Integrand size = 18, antiderivative size = 94 \[ \int x \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {\cosh (a+b x)}{8 b^2}-\frac {\cosh (3 a+3 b x)}{144 b^2}-\frac {\cosh (5 a+5 b x)}{400 b^2}-\frac {x \sinh (a+b x)}{8 b}+\frac {x \sinh (3 a+3 b x)}{48 b}+\frac {x \sinh (5 a+5 b x)}{80 b} \] Output:
1/8*cosh(b*x+a)/b^2-1/144*cosh(3*b*x+3*a)/b^2-1/400*cosh(5*b*x+5*a)/b^2-1/ 8*x*sinh(b*x+a)/b+1/48*x*sinh(3*b*x+3*a)/b+1/80*x*sinh(5*b*x+5*a)/b
Time = 0.09 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.74 \[ \int x \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {450 \cosh (a+b x)-25 \cosh (3 (a+b x))-9 \cosh (5 (a+b x))-450 b x \sinh (a+b x)+75 b x \sinh (3 (a+b x))+45 b x \sinh (5 (a+b x))}{3600 b^2} \] Input:
Integrate[x*Cosh[a + b*x]^3*Sinh[a + b*x]^2,x]
Output:
(450*Cosh[a + b*x] - 25*Cosh[3*(a + b*x)] - 9*Cosh[5*(a + b*x)] - 450*b*x* Sinh[a + b*x] + 75*b*x*Sinh[3*(a + b*x)] + 45*b*x*Sinh[5*(a + b*x)])/(3600 *b^2)
Time = 0.49 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, 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.
| \(\displaystyle \int x \sinh ^2(a+b x) \cosh ^3(a+b x) \, dx\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \int \left (-\frac {1}{8} x \cosh (a+b x)+\frac {1}{16} x \cosh (3 a+3 b x)+\frac {1}{16} x \cosh (5 a+5 b x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\cosh (a+b x)}{8 b^2}-\frac {\cosh (3 a+3 b x)}{144 b^2}-\frac {\cosh (5 a+5 b x)}{400 b^2}-\frac {x \sinh (a+b x)}{8 b}+\frac {x \sinh (3 a+3 b x)}{48 b}+\frac {x \sinh (5 a+5 b x)}{80 b}\) |
Input:
Int[x*Cosh[a + b*x]^3*Sinh[a + b*x]^2,x]
Output:
Cosh[a + b*x]/(8*b^2) - Cosh[3*a + 3*b*x]/(144*b^2) - Cosh[5*a + 5*b*x]/(4 00*b^2) - (x*Sinh[a + b*x])/(8*b) + (x*Sinh[3*a + 3*b*x])/(48*b) + (x*Sinh [5*a + 5*b*x])/(80*b)
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 = 15.72 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.24
| method | result | size |
| risch | \(\frac {\left (5 b x -1\right ) {\mathrm e}^{5 b x +5 a}}{800 b^{2}}+\frac {\left (3 b x -1\right ) {\mathrm e}^{3 b x +3 a}}{288 b^{2}}-\frac {\left (b x -1\right ) {\mathrm e}^{b x +a}}{16 b^{2}}+\frac {\left (b x +1\right ) {\mathrm e}^{-b x -a}}{16 b^{2}}-\frac {\left (3 b x +1\right ) {\mathrm e}^{-3 b x -3 a}}{288 b^{2}}-\frac {\left (5 b x +1\right ) {\mathrm e}^{-5 b x -5 a}}{800 b^{2}}\) | \(117\) |
| derivativedivides | \(\frac {\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{5}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{15}-\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{15}-\frac {\cosh \left (b x +a \right )^{5}}{25}+\frac {2 \cosh \left (b x +a \right )}{15}+\frac {\cosh \left (b x +a \right )^{3}}{45}-a \left (\frac {\sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{5}-\frac {\left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{5}\right )}{b^{2}}\) | \(129\) |
| default | \(\frac {\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{5}-\frac {2 \left (b x +a \right ) \sinh \left (b x +a \right )}{15}-\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{15}-\frac {\cosh \left (b x +a \right )^{5}}{25}+\frac {2 \cosh \left (b x +a \right )}{15}+\frac {\cosh \left (b x +a \right )^{3}}{45}-a \left (\frac {\sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{5}-\frac {\left (\frac {2}{3}+\frac {\cosh \left (b x +a \right )^{2}}{3}\right ) \sinh \left (b x +a \right )}{5}\right )}{b^{2}}\) | \(129\) |
| orering | \(-\frac {2 \left (259 b^{4} x^{4}-140 b^{2} x^{2}+72\right ) \cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )^{2}}{225 b^{6} x^{4}}+\frac {\left (259 b^{4} x^{4}-280 b^{2} x^{2}+144\right ) \left (\cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )^{2}+3 x \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )^{3} b +2 x \cosh \left (b x +a \right )^{4} \sinh \left (b x +a \right ) b \right )}{225 b^{6} x^{4}}+\frac {4 \left (35 b^{2} x^{2}-18\right ) \left (6 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )^{3} b +4 \cosh \left (b x +a \right )^{4} \sinh \left (b x +a \right ) b +6 x \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{4} b^{2}+17 x \cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )^{2} b^{2}+2 x \cosh \left (b x +a \right )^{5} b^{2}\right )}{225 b^{6} x^{3}}-\frac {\left (35 b^{2} x^{2}-24\right ) \left (18 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )^{4} b^{2}+51 \cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )^{2} b^{2}+6 \cosh \left (b x +a \right )^{5} b^{2}+6 x \,b^{3} \sinh \left (b x +a \right )^{5}+75 x \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )^{3} b^{3}+44 x \cosh \left (b x +a \right )^{4} \sinh \left (b x +a \right ) b^{3}\right )}{225 b^{6} x^{2}}-\frac {2 \left (24 b^{3} \sinh \left (b x +a \right )^{5}+300 \cosh \left (b x +a \right )^{2} \sinh \left (b x +a \right )^{3} b^{3}+176 \cosh \left (b x +a \right )^{4} \sinh \left (b x +a \right ) b^{3}+180 x \,b^{4} \sinh \left (b x +a \right )^{4} \cosh \left (b x +a \right )+401 x \cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )^{2} b^{4}+44 x \cosh \left (b x +a \right )^{5} b^{4}\right )}{75 b^{6} x}+\frac {900 b^{4} \sinh \left (b x +a \right )^{4} \cosh \left (b x +a \right )+2005 \cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )^{2} b^{4}+220 \cosh \left (b x +a \right )^{5} b^{4}+1923 x \,b^{5} \sinh \left (b x +a \right )^{3} \cosh \left (b x +a \right )^{2}+180 x \,b^{5} \sinh \left (b x +a \right )^{5}+1022 x \cosh \left (b x +a \right )^{4} \sinh \left (b x +a \right ) b^{5}}{225 b^{6}}\) | \(598\) |
Input:
int(x*cosh(b*x+a)^3*sinh(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/800*(5*b*x-1)/b^2*exp(5*b*x+5*a)+1/288*(3*b*x-1)/b^2*exp(3*b*x+3*a)-1/16 *(b*x-1)/b^2*exp(b*x+a)+1/16*(b*x+1)/b^2*exp(-b*x-a)-1/288*(3*b*x+1)/b^2*e xp(-3*b*x-3*a)-1/800*(5*b*x+1)/b^2*exp(-5*b*x-5*a)
Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.62 \[ \int x \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {45 \, b x \sinh \left (b x + a\right )^{5} - 9 \, \cosh \left (b x + a\right )^{5} - 45 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + 75 \, {\left (6 \, b x \cosh \left (b x + a\right )^{2} + b x\right )} \sinh \left (b x + a\right )^{3} - 25 \, \cosh \left (b x + a\right )^{3} - 15 \, {\left (6 \, \cosh \left (b x + a\right )^{3} + 5 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} + 225 \, {\left (b x \cosh \left (b x + a\right )^{4} + b x \cosh \left (b x + a\right )^{2} - 2 \, b x\right )} \sinh \left (b x + a\right ) + 450 \, \cosh \left (b x + a\right )}{3600 \, b^{2}} \] Input:
integrate(x*cosh(b*x+a)^3*sinh(b*x+a)^2,x, algorithm="fricas")
Output:
1/3600*(45*b*x*sinh(b*x + a)^5 - 9*cosh(b*x + a)^5 - 45*cosh(b*x + a)*sinh (b*x + a)^4 + 75*(6*b*x*cosh(b*x + a)^2 + b*x)*sinh(b*x + a)^3 - 25*cosh(b *x + a)^3 - 15*(6*cosh(b*x + a)^3 + 5*cosh(b*x + a))*sinh(b*x + a)^2 + 225 *(b*x*cosh(b*x + a)^4 + b*x*cosh(b*x + a)^2 - 2*b*x)*sinh(b*x + a) + 450*c osh(b*x + a))/b^2
Time = 0.40 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.19 \[ \int x \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\begin {cases} - \frac {2 x \sinh ^{5}{\left (a + b x \right )}}{15 b} + \frac {x \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b} + \frac {2 \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{15 b^{2}} - \frac {13 \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{45 b^{2}} + \frac {26 \cosh ^{5}{\left (a + b x \right )}}{225 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{2} \sinh ^{2}{\left (a \right )} \cosh ^{3}{\left (a \right )}}{2} & \text {otherwise} \end {cases} \] Input:
integrate(x*cosh(b*x+a)**3*sinh(b*x+a)**2,x)
Output:
Piecewise((-2*x*sinh(a + b*x)**5/(15*b) + x*sinh(a + b*x)**3*cosh(a + b*x) **2/(3*b) + 2*sinh(a + b*x)**4*cosh(a + b*x)/(15*b**2) - 13*sinh(a + b*x)* *2*cosh(a + b*x)**3/(45*b**2) + 26*cosh(a + b*x)**5/(225*b**2), Ne(b, 0)), (x**2*sinh(a)**2*cosh(a)**3/2, True))
Time = 0.06 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.37 \[ \int x \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {{\left (5 \, b x e^{\left (5 \, a\right )} - e^{\left (5 \, a\right )}\right )} e^{\left (5 \, b x\right )}}{800 \, b^{2}} + \frac {{\left (3 \, b x e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{288 \, b^{2}} - \frac {{\left (b x e^{a} - e^{a}\right )} e^{\left (b x\right )}}{16 \, b^{2}} + \frac {{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{16 \, b^{2}} - \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{288 \, b^{2}} - \frac {{\left (5 \, b x + 1\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{800 \, b^{2}} \] Input:
integrate(x*cosh(b*x+a)^3*sinh(b*x+a)^2,x, algorithm="maxima")
Output:
1/800*(5*b*x*e^(5*a) - e^(5*a))*e^(5*b*x)/b^2 + 1/288*(3*b*x*e^(3*a) - e^( 3*a))*e^(3*b*x)/b^2 - 1/16*(b*x*e^a - e^a)*e^(b*x)/b^2 + 1/16*(b*x + 1)*e^ (-b*x - a)/b^2 - 1/288*(3*b*x + 1)*e^(-3*b*x - 3*a)/b^2 - 1/800*(5*b*x + 1 )*e^(-5*b*x - 5*a)/b^2
Time = 0.12 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.23 \[ \int x \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {{\left (5 \, b x - 1\right )} e^{\left (5 \, b x + 5 \, a\right )}}{800 \, b^{2}} + \frac {{\left (3 \, b x - 1\right )} e^{\left (3 \, b x + 3 \, a\right )}}{288 \, b^{2}} - \frac {{\left (b x - 1\right )} e^{\left (b x + a\right )}}{16 \, b^{2}} + \frac {{\left (b x + 1\right )} e^{\left (-b x - a\right )}}{16 \, b^{2}} - \frac {{\left (3 \, b x + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{288 \, b^{2}} - \frac {{\left (5 \, b x + 1\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{800 \, b^{2}} \] Input:
integrate(x*cosh(b*x+a)^3*sinh(b*x+a)^2,x, algorithm="giac")
Output:
1/800*(5*b*x - 1)*e^(5*b*x + 5*a)/b^2 + 1/288*(3*b*x - 1)*e^(3*b*x + 3*a)/ b^2 - 1/16*(b*x - 1)*e^(b*x + a)/b^2 + 1/16*(b*x + 1)*e^(-b*x - a)/b^2 - 1 /288*(3*b*x + 1)*e^(-3*b*x - 3*a)/b^2 - 1/800*(5*b*x + 1)*e^(-5*b*x - 5*a) /b^2
Time = 0.16 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.88 \[ \int x \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=-\frac {b\,\left (\frac {2\,x\,{\mathrm {sinh}\left (a+b\,x\right )}^5}{15}-\frac {x\,{\mathrm {cosh}\left (a+b\,x\right )}^2\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{3}\right )-\frac {2\,\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {sinh}\left (a+b\,x\right )}^4}{15}-\frac {26\,{\mathrm {cosh}\left (a+b\,x\right )}^5}{225}+\frac {13\,{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{45}}{b^2} \] Input:
int(x*cosh(a + b*x)^3*sinh(a + b*x)^2,x)
Output:
-(b*((2*x*sinh(a + b*x)^5)/15 - (x*cosh(a + b*x)^2*sinh(a + b*x)^3)/3) - ( 2*cosh(a + b*x)*sinh(a + b*x)^4)/15 - (26*cosh(a + b*x)^5)/225 + (13*cosh( a + b*x)^3*sinh(a + b*x)^2)/45)/b^2
Time = 0.23 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.63 \[ \int x \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx=\frac {45 e^{10 b x +10 a} b x -9 e^{10 b x +10 a}+75 e^{8 b x +8 a} b x -25 e^{8 b x +8 a}-450 e^{6 b x +6 a} b x +450 e^{6 b x +6 a}+450 e^{4 b x +4 a} b x +450 e^{4 b x +4 a}-75 e^{2 b x +2 a} b x -25 e^{2 b x +2 a}-45 b x -9}{7200 e^{5 b x +5 a} b^{2}} \] Input:
int(x*cosh(b*x+a)^3*sinh(b*x+a)^2,x)
Output:
(45*e**(10*a + 10*b*x)*b*x - 9*e**(10*a + 10*b*x) + 75*e**(8*a + 8*b*x)*b* x - 25*e**(8*a + 8*b*x) - 450*e**(6*a + 6*b*x)*b*x + 450*e**(6*a + 6*b*x) + 450*e**(4*a + 4*b*x)*b*x + 450*e**(4*a + 4*b*x) - 75*e**(2*a + 2*b*x)*b* x - 25*e**(2*a + 2*b*x) - 45*b*x - 9)/(7200*e**(5*a + 5*b*x)*b**2)