### 3.198 $$\int \cosh (4 x) \sinh (x) \, dx$$

Optimal. Leaf size=17 $\frac{1}{10} \cosh (5 x)-\frac{1}{6} \cosh (3 x)$

[Out]

-Cosh[3*x]/6 + Cosh[5*x]/10

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Rubi [A]  time = 0.0108626, 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} $\frac{1}{10} \cosh (5 x)-\frac{1}{6} \cosh (3 x)$

Antiderivative was successfully veriﬁed.

[In]

Int[Cosh[4*x]*Sinh[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 (4 x) \sinh (x) \, dx &=-\frac{1}{6} \cosh (3 x)+\frac{1}{10} \cosh (5 x)\\ \end{align*}

Mathematica [A]  time = 0.0061861, size = 17, normalized size = 1. $\frac{1}{10} \cosh (5 x)-\frac{1}{6} \cosh (3 x)$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-Cosh[3*x]/6 + Cosh[5*x]/10

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Maple [A]  time = 0.027, size = 14, normalized size = 0.8 \begin{align*} -{\frac{\cosh \left ( 3\,x \right ) }{6}}+{\frac{\cosh \left ( 5\,x \right ) }{10}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*cosh(3*x)+1/10*cosh(5*x)

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Maxima [A]  time = 1.04297, 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(4*x)*sinh(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 = 2.05455, 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(4*x)*sinh(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 = 0.596578, size = 20, normalized size = 1.18 \begin{align*} \frac{4 \sinh{\left (x \right )} \sinh{\left (4 x \right )}}{15} - \frac{\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(4*x)*sinh(x),x)

[Out]

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

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Giac [A]  time = 1.16258, 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(4*x)*sinh(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)