Integrand size = 20, antiderivative size = 202 \[ \int x^3 \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx=-\frac {3 x \cosh (a+b x)}{4 b^3}-\frac {x^3 \cosh (a+b x)}{8 b}-\frac {x \cosh (3 a+3 b x)}{72 b^3}-\frac {x^3 \cosh (3 a+3 b x)}{48 b}+\frac {3 x \cosh (5 a+5 b x)}{1000 b^3}+\frac {x^3 \cosh (5 a+5 b x)}{80 b}+\frac {3 \sinh (a+b x)}{4 b^4}+\frac {3 x^2 \sinh (a+b x)}{8 b^2}+\frac {\sinh (3 a+3 b x)}{216 b^4}+\frac {x^2 \sinh (3 a+3 b x)}{48 b^2}-\frac {3 \sinh (5 a+5 b x)}{5000 b^4}-\frac {3 x^2 \sinh (5 a+5 b x)}{400 b^2} \] Output:
-3/4*x*cosh(b*x+a)/b^3-1/8*x^3*cosh(b*x+a)/b-1/72*x*cosh(3*b*x+3*a)/b^3-1/ 48*x^3*cosh(3*b*x+3*a)/b+3/1000*x*cosh(5*b*x+5*a)/b^3+1/80*x^3*cosh(5*b*x+ 5*a)/b+3/4*sinh(b*x+a)/b^4+3/8*x^2*sinh(b*x+a)/b^2+1/216*sinh(3*b*x+3*a)/b ^4+1/48*x^2*sinh(3*b*x+3*a)/b^2-3/5000*sinh(5*b*x+5*a)/b^4-3/400*x^2*sinh( 5*b*x+5*a)/b^2
Time = 0.31 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.67 \[ \int x^3 \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx=\frac {-33750 \left (b x \left (6+b^2 x^2\right ) \cosh (a+b x)-3 \left (2+b^2 x^2\right ) \sinh (a+b x)\right )-625 \left (\left (6 b x+9 b^3 x^3\right ) \cosh (3 (a+b x))-\left (2+9 b^2 x^2\right ) \sinh (3 (a+b x))\right )+27 \left (5 b x \left (6+25 b^2 x^2\right ) \cosh (5 (a+b x))-3 \left (2+25 b^2 x^2\right ) \sinh (5 (a+b x))\right )}{270000 b^4} \] Input:
Integrate[x^3*Cosh[a + b*x]^2*Sinh[a + b*x]^3,x]
Output:
(-33750*(b*x*(6 + b^2*x^2)*Cosh[a + b*x] - 3*(2 + b^2*x^2)*Sinh[a + b*x]) - 625*((6*b*x + 9*b^3*x^3)*Cosh[3*(a + b*x)] - (2 + 9*b^2*x^2)*Sinh[3*(a + b*x)]) + 27*(5*b*x*(6 + 25*b^2*x^2)*Cosh[5*(a + b*x)] - 3*(2 + 25*b^2*x^2 )*Sinh[5*(a + b*x)]))/(270000*b^4)
Time = 0.52 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5971, 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.
| \(\displaystyle \int x^3 \sinh ^3(a+b x) \cosh ^2(a+b x) \, dx\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \int \left (-\frac {1}{8} x^3 \sinh (a+b x)-\frac {1}{16} x^3 \sinh (3 a+3 b x)+\frac {1}{16} x^3 \sinh (5 a+5 b x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 \sinh (a+b x)}{4 b^4}+\frac {\sinh (3 a+3 b x)}{216 b^4}-\frac {3 \sinh (5 a+5 b x)}{5000 b^4}-\frac {3 x \cosh (a+b x)}{4 b^3}-\frac {x \cosh (3 a+3 b x)}{72 b^3}+\frac {3 x \cosh (5 a+5 b x)}{1000 b^3}+\frac {3 x^2 \sinh (a+b x)}{8 b^2}+\frac {x^2 \sinh (3 a+3 b x)}{48 b^2}-\frac {3 x^2 \sinh (5 a+5 b x)}{400 b^2}-\frac {x^3 \cosh (a+b x)}{8 b}-\frac {x^3 \cosh (3 a+3 b x)}{48 b}+\frac {x^3 \cosh (5 a+5 b x)}{80 b}\) |
Input:
Int[x^3*Cosh[a + b*x]^2*Sinh[a + b*x]^3,x]
Output:
(-3*x*Cosh[a + b*x])/(4*b^3) - (x^3*Cosh[a + b*x])/(8*b) - (x*Cosh[3*a + 3 *b*x])/(72*b^3) - (x^3*Cosh[3*a + 3*b*x])/(48*b) + (3*x*Cosh[5*a + 5*b*x]) /(1000*b^3) + (x^3*Cosh[5*a + 5*b*x])/(80*b) + (3*Sinh[a + b*x])/(4*b^4) + (3*x^2*Sinh[a + b*x])/(8*b^2) + Sinh[3*a + 3*b*x]/(216*b^4) + (x^2*Sinh[3 *a + 3*b*x])/(48*b^2) - (3*Sinh[5*a + 5*b*x])/(5000*b^4) - (3*x^2*Sinh[5*a + 5*b*x])/(400*b^2)
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Time = 40.44 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.05
| method | result | size |
| risch | \(\frac {\left (125 b^{3} x^{3}-75 b^{2} x^{2}+30 b x -6\right ) {\mathrm e}^{5 b x +5 a}}{20000 b^{4}}-\frac {\left (9 b^{3} x^{3}-9 b^{2} x^{2}+6 b x -2\right ) {\mathrm e}^{3 b x +3 a}}{864 b^{4}}-\frac {\left (b^{3} x^{3}-3 b^{2} x^{2}+6 b x -6\right ) {\mathrm e}^{b x +a}}{16 b^{4}}-\frac {\left (b^{3} x^{3}+3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x -a}}{16 b^{4}}-\frac {\left (9 b^{3} x^{3}+9 b^{2} x^{2}+6 b x +2\right ) {\mathrm e}^{-3 b x -3 a}}{864 b^{4}}+\frac {\left (125 b^{3} x^{3}+75 b^{2} x^{2}+30 b x +6\right ) {\mathrm e}^{-5 b x -5 a}}{20000 b^{4}}\) | \(213\) |
| derivativedivides | \(\frac {-a^{3} \left (\frac {\cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )^{2}}{5}-\frac {2 \cosh \left (b x +a \right )^{3}}{15}\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}{5}-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{15}-\frac {\sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{25}+\frac {26 \sinh \left (b x +a \right )}{225}+\frac {13 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{225}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}{5}-\frac {2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}{15}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{25}+\frac {52 \left (b x +a \right ) \sinh \left (b x +a \right )}{225}+\frac {26 \left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{225}+\frac {2 \cosh \left (b x +a \right )^{5}}{125}-\frac {52 \cosh \left (b x +a \right )}{225}-\frac {26 \cosh \left (b x +a \right )^{3}}{675}\right )+\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}{5}-\frac {2 \left (b x +a \right )^{3} \cosh \left (b x +a \right )^{3}}{15}-\frac {3 \left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{25}+\frac {26 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{75}+\frac {13 \left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{75}+\frac {6 \left (b x +a \right ) \cosh \left (b x +a \right )^{5}}{125}+\frac {12568 \sinh \left (b x +a \right )}{16875}-\frac {6 \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{625}+\frac {434 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{16875}-\frac {52 \left (b x +a \right ) \cosh \left (b x +a \right )}{75}-\frac {26 \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{225}}{b^{4}}\) | \(439\) |
| default | \(\frac {-a^{3} \left (\frac {\cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )^{2}}{5}-\frac {2 \cosh \left (b x +a \right )^{3}}{15}\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}{5}-\frac {2 \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{15}-\frac {\sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{25}+\frac {26 \sinh \left (b x +a \right )}{225}+\frac {13 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{225}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}{5}-\frac {2 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}{15}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{25}+\frac {52 \left (b x +a \right ) \sinh \left (b x +a \right )}{225}+\frac {26 \left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{225}+\frac {2 \cosh \left (b x +a \right )^{5}}{125}-\frac {52 \cosh \left (b x +a \right )}{225}-\frac {26 \cosh \left (b x +a \right )^{3}}{675}\right )+\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{3}}{5}-\frac {2 \left (b x +a \right )^{3} \cosh \left (b x +a \right )^{3}}{15}-\frac {3 \left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{25}+\frac {26 \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{75}+\frac {13 \left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{75}+\frac {6 \left (b x +a \right ) \cosh \left (b x +a \right )^{5}}{125}+\frac {12568 \sinh \left (b x +a \right )}{16875}-\frac {6 \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{625}+\frac {434 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )}{16875}-\frac {52 \left (b x +a \right ) \cosh \left (b x +a \right )}{75}-\frac {26 \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{225}}{b^{4}}\) | \(439\) |
| orering | \(\text {Expression too large to display}\) | \(994\) |
Input:
int(x^3*cosh(b*x+a)^2*sinh(b*x+a)^3,x,method=_RETURNVERBOSE)
Output:
1/20000*(125*b^3*x^3-75*b^2*x^2+30*b*x-6)/b^4*exp(5*b*x+5*a)-1/864*(9*b^3* x^3-9*b^2*x^2+6*b*x-2)/b^4*exp(3*b*x+3*a)-1/16*(b^3*x^3-3*b^2*x^2+6*b*x-6) /b^4*exp(b*x+a)-1/16*(b^3*x^3+3*b^2*x^2+6*b*x+6)/b^4*exp(-b*x-a)-1/864*(9* b^3*x^3+9*b^2*x^2+6*b*x+2)/b^4*exp(-3*b*x-3*a)+1/20000*(125*b^3*x^3+75*b^2 *x^2+30*b*x+6)/b^4*exp(-5*b*x-5*a)
Time = 0.09 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.36 \[ \int x^3 \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx=\frac {135 \, {\left (25 \, b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right )^{5} + 675 \, {\left (25 \, b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 81 \, {\left (25 \, b^{2} x^{2} + 2\right )} \sinh \left (b x + a\right )^{5} - 1875 \, {\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )^{3} + 5 \, {\left (1125 \, b^{2} x^{2} - 162 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} + 250\right )} \sinh \left (b x + a\right )^{3} + 225 \, {\left (6 \, {\left (25 \, b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right )^{3} - 25 \, {\left (3 \, b^{3} x^{3} + 2 \, b x\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 33750 \, {\left (b^{3} x^{3} + 6 \, b x\right )} \cosh \left (b x + a\right ) - 15 \, {\left (27 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{4} - 6750 \, b^{2} x^{2} - 125 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} - 13500\right )} \sinh \left (b x + a\right )}{270000 \, b^{4}} \] Input:
integrate(x^3*cosh(b*x+a)^2*sinh(b*x+a)^3,x, algorithm="fricas")
Output:
1/270000*(135*(25*b^3*x^3 + 6*b*x)*cosh(b*x + a)^5 + 675*(25*b^3*x^3 + 6*b *x)*cosh(b*x + a)*sinh(b*x + a)^4 - 81*(25*b^2*x^2 + 2)*sinh(b*x + a)^5 - 1875*(3*b^3*x^3 + 2*b*x)*cosh(b*x + a)^3 + 5*(1125*b^2*x^2 - 162*(25*b^2*x ^2 + 2)*cosh(b*x + a)^2 + 250)*sinh(b*x + a)^3 + 225*(6*(25*b^3*x^3 + 6*b* x)*cosh(b*x + a)^3 - 25*(3*b^3*x^3 + 2*b*x)*cosh(b*x + a))*sinh(b*x + a)^2 - 33750*(b^3*x^3 + 6*b*x)*cosh(b*x + a) - 15*(27*(25*b^2*x^2 + 2)*cosh(b* x + a)^4 - 6750*b^2*x^2 - 125*(9*b^2*x^2 + 2)*cosh(b*x + a)^2 - 13500)*sin h(b*x + a))/b^4
Time = 0.76 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.25 \[ \int x^3 \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx=\begin {cases} \frac {x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3 b} - \frac {2 x^{3} \cosh ^{5}{\left (a + b x \right )}}{15 b} + \frac {26 x^{2} \sinh ^{5}{\left (a + b x \right )}}{75 b^{2}} - \frac {13 x^{2} \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{15 b^{2}} + \frac {2 x^{2} \sinh {\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{5 b^{2}} - \frac {52 x \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{75 b^{3}} + \frac {338 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{225 b^{3}} - \frac {856 x \cosh ^{5}{\left (a + b x \right )}}{1125 b^{3}} + \frac {12568 \sinh ^{5}{\left (a + b x \right )}}{16875 b^{4}} - \frac {5114 \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3375 b^{4}} + \frac {856 \sinh {\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{1125 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh ^{3}{\left (a \right )} \cosh ^{2}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(x**3*cosh(b*x+a)**2*sinh(b*x+a)**3,x)
Output:
Piecewise((x**3*sinh(a + b*x)**2*cosh(a + b*x)**3/(3*b) - 2*x**3*cosh(a + b*x)**5/(15*b) + 26*x**2*sinh(a + b*x)**5/(75*b**2) - 13*x**2*sinh(a + b*x )**3*cosh(a + b*x)**2/(15*b**2) + 2*x**2*sinh(a + b*x)*cosh(a + b*x)**4/(5 *b**2) - 52*x*sinh(a + b*x)**4*cosh(a + b*x)/(75*b**3) + 338*x*sinh(a + b* x)**2*cosh(a + b*x)**3/(225*b**3) - 856*x*cosh(a + b*x)**5/(1125*b**3) + 1 2568*sinh(a + b*x)**5/(16875*b**4) - 5114*sinh(a + b*x)**3*cosh(a + b*x)** 2/(3375*b**4) + 856*sinh(a + b*x)*cosh(a + b*x)**4/(1125*b**4), Ne(b, 0)), (x**4*sinh(a)**3*cosh(a)**2/4, True))
Time = 0.05 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.21 \[ \int x^3 \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx=\frac {{\left (125 \, b^{3} x^{3} e^{\left (5 \, a\right )} - 75 \, b^{2} x^{2} e^{\left (5 \, a\right )} + 30 \, b x e^{\left (5 \, a\right )} - 6 \, e^{\left (5 \, a\right )}\right )} e^{\left (5 \, b x\right )}}{20000 \, b^{4}} - \frac {{\left (9 \, b^{3} x^{3} e^{\left (3 \, a\right )} - 9 \, b^{2} x^{2} e^{\left (3 \, a\right )} + 6 \, b x e^{\left (3 \, a\right )} - 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{864 \, b^{4}} - \frac {{\left (b^{3} x^{3} e^{a} - 3 \, b^{2} x^{2} e^{a} + 6 \, b x e^{a} - 6 \, e^{a}\right )} e^{\left (b x\right )}}{16 \, b^{4}} - \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{16 \, b^{4}} - \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{864 \, b^{4}} + \frac {{\left (125 \, b^{3} x^{3} + 75 \, b^{2} x^{2} + 30 \, b x + 6\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{20000 \, b^{4}} \] Input:
integrate(x^3*cosh(b*x+a)^2*sinh(b*x+a)^3,x, algorithm="maxima")
Output:
1/20000*(125*b^3*x^3*e^(5*a) - 75*b^2*x^2*e^(5*a) + 30*b*x*e^(5*a) - 6*e^( 5*a))*e^(5*b*x)/b^4 - 1/864*(9*b^3*x^3*e^(3*a) - 9*b^2*x^2*e^(3*a) + 6*b*x *e^(3*a) - 2*e^(3*a))*e^(3*b*x)/b^4 - 1/16*(b^3*x^3*e^a - 3*b^2*x^2*e^a + 6*b*x*e^a - 6*e^a)*e^(b*x)/b^4 - 1/16*(b^3*x^3 + 3*b^2*x^2 + 6*b*x + 6)*e^ (-b*x - a)/b^4 - 1/864*(9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e^(-3*b*x - 3*a )/b^4 + 1/20000*(125*b^3*x^3 + 75*b^2*x^2 + 30*b*x + 6)*e^(-5*b*x - 5*a)/b ^4
Time = 0.12 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.05 \[ \int x^3 \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx=\frac {{\left (125 \, b^{3} x^{3} - 75 \, b^{2} x^{2} + 30 \, b x - 6\right )} e^{\left (5 \, b x + 5 \, a\right )}}{20000 \, b^{4}} - \frac {{\left (9 \, b^{3} x^{3} - 9 \, b^{2} x^{2} + 6 \, b x - 2\right )} e^{\left (3 \, b x + 3 \, a\right )}}{864 \, b^{4}} - \frac {{\left (b^{3} x^{3} - 3 \, b^{2} x^{2} + 6 \, b x - 6\right )} e^{\left (b x + a\right )}}{16 \, b^{4}} - \frac {{\left (b^{3} x^{3} + 3 \, b^{2} x^{2} + 6 \, b x + 6\right )} e^{\left (-b x - a\right )}}{16 \, b^{4}} - \frac {{\left (9 \, b^{3} x^{3} + 9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{864 \, b^{4}} + \frac {{\left (125 \, b^{3} x^{3} + 75 \, b^{2} x^{2} + 30 \, b x + 6\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{20000 \, b^{4}} \] Input:
integrate(x^3*cosh(b*x+a)^2*sinh(b*x+a)^3,x, algorithm="giac")
Output:
1/20000*(125*b^3*x^3 - 75*b^2*x^2 + 30*b*x - 6)*e^(5*b*x + 5*a)/b^4 - 1/86 4*(9*b^3*x^3 - 9*b^2*x^2 + 6*b*x - 2)*e^(3*b*x + 3*a)/b^4 - 1/16*(b^3*x^3 - 3*b^2*x^2 + 6*b*x - 6)*e^(b*x + a)/b^4 - 1/16*(b^3*x^3 + 3*b^2*x^2 + 6*b *x + 6)*e^(-b*x - a)/b^4 - 1/864*(9*b^3*x^3 + 9*b^2*x^2 + 6*b*x + 2)*e^(-3 *b*x - 3*a)/b^4 + 1/20000*(125*b^3*x^3 + 75*b^2*x^2 + 30*b*x + 6)*e^(-5*b* x - 5*a)/b^4
Time = 1.07 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.86 \[ \int x^3 \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx=\frac {12568\,\mathrm {sinh}\left (a+b\,x\right )}{16875\,b^4}-\frac {\frac {x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{3}-\frac {x^3\,{\mathrm {cosh}\left (a+b\,x\right )}^5}{5}}{b}-\frac {-\frac {6\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^5}{125}+\frac {26\,x\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{225}+\frac {52\,x\,\mathrm {cosh}\left (a+b\,x\right )}{75}}{b^3}+\frac {\frac {26\,x^2\,\mathrm {sinh}\left (a+b\,x\right )}{75}+\frac {13\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{75}-\frac {3\,x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^4\,\mathrm {sinh}\left (a+b\,x\right )}{25}}{b^2}+\frac {434\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,\mathrm {sinh}\left (a+b\,x\right )}{16875\,b^4}-\frac {6\,{\mathrm {cosh}\left (a+b\,x\right )}^4\,\mathrm {sinh}\left (a+b\,x\right )}{625\,b^4} \] Input:
int(x^3*cosh(a + b*x)^2*sinh(a + b*x)^3,x)
Output:
(12568*sinh(a + b*x))/(16875*b^4) - ((x^3*cosh(a + b*x)^3)/3 - (x^3*cosh(a + b*x)^5)/5)/b - ((52*x*cosh(a + b*x))/75 + (26*x*cosh(a + b*x)^3)/225 - (6*x*cosh(a + b*x)^5)/125)/b^3 + ((26*x^2*sinh(a + b*x))/75 + (13*x^2*cosh (a + b*x)^2*sinh(a + b*x))/75 - (3*x^2*cosh(a + b*x)^4*sinh(a + b*x))/25)/ b^2 + (434*cosh(a + b*x)^2*sinh(a + b*x))/(16875*b^4) - (6*cosh(a + b*x)^4 *sinh(a + b*x))/(625*b^4)
Time = 0.22 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.73 \[ \int x^3 \cosh ^2(a+b x) \sinh ^3(a+b x) \, dx=\frac {3375 e^{10 b x +10 a} b^{3} x^{3}-2025 e^{10 b x +10 a} b^{2} x^{2}+810 e^{10 b x +10 a} b x -162 e^{10 b x +10 a}-5625 e^{8 b x +8 a} b^{3} x^{3}+5625 e^{8 b x +8 a} b^{2} x^{2}-3750 e^{8 b x +8 a} b x +1250 e^{8 b x +8 a}-33750 e^{6 b x +6 a} b^{3} x^{3}+101250 e^{6 b x +6 a} b^{2} x^{2}-202500 e^{6 b x +6 a} b x +202500 e^{6 b x +6 a}-33750 e^{4 b x +4 a} b^{3} x^{3}-101250 e^{4 b x +4 a} b^{2} x^{2}-202500 e^{4 b x +4 a} b x -202500 e^{4 b x +4 a}-5625 e^{2 b x +2 a} b^{3} x^{3}-5625 e^{2 b x +2 a} b^{2} x^{2}-3750 e^{2 b x +2 a} b x -1250 e^{2 b x +2 a}+3375 b^{3} x^{3}+2025 b^{2} x^{2}+810 b x +162}{540000 e^{5 b x +5 a} b^{4}} \] Input:
int(x^3*cosh(b*x+a)^2*sinh(b*x+a)^3,x)
Output:
(3375*e**(10*a + 10*b*x)*b**3*x**3 - 2025*e**(10*a + 10*b*x)*b**2*x**2 + 8 10*e**(10*a + 10*b*x)*b*x - 162*e**(10*a + 10*b*x) - 5625*e**(8*a + 8*b*x) *b**3*x**3 + 5625*e**(8*a + 8*b*x)*b**2*x**2 - 3750*e**(8*a + 8*b*x)*b*x + 1250*e**(8*a + 8*b*x) - 33750*e**(6*a + 6*b*x)*b**3*x**3 + 101250*e**(6*a + 6*b*x)*b**2*x**2 - 202500*e**(6*a + 6*b*x)*b*x + 202500*e**(6*a + 6*b*x ) - 33750*e**(4*a + 4*b*x)*b**3*x**3 - 101250*e**(4*a + 4*b*x)*b**2*x**2 - 202500*e**(4*a + 4*b*x)*b*x - 202500*e**(4*a + 4*b*x) - 5625*e**(2*a + 2* b*x)*b**3*x**3 - 5625*e**(2*a + 2*b*x)*b**2*x**2 - 3750*e**(2*a + 2*b*x)*b *x - 1250*e**(2*a + 2*b*x) + 3375*b**3*x**3 + 2025*b**2*x**2 + 810*b*x + 1 62)/(540000*e**(5*a + 5*b*x)*b**4)