Integrand size = 18, antiderivative size = 155 \[ \int x^3 \cosh ^3(a+b x) \sinh (a+b x) \, dx=-\frac {45 x}{256 b^3}-\frac {3 x^3}{32 b}+\frac {9 x \cosh ^2(a+b x)}{32 b^3}+\frac {3 x \cosh ^4(a+b x)}{32 b^3}+\frac {x^3 \cosh ^4(a+b x)}{4 b}-\frac {45 \cosh (a+b x) \sinh (a+b x)}{256 b^4}-\frac {9 x^2 \cosh (a+b x) \sinh (a+b x)}{32 b^2}-\frac {3 \cosh ^3(a+b x) \sinh (a+b x)}{128 b^4}-\frac {3 x^2 \cosh ^3(a+b x) \sinh (a+b x)}{16 b^2} \]
-45/256*x/b^3-3/32*x^3/b+9/32*x*cosh(b*x+a)^2/b^3+3/32*x*cosh(b*x+a)^4/b^3 +1/4*x^3*cosh(b*x+a)^4/b-45/256*cosh(b*x+a)*sinh(b*x+a)/b^4-9/32*x^2*cosh( b*x+a)*sinh(b*x+a)/b^2-3/128*cosh(b*x+a)^3*sinh(b*x+a)/b^4-3/16*x^2*cosh(b *x+a)^3*sinh(b*x+a)/b^2
Time = 0.42 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.59 \[ \int x^3 \cosh ^3(a+b x) \sinh (a+b x) \, dx=\frac {32 b x \left (3+2 b^2 x^2\right ) \cosh (2 (a+b x))+2 b x \left (3+8 b^2 x^2\right ) \cosh (4 (a+b x))-3 \left (16+32 b^2 x^2+\left (1+8 b^2 x^2\right ) \cosh (2 (a+b x))\right ) \sinh (2 (a+b x))}{512 b^4} \]
(32*b*x*(3 + 2*b^2*x^2)*Cosh[2*(a + b*x)] + 2*b*x*(3 + 8*b^2*x^2)*Cosh[4*( a + b*x)] - 3*(16 + 32*b^2*x^2 + (1 + 8*b^2*x^2)*Cosh[2*(a + b*x)])*Sinh[2 *(a + b*x)])/(512*b^4)
Time = 0.77 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.34, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {5896, 3042, 3792, 3042, 3115, 3042, 3115, 24, 3792, 15, 3042, 3115, 24}
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.
| \(\displaystyle \int x^3 \sinh (a+b x) \cosh ^3(a+b x) \, dx\) |
| \(\Big \downarrow \) 5896 |
| \(\displaystyle \frac {x^3 \cosh ^4(a+b x)}{4 b}-\frac {3 \int x^2 \cosh ^4(a+b x)dx}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x^3 \cosh ^4(a+b x)}{4 b}-\frac {3 \int x^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^4dx}{4 b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {x^3 \cosh ^4(a+b x)}{4 b}-\frac {3 \left (\frac {\int \cosh ^4(a+b x)dx}{8 b^2}+\frac {3}{4} \int x^2 \cosh ^2(a+b x)dx-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x^3 \cosh ^4(a+b x)}{4 b}-\frac {3 \left (\frac {\int \sin \left (i a+i b x+\frac {\pi }{2}\right )^4dx}{8 b^2}+\frac {3}{4} \int x^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{4 b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {x^3 \cosh ^4(a+b x)}{4 b}-\frac {3 \left (\frac {\frac {3}{4} \int \cosh ^2(a+b x)dx+\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}}{8 b^2}+\frac {3}{4} \int x^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x^3 \cosh ^4(a+b x)}{4 b}-\frac {3 \left (\frac {\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \int \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx}{8 b^2}+\frac {3}{4} \int x^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{4 b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {x^3 \cosh ^4(a+b x)}{4 b}-\frac {3 \left (\frac {\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}\right )+\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}}{8 b^2}+\frac {3}{4} \int x^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{4 b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {x^3 \cosh ^4(a+b x)}{4 b}-\frac {3 \left (\frac {3}{4} \int x^2 \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{4 b}\) |
| \(\Big \downarrow \) 3792 |
| \(\displaystyle \frac {x^3 \cosh ^4(a+b x)}{4 b}-\frac {3 \left (\frac {3}{4} \left (\frac {\int \cosh ^2(a+b x)dx}{2 b^2}+\frac {\int x^2dx}{2}-\frac {x \cosh ^2(a+b x)}{2 b^2}+\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}\right )-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{4 b}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {x^3 \cosh ^4(a+b x)}{4 b}-\frac {3 \left (\frac {3}{4} \left (\frac {\int \cosh ^2(a+b x)dx}{2 b^2}-\frac {x \cosh ^2(a+b x)}{2 b^2}+\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{4 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {x^3 \cosh ^4(a+b x)}{4 b}-\frac {3 \left (\frac {3}{4} \left (\frac {\int \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx}{2 b^2}-\frac {x \cosh ^2(a+b x)}{2 b^2}+\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{4 b}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {x^3 \cosh ^4(a+b x)}{4 b}-\frac {3 \left (\frac {3}{4} \left (\frac {\frac {\int 1dx}{2}+\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}}{2 b^2}-\frac {x \cosh ^2(a+b x)}{2 b^2}+\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{4 b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {x^3 \cosh ^4(a+b x)}{4 b}-\frac {3 \left (\frac {3}{4} \left (-\frac {x \cosh ^2(a+b x)}{2 b^2}+\frac {\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}}{2 b^2}+\frac {x^2 \sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x^3}{6}\right )-\frac {x \cosh ^4(a+b x)}{8 b^2}+\frac {\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )}{8 b^2}+\frac {x^2 \sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )}{4 b}\) |
(x^3*Cosh[a + b*x]^4)/(4*b) - (3*(-1/8*(x*Cosh[a + b*x]^4)/b^2 + (x^2*Cosh [a + b*x]^3*Sinh[a + b*x])/(4*b) + ((Cosh[a + b*x]^3*Sinh[a + b*x])/(4*b) + (3*(x/2 + (Cosh[a + b*x]*Sinh[a + b*x])/(2*b)))/4)/(8*b^2) + (3*(x^3/6 - (x*Cosh[a + b*x]^2)/(2*b^2) + (x^2*Cosh[a + b*x]*Sinh[a + b*x])/(2*b) + ( x/2 + (Cosh[a + b*x]*Sinh[a + b*x])/(2*b))/(2*b^2)))/4))/(4*b)
3.3.69.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[Cosh[(a_.) + (b_.)*(x_)^(n_.)]^(p_.)*(x_)^(m_.)*Sinh[(a_.) + (b_.)*(x_) ^(n_.)], x_Symbol] :> Simp[x^(m - n + 1)*(Cosh[a + b*x^n]^(p + 1)/(b*n*(p + 1))), x] - Simp[(m - n + 1)/(b*n*(p + 1)) Int[x^(m - n)*Cosh[a + b*x^n]^ (p + 1), x], x] /; FreeQ[{a, b, p}, x] && LtQ[0, n, m + 1] && NeQ[p, -1]
Time = 7.58 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.94
| method | result | size |
| risch | \(\frac {\left (32 x^{3} b^{3}-24 x^{2} b^{2}+12 b x -3\right ) {\mathrm e}^{4 b x +4 a}}{2048 b^{4}}+\frac {\left (4 x^{3} b^{3}-6 x^{2} b^{2}+6 b x -3\right ) {\mathrm e}^{2 b x +2 a}}{64 b^{4}}+\frac {\left (4 x^{3} b^{3}+6 x^{2} b^{2}+6 b x +3\right ) {\mathrm e}^{-2 b x -2 a}}{64 b^{4}}+\frac {\left (32 x^{3} b^{3}+24 x^{2} b^{2}+12 b x +3\right ) {\mathrm e}^{-4 b x -4 a}}{2048 b^{4}}\) | \(146\) |
| derivativedivides | \(\frac {-\frac {a^{3} \cosh \left (b x +a \right )^{4}}{4}+3 a^{2} \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{4}-\frac {\cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{16}-\frac {3 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{32}-\frac {3 b x}{32}-\frac {3 a}{32}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right )^{4}}{4}-\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{8}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{16}-\frac {3 \left (b x +a \right )^{2}}{32}+\frac {\cosh \left (b x +a \right )^{4}}{32}+\frac {3 \cosh \left (b x +a \right )^{2}}{32}\right )+\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right )^{4}}{4}-\frac {3 \left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{16}-\frac {9 \left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{32}-\frac {3 \left (b x +a \right )^{3}}{32}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{32}-\frac {3 \cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{128}-\frac {45 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{256}-\frac {45 b x}{256}-\frac {45 a}{256}+\frac {9 \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{32}}{b^{4}}\) | \(304\) |
| default | \(\frac {-\frac {a^{3} \cosh \left (b x +a \right )^{4}}{4}+3 a^{2} \left (\frac {\left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{4}-\frac {\cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{16}-\frac {3 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{32}-\frac {3 b x}{32}-\frac {3 a}{32}\right )-3 a \left (\frac {\left (b x +a \right )^{2} \cosh \left (b x +a \right )^{4}}{4}-\frac {\left (b x +a \right ) \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{8}-\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{16}-\frac {3 \left (b x +a \right )^{2}}{32}+\frac {\cosh \left (b x +a \right )^{4}}{32}+\frac {3 \cosh \left (b x +a \right )^{2}}{32}\right )+\frac {\left (b x +a \right )^{3} \cosh \left (b x +a \right )^{4}}{4}-\frac {3 \left (b x +a \right )^{2} \sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}{16}-\frac {9 \left (b x +a \right )^{2} \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{32}-\frac {3 \left (b x +a \right )^{3}}{32}+\frac {3 \left (b x +a \right ) \cosh \left (b x +a \right )^{4}}{32}-\frac {3 \cosh \left (b x +a \right )^{3} \sinh \left (b x +a \right )}{128}-\frac {45 \cosh \left (b x +a \right ) \sinh \left (b x +a \right )}{256}-\frac {45 b x}{256}-\frac {45 a}{256}+\frac {9 \left (b x +a \right ) \cosh \left (b x +a \right )^{2}}{32}}{b^{4}}\) | \(304\) |
1/2048*(32*b^3*x^3-24*b^2*x^2+12*b*x-3)/b^4*exp(4*b*x+4*a)+1/64*(4*b^3*x^3 -6*b^2*x^2+6*b*x-3)/b^4*exp(2*b*x+2*a)+1/64*(4*b^3*x^3+6*b^2*x^2+6*b*x+3)/ b^4*exp(-2*b*x-2*a)+1/2048*(32*b^3*x^3+24*b^2*x^2+12*b*x+3)/b^4*exp(-4*b*x -4*a)
Time = 0.26 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.23 \[ \int x^3 \cosh ^3(a+b x) \sinh (a+b x) \, dx=\frac {{\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{4} - 3 \, {\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \sinh \left (b x + a\right )^{4} + 16 \, {\left (2 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (16 \, b^{3} x^{3} + 3 \, {\left (8 \, b^{3} x^{3} + 3 \, b x\right )} \cosh \left (b x + a\right )^{2} + 24 \, b x\right )} \sinh \left (b x + a\right )^{2} - 3 \, {\left ({\left (8 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )^{3} + 16 \, {\left (2 \, b^{2} x^{2} + 1\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{256 \, b^{4}} \]
1/256*((8*b^3*x^3 + 3*b*x)*cosh(b*x + a)^4 - 3*(8*b^2*x^2 + 1)*cosh(b*x + a)*sinh(b*x + a)^3 + (8*b^3*x^3 + 3*b*x)*sinh(b*x + a)^4 + 16*(2*b^3*x^3 + 3*b*x)*cosh(b*x + a)^2 + 2*(16*b^3*x^3 + 3*(8*b^3*x^3 + 3*b*x)*cosh(b*x + a)^2 + 24*b*x)*sinh(b*x + a)^2 - 3*((8*b^2*x^2 + 1)*cosh(b*x + a)^3 + 16* (2*b^2*x^2 + 1)*cosh(b*x + a))*sinh(b*x + a))/b^4
Time = 0.69 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.46 \[ \int x^3 \cosh ^3(a+b x) \sinh (a+b x) \, dx=\begin {cases} - \frac {3 x^{3} \sinh ^{4}{\left (a + b x \right )}}{32 b} + \frac {3 x^{3} \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{16 b} + \frac {5 x^{3} \cosh ^{4}{\left (a + b x \right )}}{32 b} + \frac {9 x^{2} \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{32 b^{2}} - \frac {15 x^{2} \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{32 b^{2}} - \frac {45 x \sinh ^{4}{\left (a + b x \right )}}{256 b^{3}} + \frac {9 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{128 b^{3}} + \frac {51 x \cosh ^{4}{\left (a + b x \right )}}{256 b^{3}} + \frac {45 \sinh ^{3}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{256 b^{4}} - \frac {51 \sinh {\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{256 b^{4}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh {\left (a \right )} \cosh ^{3}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \]
Piecewise((-3*x**3*sinh(a + b*x)**4/(32*b) + 3*x**3*sinh(a + b*x)**2*cosh( a + b*x)**2/(16*b) + 5*x**3*cosh(a + b*x)**4/(32*b) + 9*x**2*sinh(a + b*x) **3*cosh(a + b*x)/(32*b**2) - 15*x**2*sinh(a + b*x)*cosh(a + b*x)**3/(32*b **2) - 45*x*sinh(a + b*x)**4/(256*b**3) + 9*x*sinh(a + b*x)**2*cosh(a + b* x)**2/(128*b**3) + 51*x*cosh(a + b*x)**4/(256*b**3) + 45*sinh(a + b*x)**3* cosh(a + b*x)/(256*b**4) - 51*sinh(a + b*x)*cosh(a + b*x)**3/(256*b**4), N e(b, 0)), (x**4*sinh(a)*cosh(a)**3/4, True))
Time = 0.20 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.10 \[ \int x^3 \cosh ^3(a+b x) \sinh (a+b x) \, dx=\frac {{\left (32 \, b^{3} x^{3} e^{\left (4 \, a\right )} - 24 \, b^{2} x^{2} e^{\left (4 \, a\right )} + 12 \, b x e^{\left (4 \, a\right )} - 3 \, e^{\left (4 \, a\right )}\right )} e^{\left (4 \, b x\right )}}{2048 \, b^{4}} + \frac {{\left (4 \, b^{3} x^{3} e^{\left (2 \, a\right )} - 6 \, b^{2} x^{2} e^{\left (2 \, a\right )} + 6 \, b x e^{\left (2 \, a\right )} - 3 \, e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}}{64 \, b^{4}} + \frac {{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{64 \, b^{4}} + \frac {{\left (32 \, b^{3} x^{3} + 24 \, b^{2} x^{2} + 12 \, b x + 3\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{2048 \, b^{4}} \]
1/2048*(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 + 1/64*(4*b^3*x^3*e^(2*a) - 6*b^2*x^2*e^(2*a) + 6*b*x*e^ (2*a) - 3*e^(2*a))*e^(2*b*x)/b^4 + 1/64*(4*b^3*x^3 + 6*b^2*x^2 + 6*b*x + 3 )*e^(-2*b*x - 2*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 = 0.29 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.94 \[ \int x^3 \cosh ^3(a+b x) \sinh (a+b x) \, dx=\frac {{\left (32 \, b^{3} x^{3} - 24 \, b^{2} x^{2} + 12 \, b x - 3\right )} e^{\left (4 \, b x + 4 \, a\right )}}{2048 \, b^{4}} + \frac {{\left (4 \, b^{3} x^{3} - 6 \, b^{2} x^{2} + 6 \, b x - 3\right )} e^{\left (2 \, b x + 2 \, a\right )}}{64 \, b^{4}} + \frac {{\left (4 \, b^{3} x^{3} + 6 \, b^{2} x^{2} + 6 \, b x + 3\right )} e^{\left (-2 \, b x - 2 \, a\right )}}{64 \, b^{4}} + \frac {{\left (32 \, b^{3} x^{3} + 24 \, b^{2} x^{2} + 12 \, b x + 3\right )} e^{\left (-4 \, b x - 4 \, a\right )}}{2048 \, b^{4}} \]
1/2048*(32*b^3*x^3 - 24*b^2*x^2 + 12*b*x - 3)*e^(4*b*x + 4*a)/b^4 + 1/64*( 4*b^3*x^3 - 6*b^2*x^2 + 6*b*x - 3)*e^(2*b*x + 2*a)/b^4 + 1/64*(4*b^3*x^3 + 6*b^2*x^2 + 6*b*x + 3)*e^(-2*b*x - 2*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 = 2.35 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.81 \[ \int x^3 \cosh ^3(a+b x) \sinh (a+b x) \, dx=\frac {\frac {x^3\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{8}+\frac {x^3\,\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{32}}{b}-\frac {\frac {3\,x^2\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{16}+\frac {3\,x^2\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{128}}{b^2}+\frac {\frac {3\,x\,\mathrm {cosh}\left (2\,a+2\,b\,x\right )}{16}+\frac {3\,x\,\mathrm {cosh}\left (4\,a+4\,b\,x\right )}{256}}{b^3}-\frac {3\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )}{32\,b^4}-\frac {3\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{1024\,b^4} \]