Integrand size = 20, antiderivative size = 79 \[ \int x^3 \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx=-\frac {x^4}{32}-\frac {3 \cosh (4 a+4 b x)}{1024 b^4}-\frac {3 x^2 \cosh (4 a+4 b x)}{128 b^2}+\frac {3 x \sinh (4 a+4 b x)}{256 b^3}+\frac {x^3 \sinh (4 a+4 b x)}{32 b} \]
-1/32*x^4-3/1024*cosh(4*b*x+4*a)/b^4-3/128*x^2*cosh(4*b*x+4*a)/b^2+3/256*x *sinh(4*b*x+4*a)/b^3+1/32*x^3*sinh(4*b*x+4*a)/b
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.73 \[ \int x^3 \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx=\frac {-32 b^4 x^4-3 \left (1+8 b^2 x^2\right ) \cosh (4 (a+b x))+4 b x \left (3+8 b^2 x^2\right ) \sinh (4 (a+b x))}{1024 b^4} \]
(-32*b^4*x^4 - 3*(1 + 8*b^2*x^2)*Cosh[4*(a + b*x)] + 4*b*x*(3 + 8*b^2*x^2) *Sinh[4*(a + b*x)])/(1024*b^4)
Time = 0.30 (sec) , antiderivative size = 79, 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.
\(\displaystyle \int x^3 \sinh ^2(a+b x) \cosh ^2(a+b x) \, dx\) |
\(\Big \downarrow \) 5971 |
\(\displaystyle \int \left (\frac {1}{8} x^3 \cosh (4 a+4 b x)-\frac {x^3}{8}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 \cosh (4 a+4 b x)}{1024 b^4}+\frac {3 x \sinh (4 a+4 b x)}{256 b^3}-\frac {3 x^2 \cosh (4 a+4 b x)}{128 b^2}+\frac {x^3 \sinh (4 a+4 b x)}{32 b}-\frac {x^4}{32}\) |
-1/32*x^4 - (3*Cosh[4*a + 4*b*x])/(1024*b^4) - (3*x^2*Cosh[4*a + 4*b*x])/( 128*b^2) + (3*x*Sinh[4*a + 4*b*x])/(256*b^3) + (x^3*Sinh[4*a + 4*b*x])/(32 *b)
3.3.90.3.1 Defintions of rubi rules used
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 = 11.06 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {x^{4}}{32}+\frac {\left (32 x^{3} b^{3}-24 x^{2} b^{2}+12 b x -3\right ) {\mathrm e}^{4 b x +4 a}}{2048 b^{4}}-\frac {\left (32 x^{3} b^{3}+24 x^{2} b^{2}+12 b x +3\right ) {\mathrm e}^{-4 b x -4 a}}{2048 b^{4}}\) | \(79\) |
derivativedivides | \(\frac {-a^{3} \left (\frac {\cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{4}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {b x}{8}-\frac {a}{8}\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {\left (b x +a \right )^{2}}{16}-\frac {\cosh \left (b x +a \right )^{4}}{16}+\frac {\cosh \left (b x +a \right )^{2}}{16}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {\left (b x +a \right )^{3}}{24}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{8}+\frac {\cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{32}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}-\frac {b x}{64}-\frac {a}{64}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{8}\right )+\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}-\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {\left (b x +a \right )^{4}}{32}-\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{4}}{16}+\frac {3 \left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{32}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}-\frac {3 \left (b x +a \right )^{2}}{128}-\frac {3 \cosh \left (b x +a \right )^{4}}{128}+\frac {3 \cosh \left (b x +a \right )^{2}}{128}+\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}{16}}{b^{4}}\) | \(404\) |
default | \(\frac {-a^{3} \left (\frac {\cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{4}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {b x}{8}-\frac {a}{8}\right )+3 a^{2} \left (\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {\left (b x +a \right )^{2}}{16}-\frac {\cosh \left (b x +a \right )^{4}}{16}+\frac {\cosh \left (b x +a \right )^{2}}{16}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}-\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {\left (b x +a \right )^{3}}{24}-\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{8}+\frac {\cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{32}-\frac {\cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}-\frac {b x}{64}-\frac {a}{64}+\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{8}\right )+\frac {\left (b x +a \right )^{3} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{4}-\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{8}-\frac {\left (b x +a \right )^{4}}{32}-\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{4}}{16}+\frac {3 \left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{32}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{64}-\frac {3 \left (b x +a \right )^{2}}{128}-\frac {3 \cosh \left (b x +a \right )^{4}}{128}+\frac {3 \cosh \left (b x +a \right )^{2}}{128}+\frac {3 \left (b x +a \right )^{2} \cosh \left (b x +a \right )^{2}}{16}}{b^{4}}\) | \(404\) |
-1/32*x^4+1/2048*(32*b^3*x^3-24*b^2*x^2+12*b*x-3)/b^4*exp(4*b*x+4*a)-1/204 8*(32*b^3*x^3+24*b^2*x^2+12*b*x+3)/b^4*exp(-4*b*x-4*a)
Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (69) = 138\).
Time = 0.25 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.77 \[ \int x^3 \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx=-\frac {32 \, b^{4} x^{4} + 3 \, {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{4} - 16 \, {\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 18 \, {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} - 16 \, {\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 3 \, {\left (8 \, b^{2} x^{2} + 1\right )} \sinh \left (b x + a\right )^{4}}{1024 \, b^{4}} \]
-1/1024*(32*b^4*x^4 + 3*(8*b^2*x^2 + 1)*cosh(b*x + a)^4 - 16*(8*b^3*x^3 + 3*b*x)*cosh(b*x + a)^3*sinh(b*x + a) + 18*(8*b^2*x^2 + 1)*cosh(b*x + a)^2* sinh(b*x + a)^2 - 16*(8*b^3*x^3 + 3*b*x)*cosh(b*x + a)*sinh(b*x + a)^3 + 3 *(8*b^2*x^2 + 1)*sinh(b*x + a)^4)/b^4
Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (76) = 152\).
Time = 0.55 (sec) , antiderivative size = 250, normalized size of antiderivative = 3.16 \[ \int x^3 \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx=\begin {cases} - \frac {x^{4} \sinh ^{4}{\left (a + b x \right )}}{32} + \frac {x^{4} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16} - \frac {x^{4} \cosh ^{4}{\left (a + b x \right )}}{32} + \frac {x^{3} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{8 b} + \frac {x^{3} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{8 b} - \frac {3 x^{2} \sinh ^{4}{\left (a + b x \right )}}{128 b^{2}} - \frac {9 x^{2} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{64 b^{2}} - \frac {3 x^{2} \cosh ^{4}{\left (a + b x \right )}}{128 b^{2}} + \frac {3 x \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{64 b^{3}} + \frac {3 x \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{64 b^{3}} - \frac {3 \sinh ^{4}{\left (a + b x \right )}}{256 b^{4}} - \frac {3 \cosh ^{4}{\left (a + b x \right )}}{256 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh ^{2}{\left (a \right )} \cosh ^{2}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \]
Piecewise((-x**4*sinh(a + b*x)**4/32 + x**4*sinh(a + b*x)**2*cosh(a + b*x) **2/16 - x**4*cosh(a + b*x)**4/32 + x**3*sinh(a + b*x)**3*cosh(a + b*x)/(8 *b) + x**3*sinh(a + b*x)*cosh(a + b*x)**3/(8*b) - 3*x**2*sinh(a + b*x)**4/ (128*b**2) - 9*x**2*sinh(a + b*x)**2*cosh(a + b*x)**2/(64*b**2) - 3*x**2*c osh(a + b*x)**4/(128*b**2) + 3*x*sinh(a + b*x)**3*cosh(a + b*x)/(64*b**3) + 3*x*sinh(a + b*x)*cosh(a + b*x)**3/(64*b**3) - 3*sinh(a + b*x)**4/(256*b **4) - 3*cosh(a + b*x)**4/(256*b**4), Ne(b, 0)), (x**4*sinh(a)**2*cosh(a)* *2/4, True))
Time = 0.21 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.15 \[ \int x^3 \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx=-\frac {1}{32} \, x^{4} + \frac {{\left (32 \, b^{3} x^{3} e^{\left (4 \, a\right )} - 24 \, b^{2} x^{2} e^{\left (4 \, a\right )} + 12 \, b x e^{\left (4 \, a\right )} - 3 \, e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{2048 \, b^{4}} - \frac {{\left (32 \, b^{3} x^{3} + 24 \, b^{2} x^{2} + 12 \, b x + 3\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{2048 \, b^{4}} \]
-1/32*x^4 + 1/2048*(32*b^3*x^3*e^(4*a) - 24*b^2*x^2*e^(4*a) + 12*b*x*e^(4* a) - 3*e^(4*a))*e^(4*b*x)/b^4 - 1/2048*(32*b^3*x^3 + 24*b^2*x^2 + 12*b*x + 3)*e^(-4*b*x - 4*a)/b^4
Time = 0.27 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.99 \[ \int x^3 \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx=-\frac {1}{32} \, x^{4} + \frac {{\left (32 \, b^{3} x^{3} - 24 \, b^{2} x^{2} + 12 \, b x - 3\right )} e^{\left (4 \, b x + 4 \, a\right )}}{2048 \, b^{4}} - \frac {{\left (32 \, b^{3} x^{3} + 24 \, b^{2} x^{2} + 12 \, b x + 3\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{2048 \, b^{4}} \]
-1/32*x^4 + 1/2048*(32*b^3*x^3 - 24*b^2*x^2 + 12*b*x - 3)*e^(4*b*x + 4*a)/ b^4 - 1/2048*(32*b^3*x^3 + 24*b^2*x^2 + 12*b*x + 3)*e^(-4*b*x - 4*a)/b^4
Time = 2.28 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int x^3 \cosh ^2(a+b x) \sinh ^2(a+b x) \, dx=-\frac {\frac {3\,\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{1024}-\frac {3\,b\,x\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{256}+\frac {3\,b^2\,x^2\,\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{128}-\frac {b^3\,x^3\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{32}}{b^4}-\frac {x^4}{32} \]