \, e^{\left (-3 \, x\right )} + \frac{1}{20} \, e^{\left (-5 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(4*x),x, algorithm="maxima")

[Out]

1/60*(5*e^(-2*x) + 3)*e^(5*x) + 1/12*e^(-3*x) + 1/20*e^(-5*x)

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Fricas [B]  time = 2.0702, size = 130, normalized size = 7.65 \begin{align*} \frac{1}{10} \, \cosh \left (x\right )^{5} + \frac{1}{2} \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \frac{1}{6} \, \cosh \left (x\right )^{3} + \frac{1}{2} \,{\left (2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(4*x),x, algorithm="fricas")

[Out]

1/10*cosh(x)^5 + 1/2*cosh(x)*sinh(x)^4 + 1/6*cosh(x)^3 + 1/2*(2*cosh(x)^3 + cosh(x))*sinh(x)^2

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Sympy [A]  time = 0.586667, size = 20, normalized size = 1.18 \begin{align*} - \frac{\sinh{\left (x \right )} \sinh{\left (4 x \right )}}{15} + \frac{4 \cosh{\left (x \right )} \cosh{\left (4 x \right )}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(4*x),x)

[Out]

-sinh(x)*sinh(4*x)/15 + 4*cosh(x)*cosh(4*x)/15

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Giac [A]  time = 1.15576, size = 36, normalized size = 2.12 \begin{align*} \frac{1}{60} \,{\left (5 \, e^{\left (2 \, x\right )} + 3\right )} e^{\left (-5 \, x\right )} + \frac{1}{20} \, e^{\left (5 \, x\right )} + \frac{1}{12} \, e^{\left (3 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)*sinh(4*x),x, algorithm="giac")

[Out]

1/60*(5*e^(2*x) + 3)*e^(-5*x) + 1/20*e^(5*x) + 1/12*e^(3*x)