3.6.91 \(\int \cosh (\frac {3 x}{2}) \sinh (x) \sinh (\frac {5 x}{2}) \, dx\) [591]

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]

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

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Rubi [A]
time = 0.02, antiderivative size = 30, normalized size of antiderivative = 1.00, 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 = {4440, 2717} \begin {gather*} -\frac {x}{4}+\frac {1}{8} \sinh (2 x)-\frac {1}{12} \sinh (3 x)+\frac {1}{20} \sinh (5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*x + Sinh[2*x]/8 - Sinh[3*x]/12 + Sinh[5*x]/20

Rule 2717

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

Rule 4440

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]

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*}

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Mathematica [A]
time = 0.02, size = 30, normalized size = 1.00 \begin {gather*} -\frac {x}{4}+\frac {1}{8} \sinh (2 x)-\frac {1}{12} \sinh (3 x)+\frac {1}{20} \sinh (5 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*x + Sinh[2*x]/8 - Sinh[3*x]/12 + Sinh[5*x]/20

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Maple [A]
time = 0.13, size = 23, normalized size = 0.77

method	result	size
default	\(-\frac {x}{4}+\frac {\sinh \left (2 x \right )}{8}-\frac {\sinh \left (3 x \right )}{12}+\frac {\sinh \left (5 x \right )}{20}\)	\(23\)
risch	\(-\frac {x}{4}+\frac {{\mathrm e}^{5 x}}{40}-\frac {{\mathrm e}^{3 x}}{24}+\frac {{\mathrm e}^{2 x}}{16}-\frac {{\mathrm e}^{-2 x}}{16}+\frac {{\mathrm e}^{-3 x}}{24}-\frac {{\mathrm e}^{-5 x}}{40}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 2.08, size = 42, normalized size = 1.40 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (22) = 44\).
time = 0.68, size = 111, normalized size = 3.70 \begin {gather*} 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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (22) = 44\).
time = 1.20, size = 139, normalized size = 4.63 \begin {gather*} - \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 {4 \sinh {\left (x \right )} \cosh {\left (\frac {3 x}{2} \right )} \cosh {\left (\frac {5 x}{2} \right )}}{15} - \frac {3 \sinh {\left (\frac {3 x}{2} \right )} \cosh {\left (x \right )} \cosh {\left (\frac {5 x}{2} \right )}}{20} + \frac {\sinh {\left (\frac {5 x}{2} \right )} \cosh {\left (x \right )} \cosh {\left (\frac {3 x}{2} \right )}}{12} \end {gather*}

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (22) = 44\).
time = 1.50, size = 48, normalized size = 1.60 \begin {gather*} \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 {gather*}

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
)

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Mupad [B]
time = 0.42, size = 40, normalized size = 1.33 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}}{16}-\frac {{\mathrm {e}}^{-2\,x}}{16}-\frac {x}{4}+\frac {{\mathrm {e}}^{-3\,x}}{24}-\frac {{\mathrm {e}}^{3\,x}}{24}-\frac {{\mathrm {e}}^{-5\,x}}{40}+\frac {{\mathrm {e}}^{5\,x}}{40} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(2*x)/16 - exp(-2*x)/16 - x/4 + exp(-3*x)/24 - exp(3*x)/24 - exp(-5*x)/40 + exp(5*x)/40

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