Integrand size = 15, antiderivative size = 30 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=-\frac {x}{4}+\frac {1}{8} \sinh (2 x)-\frac {1}{12} \sinh (3 x)+\frac {1}{20} \sinh (5 x) \] Output:
-1/4*x+1/8*sinh(2*x)-1/12*sinh(3*x)+1/20*sinh(5*x)
Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=-\frac {x}{4}+\frac {1}{8} \sinh (2 x)-\frac {1}{12} \sinh (3 x)+\frac {1}{20} \sinh (5 x) \] Input:
Integrate[Cosh[(3*x)/2]*Sinh[x]*Sinh[(5*x)/2],x]
Output:
-1/4*x + Sinh[2*x]/8 - Sinh[3*x]/12 + Sinh[5*x]/20
Time = 0.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {3042, 25, 4855, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \cosh \left (\frac {3 x}{2}\right ) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (i x) \sin \left (\frac {5 i x}{2}\right ) \left (-\cos \left (\frac {3 i x}{2}\right )\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \cos \left (\frac {3 i x}{2}\right ) \sin (i x) \sin \left (\frac {5 i x}{2}\right )dx\) |
\(\Big \downarrow \) 4855 |
\(\displaystyle -\int \left (-\frac {1}{4} \cosh (2 x)+\frac {1}{4} \cosh (3 x)-\frac {1}{4} \cosh (5 x)+\frac {1}{4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {x}{4}+\frac {1}{8} \sinh (2 x)-\frac {1}{12} \sinh (3 x)+\frac {1}{20} \sinh (5 x)\) |
Input:
Int[Cosh[(3*x)/2]*Sinh[x]*Sinh[(5*x)/2],x]
Output:
-1/4*x + Sinh[2*x]/8 - Sinh[3*x]/12 + Sinh[5*x]/20
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]
Time = 3.03 (sec) , antiderivative size = 23, normalized size of antiderivative = 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 | \(-\frac {x}{4}+\frac {{\mathrm e}^{5 x}}{40}-\frac {{\mathrm e}^{3 x}}{24}+\frac {{\mathrm e}^{2 x}}{16}-\frac {{\mathrm e}^{-2 x}}{16}+\frac {{\mathrm e}^{-3 x}}{24}-\frac {{\mathrm e}^{-5 x}}{40}\) | \(41\) |
parallelrisch | \(-\frac {x}{2}+\ln \left (\frac {1}{\left (1-\tanh \left (\frac {3 x}{4}\right )\right )^{\frac {1}{6}}}\right )+\ln \left (\left (1+\tanh \left (\frac {3 x}{4}\right )\right )^{\frac {1}{6}}\right )+\frac {\sinh \left (2 x \right )}{8}-\frac {\sinh \left (3 x \right )}{12}+\frac {\sinh \left (5 x \right )}{20}\) | \(43\) |
orering | \(x \cosh \left (\frac {3 x}{2}\right ) \sinh \left (x \right ) \sinh \left (\frac {5 x}{2}\right )+\frac {\sinh \left (\frac {3 x}{2}\right ) \sinh \left (x \right ) \sinh \left (\frac {5 x}{2}\right )}{120}+\frac {11 \cosh \left (\frac {3 x}{2}\right ) \cosh \left (x \right ) \sinh \left (\frac {5 x}{2}\right )}{120}+\frac {31 \cosh \left (\frac {3 x}{2}\right ) \sinh \left (x \right ) \cosh \left (\frac {5 x}{2}\right )}{120}-\frac {361 x \left (\frac {19 \cosh \left (\frac {3 x}{2}\right ) \sinh \left (x \right ) \sinh \left (\frac {5 x}{2}\right )}{2}+3 \sinh \left (\frac {3 x}{2}\right ) \cosh \left (x \right ) \sinh \left (\frac {5 x}{2}\right )+\frac {15 \sinh \left (\frac {3 x}{2}\right ) \sinh \left (x \right ) \cosh \left (\frac {5 x}{2}\right )}{2}+5 \cosh \left (\frac {3 x}{2}\right ) \cosh \left (x \right ) \cosh \left (\frac {5 x}{2}\right )\right )}{900}-\frac {19 \sinh \left (\frac {3 x}{2}\right ) \cosh \left (x \right ) \cosh \left (\frac {5 x}{2}\right )}{120}+\frac {19 x \left (\frac {361 \cosh \left (\frac {3 x}{2}\right ) \sinh \left (x \right ) \sinh \left (\frac {5 x}{2}\right )}{2}+132 \sinh \left (\frac {3 x}{2}\right ) \cosh \left (x \right ) \sinh \left (\frac {5 x}{2}\right )+\frac {345 \sinh \left (\frac {3 x}{2}\right ) \sinh \left (x \right ) \cosh \left (\frac {5 x}{2}\right )}{2}+140 \cosh \left (\frac {3 x}{2}\right ) \cosh \left (x \right ) \cosh \left (\frac {5 x}{2}\right )\right )}{450}-\frac {x \left (\frac {8209 \cosh \left (\frac {3 x}{2}\right ) \sinh \left (x \right ) \sinh \left (\frac {5 x}{2}\right )}{2}+3708 \sinh \left (\frac {3 x}{2}\right ) \cosh \left (x \right ) \sinh \left (\frac {5 x}{2}\right )+\frac {8145 \sinh \left (\frac {3 x}{2}\right ) \sinh \left (x \right ) \cosh \left (\frac {5 x}{2}\right )}{2}+3740 \cosh \left (\frac {3 x}{2}\right ) \cosh \left (x \right ) \cosh \left (\frac {5 x}{2}\right )\right )}{900}\) | \(218\) |
Input:
int(cosh(3/2*x)*sinh(x)*sinh(5/2*x),x,method=_RETURNVERBOSE)
Output:
-1/4*x+1/8*sinh(2*x)-1/12*sinh(3*x)+1/20*sinh(5*x)
Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (22) = 44\).
Time = 0.06 (sec) , antiderivative size = 111, normalized size of antiderivative = 3.70 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=6 \, \cosh \left (\frac {1}{2} \, x\right )^{3} \sinh \left (\frac {1}{2} \, x\right )^{7} + \frac {1}{2} \, \cosh \left (\frac {1}{2} \, x\right ) \sinh \left (\frac {1}{2} \, x\right )^{9} + \frac {1}{10} \, {\left (126 \, \cosh \left (\frac {1}{2} \, x\right )^{5} - 5 \, \cosh \left (\frac {1}{2} \, x\right )\right )} \sinh \left (\frac {1}{2} \, x\right )^{5} + \frac {1}{6} \, {\left (36 \, \cosh \left (\frac {1}{2} \, x\right )^{7} - 10 \, \cosh \left (\frac {1}{2} \, x\right )^{3} + 3 \, \cosh \left (\frac {1}{2} \, x\right )\right )} \sinh \left (\frac {1}{2} \, x\right )^{3} + \frac {1}{2} \, {\left (\cosh \left (\frac {1}{2} \, x\right )^{9} - \cosh \left (\frac {1}{2} \, x\right )^{5} + \cosh \left (\frac {1}{2} \, x\right )^{3}\right )} \sinh \left (\frac {1}{2} \, x\right ) - \frac {1}{4} \, x \] Input:
integrate(cosh(3/2*x)*sinh(x)*sinh(5/2*x),x, algorithm="fricas")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (22) = 44\).
Time = 0.71 (sec) , antiderivative size = 139, normalized size of antiderivative = 4.63 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=- \frac {x \sinh {\left (x \right )} \sinh {\left (\frac {3 x}{2} \right )} \cosh {\left (\frac {5 x}{2} \right )}}{4} + \frac {x \sinh {\left (x \right )} \sinh {\left (\frac {5 x}{2} \right )} \cosh {\left (\frac {3 x}{2} \right )}}{4} + \frac {x \sinh {\left (\frac {3 x}{2} \right )} \sinh {\left (\frac {5 x}{2} \right )} \cosh {\left (x \right )}}{4} - \frac {x \cosh {\left (x \right )} \cosh {\left (\frac {3 x}{2} \right )} \cosh {\left (\frac {5 x}{2} \right )}}{4} + \frac {4 \sinh {\left (x \right )} \cosh {\left (\frac {3 x}{2} \right )} \cosh {\left (\frac {5 x}{2} \right )}}{15} - \frac {3 \sinh {\left (\frac {3 x}{2} \right )} \cosh {\left (x \right )} \cosh {\left (\frac {5 x}{2} \right )}}{20} + \frac {\sinh {\left (\frac {5 x}{2} \right )} \cosh {\left (x \right )} \cosh {\left (\frac {3 x}{2} \right )}}{12} \] Input:
integrate(cosh(3/2*x)*sinh(x)*sinh(5/2*x),x)
Output:
-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
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=-\frac {1}{240} \, {\left (10 \, e^{\left (-2 \, x\right )} - 15 \, e^{\left (-3 \, x\right )} - 6\right )} e^{\left (5 \, x\right )} - \frac {1}{4} \, x - \frac {1}{16} \, e^{\left (-2 \, x\right )} + \frac {1}{24} \, e^{\left (-3 \, x\right )} - \frac {1}{40} \, e^{\left (-5 \, x\right )} \] Input:
integrate(cosh(3/2*x)*sinh(x)*sinh(5/2*x),x, algorithm="maxima")
Output:
-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)
Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (22) = 44\).
Time = 0.12 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.60 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=\frac {1}{240} \, {\left (137 \, e^{\left (5 \, x\right )} - 15 \, e^{\left (3 \, x\right )} + 10 \, e^{\left (2 \, x\right )} - 6\right )} e^{\left (-5 \, x\right )} - \frac {1}{4} \, x + \frac {1}{40} \, e^{\left (5 \, x\right )} - \frac {1}{24} \, e^{\left (3 \, x\right )} + \frac {1}{16} \, e^{\left (2 \, x\right )} \] Input:
integrate(cosh(3/2*x)*sinh(x)*sinh(5/2*x),x, algorithm="giac")
Output:
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)
Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.33 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=\frac {{\mathrm {e}}^{2\,x}}{16}-\frac {{\mathrm {e}}^{-2\,x}}{16}-\frac {x}{4}+\frac {{\mathrm {e}}^{-3\,x}}{24}-\frac {{\mathrm {e}}^{3\,x}}{24}-\frac {{\mathrm {e}}^{-5\,x}}{40}+\frac {{\mathrm {e}}^{5\,x}}{40} \] Input:
int(cosh((3*x)/2)*sinh((5*x)/2)*sinh(x),x)
Output:
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
Time = 0.15 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.97 \[ \int \cosh \left (\frac {3 x}{2}\right ) \sinh (x) \sinh \left (\frac {5 x}{2}\right ) \, dx=-\frac {\cosh \left (\frac {5 x}{2}\right ) \cosh \left (\frac {3 x}{2}\right ) \cosh \left (x \right ) x}{4}+\frac {5 \cosh \left (\frac {5 x}{2}\right ) \cosh \left (\frac {3 x}{2}\right ) \sinh \left (x \right )}{12}-\frac {\cosh \left (\frac {5 x}{2}\right ) \sinh \left (\frac {3 x}{2}\right ) \sinh \left (x \right ) x}{4}-\frac {\cosh \left (\frac {3 x}{2}\right ) \cosh \left (x \right ) \sinh \left (\frac {5 x}{2}\right )}{15}+\frac {\cosh \left (\frac {3 x}{2}\right ) \sinh \left (\frac {5 x}{2}\right ) \sinh \left (x \right ) x}{4}+\frac {\cosh \left (x \right ) \sinh \left (\frac {5 x}{2}\right ) \sinh \left (\frac {3 x}{2}\right ) x}{4}-\frac {3 \sinh \left (\frac {5 x}{2}\right ) \sinh \left (\frac {3 x}{2}\right ) \sinh \left (x \right )}{20} \] Input:
int(cosh(3/2*x)*sinh(x)*sinh(5/2*x),x)
Output:
( - 15*cosh((5*x)/2)*cosh((3*x)/2)*cosh(x)*x + 25*cosh((5*x)/2)*cosh((3*x) /2)*sinh(x) - 15*cosh((5*x)/2)*sinh((3*x)/2)*sinh(x)*x - 4*cosh((3*x)/2)*c osh(x)*sinh((5*x)/2) + 15*cosh((3*x)/2)*sinh((5*x)/2)*sinh(x)*x + 15*cosh( x)*sinh((5*x)/2)*sinh((3*x)/2)*x - 9*sinh((5*x)/2)*sinh((3*x)/2)*sinh(x))/ 60