Integrand size = 17, antiderivative size = 134 \[ \int \cosh ^6(a+b x) \sinh ^6(a+b x) \, dx=-\frac {5 x}{1024}-\frac {5 \cosh (a+b x) \sinh (a+b x)}{1024 b}-\frac {5 \cosh ^3(a+b x) \sinh (a+b x)}{1536 b}-\frac {\cosh ^5(a+b x) \sinh (a+b x)}{384 b}+\frac {\cosh ^7(a+b x) \sinh (a+b x)}{64 b}-\frac {\cosh ^7(a+b x) \sinh ^3(a+b x)}{24 b}+\frac {\cosh ^7(a+b x) \sinh ^5(a+b x)}{12 b} \]
-5/1024*x-5/1024*cosh(b*x+a)*sinh(b*x+a)/b-5/1536*cosh(b*x+a)^3*sinh(b*x+a )/b-1/384*cosh(b*x+a)^5*sinh(b*x+a)/b+1/64*cosh(b*x+a)^7*sinh(b*x+a)/b-1/2 4*cosh(b*x+a)^7*sinh(b*x+a)^3/b+1/12*cosh(b*x+a)^7*sinh(b*x+a)^5/b
Time = 0.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.32 \[ \int \cosh ^6(a+b x) \sinh ^6(a+b x) \, dx=\frac {-120 a-120 b x+45 \sinh (4 (a+b x))-9 \sinh (8 (a+b x))+\sinh (12 (a+b x))}{24576 b} \]
(-120*a - 120*b*x + 45*Sinh[4*(a + b*x)] - 9*Sinh[8*(a + b*x)] + Sinh[12*( a + b*x)])/(24576*b)
Time = 0.70 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.19, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.941, Rules used = {3042, 25, 3048, 3042, 3048, 25, 3042, 25, 3048, 3042, 3115, 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.
| \(\displaystyle \int \sinh ^6(a+b x) \cosh ^6(a+b x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (i a+i b x)^6 \left (-\cos (i a+i b x)^6\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \cos (i a+i b x)^6 \sin (i a+i b x)^6dx\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {5}{12} \int \cosh ^6(a+b x) \sinh ^4(a+b x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {5}{12} \int \cos (i a+i b x)^6 \sin (i a+i b x)^4dx\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {5}{12} \left (\frac {3}{10} \int -\cosh ^6(a+b x) \sinh ^2(a+b x)dx+\frac {\sinh ^3(a+b x) \cosh ^7(a+b x)}{10 b}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {5}{12} \left (\frac {\sinh ^3(a+b x) \cosh ^7(a+b x)}{10 b}-\frac {3}{10} \int \cosh ^6(a+b x) \sinh ^2(a+b x)dx\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {5}{12} \left (\frac {\sinh ^3(a+b x) \cosh ^7(a+b x)}{10 b}-\frac {3}{10} \int -\cos (i a+i b x)^6 \sin (i a+i b x)^2dx\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {5}{12} \left (\frac {\sinh ^3(a+b x) \cosh ^7(a+b x)}{10 b}+\frac {3}{10} \int \cos (i a+i b x)^6 \sin (i a+i b x)^2dx\right )\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \int \cosh ^6(a+b x)dx-\frac {\sinh (a+b x) \cosh ^7(a+b x)}{8 b}\right )+\frac {\sinh ^3(a+b x) \cosh ^7(a+b x)}{10 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {5}{12} \left (\frac {\sinh ^3(a+b x) \cosh ^7(a+b x)}{10 b}+\frac {3}{10} \left (-\frac {\sinh (a+b x) \cosh ^7(a+b x)}{8 b}+\frac {1}{8} \int \sin \left (i a+i b x+\frac {\pi }{2}\right )^6dx\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \cosh ^4(a+b x)dx+\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}\right )-\frac {\sinh (a+b x) \cosh ^7(a+b x)}{8 b}\right )+\frac {\sinh ^3(a+b x) \cosh ^7(a+b x)}{10 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {5}{12} \left (\frac {\sinh ^3(a+b x) \cosh ^7(a+b x)}{10 b}+\frac {3}{10} \left (-\frac {\sinh (a+b x) \cosh ^7(a+b x)}{8 b}+\frac {1}{8} \left (\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}+\frac {5}{6} \int \sin \left (i a+i b x+\frac {\pi }{2}\right )^4dx\right )\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cosh ^2(a+b x)dx+\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )+\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}\right )-\frac {\sinh (a+b x) \cosh ^7(a+b x)}{8 b}\right )+\frac {\sinh ^3(a+b x) \cosh ^7(a+b x)}{10 b}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {5}{12} \left (\frac {\sinh ^3(a+b x) \cosh ^7(a+b x)}{10 b}+\frac {3}{10} \left (-\frac {\sinh (a+b x) \cosh ^7(a+b x)}{8 b}+\frac {1}{8} \left (\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}+\frac {5}{6} \left (\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \int \sin \left (i a+i b x+\frac {\pi }{2}\right )^2dx\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}\right )+\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}\right )+\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}\right )-\frac {\sinh (a+b x) \cosh ^7(a+b x)}{8 b}\right )+\frac {\sinh ^3(a+b x) \cosh ^7(a+b x)}{10 b}\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\sinh ^5(a+b x) \cosh ^7(a+b x)}{12 b}-\frac {5}{12} \left (\frac {\sinh ^3(a+b x) \cosh ^7(a+b x)}{10 b}+\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sinh (a+b x) \cosh ^5(a+b x)}{6 b}+\frac {5}{6} \left (\frac {\sinh (a+b x) \cosh ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sinh (a+b x) \cosh (a+b x)}{2 b}+\frac {x}{2}\right )\right )\right )-\frac {\sinh (a+b x) \cosh ^7(a+b x)}{8 b}\right )\right )\) |
(Cosh[a + b*x]^7*Sinh[a + b*x]^5)/(12*b) - (5*((Cosh[a + b*x]^7*Sinh[a + b *x]^3)/(10*b) + (3*(-1/8*(Cosh[a + b*x]^7*Sinh[a + b*x])/b + ((Cosh[a + b* x]^5*Sinh[a + b*x])/(6*b) + (5*((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))/10))/12
3.1.23.3.1 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 = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.76
\[\frac {\frac {\sinh \left (b x +a \right )^{5} \cosh \left (b x +a \right )^{7}}{12}-\frac {\sinh \left (b x +a \right )^{3} \cosh \left (b x +a \right )^{7}}{24}+\frac {\sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{7}}{64}-\frac {\left (\frac {\cosh \left (b x +a \right )^{5}}{6}+\frac {5 \cosh \left (b x +a \right )^{3}}{24}+\frac {5 \cosh \left (b x +a \right )}{16}\right ) \sinh \left (b x +a \right )}{64}-\frac {5 b x}{1024}-\frac {5 a}{1024}}{b}\]
1/b*(1/12*sinh(b*x+a)^5*cosh(b*x+a)^7-1/24*sinh(b*x+a)^3*cosh(b*x+a)^7+1/6 4*sinh(b*x+a)*cosh(b*x+a)^7-1/64*(1/6*cosh(b*x+a)^5+5/24*cosh(b*x+a)^3+5/1 6*cosh(b*x+a))*sinh(b*x+a)-5/1024*b*x-5/1024*a)
Time = 0.25 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.34 \[ \int \cosh ^6(a+b x) \sinh ^6(a+b x) \, dx=\frac {55 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right )^{9} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{11} + 18 \, {\left (11 \, \cosh \left (b x + a\right )^{5} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{7} + 18 \, {\left (11 \, \cosh \left (b x + a\right )^{7} - 7 \, \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )^{5} + {\left (55 \, \cosh \left (b x + a\right )^{9} - 126 \, \cosh \left (b x + a\right )^{5} + 45 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} - 30 \, b x + 3 \, {\left (\cosh \left (b x + a\right )^{11} - 6 \, \cosh \left (b x + a\right )^{7} + 15 \, \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )}{6144 \, b} \]
1/6144*(55*cosh(b*x + a)^3*sinh(b*x + a)^9 + 3*cosh(b*x + a)*sinh(b*x + a) ^11 + 18*(11*cosh(b*x + a)^5 - cosh(b*x + a))*sinh(b*x + a)^7 + 18*(11*cos h(b*x + a)^7 - 7*cosh(b*x + a)^3)*sinh(b*x + a)^5 + (55*cosh(b*x + a)^9 - 126*cosh(b*x + a)^5 + 45*cosh(b*x + a))*sinh(b*x + a)^3 - 30*b*x + 3*(cosh (b*x + a)^11 - 6*cosh(b*x + a)^7 + 15*cosh(b*x + a)^3)*sinh(b*x + a))/b
Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (121) = 242\).
Time = 2.62 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.07 \[ \int \cosh ^6(a+b x) \sinh ^6(a+b x) \, dx=\begin {cases} - \frac {5 x \sinh ^{12}{\left (a + b x \right )}}{1024} + \frac {15 x \sinh ^{10}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{512} - \frac {75 x \sinh ^{8}{\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{1024} + \frac {25 x \sinh ^{6}{\left (a + b x \right )} \cosh ^{6}{\left (a + b x \right )}}{256} - \frac {75 x \sinh ^{4}{\left (a + b x \right )} \cosh ^{8}{\left (a + b x \right )}}{1024} + \frac {15 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{10}{\left (a + b x \right )}}{512} - \frac {5 x \cosh ^{12}{\left (a + b x \right )}}{1024} + \frac {5 \sinh ^{11}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{1024 b} - \frac {85 \sinh ^{9}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{3072 b} + \frac {33 \sinh ^{7}{\left (a + b x \right )} \cosh ^{5}{\left (a + b x \right )}}{512 b} + \frac {33 \sinh ^{5}{\left (a + b x \right )} \cosh ^{7}{\left (a + b x \right )}}{512 b} - \frac {85 \sinh ^{3}{\left (a + b x \right )} \cosh ^{9}{\left (a + b x \right )}}{3072 b} + \frac {5 \sinh {\left (a + b x \right )} \cosh ^{11}{\left (a + b x \right )}}{1024 b} & \text {for}\: b \neq 0 \\x \sinh ^{6}{\left (a \right )} \cosh ^{6}{\left (a \right )} & \text {otherwise} \end {cases} \]
Piecewise((-5*x*sinh(a + b*x)**12/1024 + 15*x*sinh(a + b*x)**10*cosh(a + b *x)**2/512 - 75*x*sinh(a + b*x)**8*cosh(a + b*x)**4/1024 + 25*x*sinh(a + b *x)**6*cosh(a + b*x)**6/256 - 75*x*sinh(a + b*x)**4*cosh(a + b*x)**8/1024 + 15*x*sinh(a + b*x)**2*cosh(a + b*x)**10/512 - 5*x*cosh(a + b*x)**12/1024 + 5*sinh(a + b*x)**11*cosh(a + b*x)/(1024*b) - 85*sinh(a + b*x)**9*cosh(a + b*x)**3/(3072*b) + 33*sinh(a + b*x)**7*cosh(a + b*x)**5/(512*b) + 33*si nh(a + b*x)**5*cosh(a + b*x)**7/(512*b) - 85*sinh(a + b*x)**3*cosh(a + b*x )**9/(3072*b) + 5*sinh(a + b*x)*cosh(a + b*x)**11/(1024*b), Ne(b, 0)), (x* sinh(a)**6*cosh(a)**6, True))
Time = 0.17 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.64 \[ \int \cosh ^6(a+b x) \sinh ^6(a+b x) \, dx=-\frac {{\left (9 \, e^{\left (-4 \, b x - 4 \, a\right )} - 45 \, e^{\left (-8 \, b x - 8 \, a\right )} - 1\right )} e^{\left (12 \, b x + 12 \, a\right )}}{49152 \, b} - \frac {5 \, {\left (b x + a\right )}}{1024 \, b} - \frac {45 \, e^{\left (-4 \, b x - 4 \, a\right )} - 9 \, e^{\left (-8 \, b x - 8 \, a\right )} + e^{\left (-12 \, b x - 12 \, a\right )}}{49152 \, b} \]
-1/49152*(9*e^(-4*b*x - 4*a) - 45*e^(-8*b*x - 8*a) - 1)*e^(12*b*x + 12*a)/ b - 5/1024*(b*x + a)/b - 1/49152*(45*e^(-4*b*x - 4*a) - 9*e^(-8*b*x - 8*a) + e^(-12*b*x - 12*a))/b
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.66 \[ \int \cosh ^6(a+b x) \sinh ^6(a+b x) \, dx=-\frac {5}{1024} \, x + \frac {e^{\left (12 \, b x + 12 \, a\right )}}{49152 \, b} - \frac {3 \, e^{\left (8 \, b x + 8 \, a\right )}}{16384 \, b} + \frac {15 \, e^{\left (4 \, b x + 4 \, a\right )}}{16384 \, b} - \frac {15 \, e^{\left (-4 \, b x - 4 \, a\right )}}{16384 \, b} + \frac {3 \, e^{\left (-8 \, b x - 8 \, a\right )}}{16384 \, b} - \frac {e^{\left (-12 \, b x - 12 \, a\right )}}{49152 \, b} \]
-5/1024*x + 1/49152*e^(12*b*x + 12*a)/b - 3/16384*e^(8*b*x + 8*a)/b + 15/1 6384*e^(4*b*x + 4*a)/b - 15/16384*e^(-4*b*x - 4*a)/b + 3/16384*e^(-8*b*x - 8*a)/b - 1/49152*e^(-12*b*x - 12*a)/b
Time = 2.69 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.31 \[ \int \cosh ^6(a+b x) \sinh ^6(a+b x) \, dx=\frac {\frac {15\,\mathrm {sinh}\left (4\,a+4\,b\,x\right )}{8192}-\frac {3\,\mathrm {sinh}\left (8\,a+8\,b\,x\right )}{8192}+\frac {\mathrm {sinh}\left (12\,a+12\,b\,x\right )}{24576}}{b}-\frac {5\,x}{1024} \]