Integrand size = 17, antiderivative size = 113 \[ \int \cosh ^4(a+b x) \sinh ^6(a+b x) \, dx=-\frac {3 x}{256}-\frac {3 \cosh (a+b x) \sinh (a+b x)}{256 b}-\frac {\cosh ^3(a+b x) \sinh (a+b x)}{128 b}+\frac {\cosh ^5(a+b x) \sinh (a+b x)}{32 b}-\frac {\cosh ^5(a+b x) \sinh ^3(a+b x)}{16 b}+\frac {\cosh ^5(a+b x) \sinh ^5(a+b x)}{10 b} \]
-3/256*x-3/256*cosh(b*x+a)*sinh(b*x+a)/b-1/128*cosh(b*x+a)^3*sinh(b*x+a)/b +1/32*cosh(b*x+a)^5*sinh(b*x+a)/b-1/16*cosh(b*x+a)^5*sinh(b*x+a)^3/b+1/10* cosh(b*x+a)^5*sinh(b*x+a)^5/b
Time = 0.12 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.55 \[ \int \cosh ^4(a+b x) \sinh ^6(a+b x) \, dx=\frac {-120 b x+20 \sinh (2 (a+b x))+40 \sinh (4 (a+b x))-10 \sinh (6 (a+b x))-5 \sinh (8 (a+b x))+2 \sinh (10 (a+b x))}{10240 b} \]
(-120*b*x + 20*Sinh[2*(a + b*x)] + 40*Sinh[4*(a + b*x)] - 10*Sinh[6*(a + b *x)] - 5*Sinh[8*(a + b*x)] + 2*Sinh[10*(a + b*x)])/(10240*b)
Time = 0.58 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.18, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.824, Rules used = {3042, 25, 3048, 3042, 3048, 25, 3042, 25, 3048, 3042, 3115, 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.
| \(\displaystyle \int \sinh ^6(a+b x) \cosh ^4(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (i a+i b x)^6 \left (-\cos (i a+i b x)^4\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \cos (i a+i b x)^4 \sin (i a+i b x)^6dx\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}-\frac {1}{2} \int \cosh ^4(a+b x) \sinh ^4(a+b x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}-\frac {1}{2} \int \cos (i a+i b x)^4 \sin (i a+i b x)^4dx\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \int -\cosh ^4(a+b x) \sinh ^2(a+b x)dx-\frac {\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}\right )+\frac {\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{2} \left (\frac {3}{8} \int \cosh ^4(a+b x) \sinh ^2(a+b x)dx-\frac {\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}\right )+\frac {\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}+\frac {1}{2} \left (-\frac {\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}+\frac {3}{8} \int -\cos (i a+i b x)^4 \sin (i a+i b x)^2dx\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}+\frac {1}{2} \left (-\frac {\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}-\frac {3}{8} \int \cos (i a+i b x)^4 \sin (i a+i b x)^2dx\right )\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (\frac {1}{6} \int \cosh ^4(a+b x)dx-\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}\right )-\frac {\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}\right )+\frac {\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}+\frac {1}{2} \left (-\frac {\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}-\frac {3}{8} \left (-\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}+\frac {1}{6} \int \sin \left (i a+i b x+\frac {\pi }{2}\right )^4dx\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \int \cosh ^2(a+b x)dx+\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )-\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}\right )-\frac {\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}\right )+\frac {\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}+\frac {1}{2} \left (-\frac {\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}-\frac {3}{8} \left (-\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}+\frac {1}{6} \left (\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \int \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx\right )\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{2} \left (-\frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}\right )+\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )-\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}\right )-\frac {\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}\right )+\frac {\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^5(a+b x)}{10 b}+\frac {1}{2} \left (-\frac {\sinh ^3(a+b x) \cosh ^5(a+b x)}{8 b}-\frac {3}{8} \left (\frac {1}{6} \left (\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )\right )-\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}\right )\right )\) |
(Cosh[a + b*x]^5*Sinh[a + b*x]^5)/(10*b) + (-1/8*(Cosh[a + b*x]^5*Sinh[a + b*x]^3)/b - (3*(-1/6*(Cosh[a + b*x]^5*Sinh[a + b*x])/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 )/6))/8)/2
3.1.20.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
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]
Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.81
\[\frac {\frac {\sinh \left (b x +a \right )^{5} \cosh \left (b x +a \right )^{5}}{10}-\frac {\sinh \left (b x +a \right )^{3} \cosh \left (b x +a \right )^{5}}{16}+\frac {\sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{5}}{32}-\frac {\left (\frac {\cosh \left (b x +a \right )^{3}}{4}+\frac {3 \cosh \left (b x +a \right )}{8}\right ) \sinh \left (b x +a \right )}{32}-\frac {3 b x}{256}-\frac {3 a}{256}}{b}\]
1/b*(1/10*sinh(b*x+a)^5*cosh(b*x+a)^5-1/16*sinh(b*x+a)^3*cosh(b*x+a)^5+1/3 2*sinh(b*x+a)*cosh(b*x+a)^5-1/32*(1/4*cosh(b*x+a)^3+3/8*cosh(b*x+a))*sinh( b*x+a)-3/256*b*x-3/256*a)
Time = 0.25 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.74 \[ \int \cosh ^4(a+b x) \sinh ^6(a+b x) \, dx=\frac {5 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{9} + 10 \, {\left (6 \, \cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{7} + {\left (126 \, \cosh \left (b x + a\right )^{5} - 70 \, \cosh \left (b x + a\right )^{3} - 15 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{5} + 10 \, {\left (6 \, \cosh \left (b x + a\right )^{7} - 7 \, \cosh \left (b x + a\right )^{5} - 5 \, \cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 30 \, b x + 5 \, {\left (\cosh \left (b x + a\right )^{9} - 2 \, \cosh \left (b x + a\right )^{7} - 3 \, \cosh \left (b x + a\right )^{5} + 8 \, \cosh \left (b x + a\right )^{3} + 2 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )}{2560 \, b} \]
1/2560*(5*cosh(b*x + a)*sinh(b*x + a)^9 + 10*(6*cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a)^7 + (126*cosh(b*x + a)^5 - 70*cosh(b*x + a)^3 - 15*co sh(b*x + a))*sinh(b*x + a)^5 + 10*(6*cosh(b*x + a)^7 - 7*cosh(b*x + a)^5 - 5*cosh(b*x + a)^3 + 4*cosh(b*x + a))*sinh(b*x + a)^3 - 30*b*x + 5*(cosh(b *x + a)^9 - 2*cosh(b*x + a)^7 - 3*cosh(b*x + a)^5 + 8*cosh(b*x + a)^3 + 2* cosh(b*x + a))*sinh(b*x + a))/b
Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (100) = 200\).
Time = 1.30 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.04 \[ \int \cosh ^4(a+b x) \sinh ^6(a+b x) \, dx=\begin {cases} \frac {3 x \sinh ^{10}{\left (a + b x \right )}}{256} - \frac {15 x \sinh ^{8}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{256} + \frac {15 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{128} - \frac {15 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{128} + \frac {15 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{8}{\left (a + b x \right )}}{256} - \frac {3 x \cosh ^{10}{\left (a + b x \right )}}{256} - \frac {3 \sinh ^{9}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{256 b} + \frac {7 \sinh ^{7}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{128 b} + \frac {\sinh ^{5}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{10 b} - \frac {7 \sinh ^{3}{\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{128 b} + \frac {3 \sinh {\left (a + b x \right )} \cosh ^{9}{\left (a + b x \right )}}{256 b} & \text {for}\: b \neq 0 \\x \sinh ^{6}{\left (a \right )} \cosh ^{4}{\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((3*x*sinh(a + b*x)**10/256 - 15*x*sinh(a + b*x)**8*cosh(a + b*x) **2/256 + 15*x*sinh(a + b*x)**6*cosh(a + b*x)**4/128 - 15*x*sinh(a + b*x)* *4*cosh(a + b*x)**6/128 + 15*x*sinh(a + b*x)**2*cosh(a + b*x)**8/256 - 3*x *cosh(a + b*x)**10/256 - 3*sinh(a + b*x)**9*cosh(a + b*x)/(256*b) + 7*sinh (a + b*x)**7*cosh(a + b*x)**3/(128*b) + sinh(a + b*x)**5*cosh(a + b*x)**5/ (10*b) - 7*sinh(a + b*x)**3*cosh(a + b*x)**7/(128*b) + 3*sinh(a + b*x)*cos h(a + b*x)**9/(256*b), Ne(b, 0)), (x*sinh(a)**6*cosh(a)**4, True))
Time = 0.18 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.17 \[ \int \cosh ^4(a+b x) \sinh ^6(a+b x) \, dx=-\frac {{\left (5 \, e^{\left (-2 \, b x - 2 \, a\right )} + 10 \, e^{\left (-4 \, b x - 4 \, a\right )} - 40 \, e^{\left (-6 \, b x - 6 \, a\right )} - 20 \, e^{\left (-8 \, b x - 8 \, a\right )} - 2\right )} e^{\left (10 \, b x + 10 \, a\right )}}{20480 \, b} - \frac {3 \, {\left (b x + a\right )}}{256 \, b} - \frac {20 \, e^{\left (-2 \, b x - 2 \, a\right )} + 40 \, e^{\left (-4 \, b x - 4 \, a\right )} - 10 \, e^{\left (-6 \, b x - 6 \, a\right )} - 5 \, e^{\left (-8 \, b x - 8 \, a\right )} + 2 \, e^{\left (-10 \, b x - 10 \, a\right )}}{20480 \, b} \]
-1/20480*(5*e^(-2*b*x - 2*a) + 10*e^(-4*b*x - 4*a) - 40*e^(-6*b*x - 6*a) - 20*e^(-8*b*x - 8*a) - 2)*e^(10*b*x + 10*a)/b - 3/256*(b*x + a)/b - 1/2048 0*(20*e^(-2*b*x - 2*a) + 40*e^(-4*b*x - 4*a) - 10*e^(-6*b*x - 6*a) - 5*e^( -8*b*x - 8*a) + 2*e^(-10*b*x - 10*a))/b
Time = 0.28 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.27 \[ \int \cosh ^4(a+b x) \sinh ^6(a+b x) \, dx=-\frac {3}{256} \, x + \frac {e^{\left (10 \, b x + 10 \, a\right )}}{10240 \, b} - \frac {e^{\left (8 \, b x + 8 \, a\right )}}{4096 \, b} - \frac {e^{\left (6 \, b x + 6 \, a\right )}}{2048 \, b} + \frac {e^{\left (4 \, b x + 4 \, a\right )}}{512 \, b} + \frac {e^{\left (2 \, b x + 2 \, a\right )}}{1024 \, b} - \frac {e^{\left (-2 \, b x - 2 \, a\right )}}{1024 \, b} - \frac {e^{\left (-4 \, b x - 4 \, a\right )}}{512 \, b} + \frac {e^{\left (-6 \, b x - 6 \, a\right )}}{2048 \, b} + \frac {e^{\left (-8 \, b x - 8 \, a\right )}}{4096 \, b} - \frac {e^{\left (-10 \, b x - 10 \, a\right )}}{10240 \, b} \]
-3/256*x + 1/10240*e^(10*b*x + 10*a)/b - 1/4096*e^(8*b*x + 8*a)/b - 1/2048 *e^(6*b*x + 6*a)/b + 1/512*e^(4*b*x + 4*a)/b + 1/1024*e^(2*b*x + 2*a)/b - 1/1024*e^(-2*b*x - 2*a)/b - 1/512*e^(-4*b*x - 4*a)/b + 1/2048*e^(-6*b*x - 6*a)/b + 1/4096*e^(-8*b*x - 8*a)/b - 1/10240*e^(-10*b*x - 10*a)/b
Time = 2.39 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.58 \[ \int \cosh ^4(a+b x) \sinh ^6(a+b x) \, dx=\frac {20\,\mathrm {sinh}\left (2\,a+2\,b\,x\right )+40\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )-10\,\mathrm {sinh}\left (6\,a+6\,b\,x\right )-5\,\mathrm {sinh}\left (8\,a+8\,b\,x\right )+2\,\mathrm {sinh}\left (10\,a+10\,b\,x\right )-120\,b\,x}{10240\,b} \]