3.591 \(\int \cosh (\frac{3 x}{2}) \sinh (x) \sinh (\frac{5 x}{2}) \, dx\)

Optimal. Leaf size=30 \[ -\frac{x}{4}+\frac{1}{8} \sinh (2 x)-\frac{1}{12} \sinh (3 x)+\frac{1}{20} \sinh (5 x) \]

[Out]

-x/4 + Sinh[2*x]/8 - Sinh[3*x]/12 + Sinh[5*x]/20

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Rubi [A]  time = 0.0343048, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {4355, 2637} \[ -\frac{x}{4}+\frac{1}{8} \sinh (2 x)-\frac{1}{12} \sinh (3 x)+\frac{1}{20} \sinh (5 x) \]

Antiderivative was successfully verified.

[In]

Int[Cosh[(3*x)/2]*Sinh[x]*Sinh[(5*x)/2],x]

[Out]

-x/4 + Sinh[2*x]/8 - Sinh[3*x]/12 + Sinh[5*x]/20

Rule 4355

Int[(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.)*(H_)[(e_.) + (f_.)*(x_)]^(r_.), x_Symbol] :>
Int[ExpandTrigReduce[ActivateTrig[F[a + b*x]^p*G[c + d*x]^q*H[e + f*x]^r], x], x] /; FreeQ[{a, b, c, d, e, f}
, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, sin] || EqQ[G, cos]) && (EqQ[H, sin] || EqQ[H, cos]) && IGtQ[p
, 0] && IGtQ[q, 0] && IGtQ[r, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \cosh \left (\frac{3 x}{2}\right ) \sinh (x) \sinh \left (\frac{5 x}{2}\right ) \, dx &=-\int \left (\frac{1}{4}-\frac{1}{4} \cosh (2 x)+\frac{1}{4} \cosh (3 x)-\frac{1}{4} \cosh (5 x)\right ) \, dx\\ &=-\frac{x}{4}+\frac{1}{4} \int \cosh (2 x) \, dx-\frac{1}{4} \int \cosh (3 x) \, dx+\frac{1}{4} \int \cosh (5 x) \, dx\\ &=-\frac{x}{4}+\frac{1}{8} \sinh (2 x)-\frac{1}{12} \sinh (3 x)+\frac{1}{20} \sinh (5 x)\\ \end{align*}

Mathematica [A]  time = 0.0098559, size = 30, normalized size = 1. \[ -\frac{x}{4}+\frac{1}{8} \sinh (2 x)-\frac{1}{12} \sinh (3 x)+\frac{1}{20} \sinh (5 x) \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[(3*x)/2]*Sinh[x]*Sinh[(5*x)/2],x]

[Out]

-x/4 + Sinh[2*x]/8 - Sinh[3*x]/12 + Sinh[5*x]/20

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Maple [A]  time = 0.099, size = 23, normalized size = 0.8 \begin{align*} -{\frac{x}{4}}+{\frac{\sinh \left ( 2\,x \right ) }{8}}-{\frac{\sinh \left ( 3\,x \right ) }{12}}+{\frac{\sinh \left ( 5\,x \right ) }{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(3/2*x)*sinh(x)*sinh(5/2*x),x)

[Out]

-1/4*x+1/8*sinh(2*x)-1/12*sinh(3*x)+1/20*sinh(5*x)

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Maxima [A]  time = 0.951566, size = 57, normalized size = 1.9 \begin{align*} -\frac{1}{240} \,{\left (10 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-3 \, x\right )} - 6\right )} e^{\left (5 \, x\right )} - \frac{1}{4} \, x - \frac{1}{16} \, e^{\left (-2 \, x\right )} + \frac{1}{24} \, e^{\left (-3 \, x\right )} - \frac{1}{40} \, e^{\left (-5 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(3/2*x)*sinh(x)*sinh(5/2*x),x, algorithm="maxima")

[Out]

-1/240*(10*e^(-2*x) - 15*e^(-3*x) - 6)*e^(5*x) - 1/4*x - 1/16*e^(-2*x) + 1/24*e^(-3*x) - 1/40*e^(-5*x)

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Fricas [B]  time = 2.20653, size = 362, normalized size = 12.07 \begin{align*} 6 \, \cosh \left (\frac{1}{2} \, x\right )^{3} \sinh \left (\frac{1}{2} \, x\right )^{7} + \frac{1}{2} \, \cosh \left (\frac{1}{2} \, x\right ) \sinh \left (\frac{1}{2} \, x\right )^{9} + \frac{1}{10} \,{\left (126 \, \cosh \left (\frac{1}{2} \, x\right )^{5} - 5 \, \cosh \left (\frac{1}{2} \, x\right )\right )} \sinh \left (\frac{1}{2} \, x\right )^{5} + \frac{1}{6} \,{\left (36 \, \cosh \left (\frac{1}{2} \, x\right )^{7} - 10 \, \cosh \left (\frac{1}{2} \, x\right )^{3} + 3 \, \cosh \left (\frac{1}{2} \, x\right )\right )} \sinh \left (\frac{1}{2} \, x\right )^{3} + \frac{1}{2} \,{\left (\cosh \left (\frac{1}{2} \, x\right )^{9} - \cosh \left (\frac{1}{2} \, x\right )^{5} + \cosh \left (\frac{1}{2} \, x\right )^{3}\right )} \sinh \left (\frac{1}{2} \, x\right ) - \frac{1}{4} \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(3/2*x)*sinh(x)*sinh(5/2*x),x, algorithm="fricas")

[Out]

6*cosh(1/2*x)^3*sinh(1/2*x)^7 + 1/2*cosh(1/2*x)*sinh(1/2*x)^9 + 1/10*(126*cosh(1/2*x)^5 - 5*cosh(1/2*x))*sinh(
1/2*x)^5 + 1/6*(36*cosh(1/2*x)^7 - 10*cosh(1/2*x)^3 + 3*cosh(1/2*x))*sinh(1/2*x)^3 + 1/2*(cosh(1/2*x)^9 - cosh
(1/2*x)^5 + cosh(1/2*x)^3)*sinh(1/2*x) - 1/4*x

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Sympy [B]  time = 12.0042, size = 138, normalized size = 4.6 \begin{align*} - \frac{x \sinh{\left (x \right )} \sinh{\left (\frac{3 x}{2} \right )} \cosh{\left (\frac{5 x}{2} \right )}}{4} + \frac{x \sinh{\left (x \right )} \sinh{\left (\frac{5 x}{2} \right )} \cosh{\left (\frac{3 x}{2} \right )}}{4} + \frac{x \sinh{\left (\frac{3 x}{2} \right )} \sinh{\left (\frac{5 x}{2} \right )} \cosh{\left (x \right )}}{4} - \frac{x \cosh{\left (x \right )} \cosh{\left (\frac{3 x}{2} \right )} \cosh{\left (\frac{5 x}{2} \right )}}{4} - \frac{\sinh{\left (x \right )} \sinh{\left (\frac{3 x}{2} \right )} \sinh{\left (\frac{5 x}{2} \right )}}{12} + \frac{7 \sinh{\left (x \right )} \cosh{\left (\frac{3 x}{2} \right )} \cosh{\left (\frac{5 x}{2} \right )}}{20} - \frac{\sinh{\left (\frac{3 x}{2} \right )} \cosh{\left (x \right )} \cosh{\left (\frac{5 x}{2} \right )}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(3/2*x)*sinh(x)*sinh(5/2*x),x)

[Out]

-x*sinh(x)*sinh(3*x/2)*cosh(5*x/2)/4 + x*sinh(x)*sinh(5*x/2)*cosh(3*x/2)/4 + x*sinh(3*x/2)*sinh(5*x/2)*cosh(x)
/4 - x*cosh(x)*cosh(3*x/2)*cosh(5*x/2)/4 - sinh(x)*sinh(3*x/2)*sinh(5*x/2)/12 + 7*sinh(x)*cosh(3*x/2)*cosh(5*x
/2)/20 - sinh(3*x/2)*cosh(x)*cosh(5*x/2)/15

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Giac [B]  time = 1.08488, size = 65, normalized size = 2.17 \begin{align*} \frac{1}{240} \,{\left (137 \, e^{\left (5 \, x\right )} - 15 \, e^{\left (3 \, x\right )} + 10 \, e^{\left (2 \, x\right )} - 6\right )} e^{\left (-5 \, x\right )} - \frac{1}{4} \, x + \frac{1}{40} \, e^{\left (5 \, x\right )} - \frac{1}{24} \, e^{\left (3 \, x\right )} + \frac{1}{16} \, e^{\left (2 \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(3/2*x)*sinh(x)*sinh(5/2*x),x, algorithm="giac")

[Out]

1/240*(137*e^(5*x) - 15*e^(3*x) + 10*e^(2*x) - 6)*e^(-5*x) - 1/4*x + 1/40*e^(5*x) - 1/24*e^(3*x) + 1/16*e^(2*x
)