Integrand size = 13, antiderivative size = 57 \[ \int \frac {\sinh ^8(x)}{a+a \cosh (x)} \, dx=\frac {5 x}{16 a}-\frac {5 \cosh (x) \sinh (x)}{16 a}+\frac {5 \cosh (x) \sinh ^3(x)}{24 a}-\frac {\cosh (x) \sinh ^5(x)}{6 a}+\frac {\sinh ^7(x)}{7 a} \]
5/16*x/a-5/16*cosh(x)*sinh(x)/a+5/24*cosh(x)*sinh(x)^3/a-1/6*cosh(x)*sinh( x)^5/a+1/7*sinh(x)^7/a
Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {\sinh ^8(x)}{a+a \cosh (x)} \, dx=\frac {420 x-105 \sinh (x)-315 \sinh (2 x)+63 \sinh (3 x)+63 \sinh (4 x)-21 \sinh (5 x)-7 \sinh (6 x)+3 \sinh (7 x)}{1344 a} \]
(420*x - 105*Sinh[x] - 315*Sinh[2*x] + 63*Sinh[3*x] + 63*Sinh[4*x] - 21*Si nh[5*x] - 7*Sinh[6*x] + 3*Sinh[7*x])/(1344*a)
Time = 0.37 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.05, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3161, 25, 3042, 25, 3115, 3042, 3115, 25, 3042, 25, 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.
\(\displaystyle \int \frac {\sinh ^8(x)}{a \cosh (x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos \left (-\frac {\pi }{2}+i x\right )^8}{a-a \sin \left (-\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 3161 |
\(\displaystyle \frac {\int -\sinh ^6(x)dx}{a}+\frac {\sinh ^7(x)}{7 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sinh ^7(x)}{7 a}-\frac {\int \sinh ^6(x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh ^7(x)}{7 a}-\frac {\int -\sin (i x)^6dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sinh ^7(x)}{7 a}+\frac {\int \sin (i x)^6dx}{a}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\frac {5}{6} \int \sinh ^4(x)dx-\frac {1}{6} \sinh ^5(x) \cosh (x)}{a}+\frac {\sinh ^7(x)}{7 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh ^7(x)}{7 a}+\frac {-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \int \sin (i x)^4dx}{a}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\frac {5}{6} \left (\frac {3}{4} \int -\sinh ^2(x)dx+\frac {1}{4} \sinh ^3(x) \cosh (x)\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)}{a}+\frac {\sinh ^7(x)}{7 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)-\frac {3}{4} \int \sinh ^2(x)dx\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)}{a}+\frac {\sinh ^7(x)}{7 a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh ^7(x)}{7 a}+\frac {-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)-\frac {3}{4} \int -\sin (i x)^2dx\right )}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sinh ^7(x)}{7 a}+\frac {-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)+\frac {3}{4} \int \sin (i x)^2dx\right )}{a}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {1}{4} \sinh ^3(x) \cosh (x)\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)}{a}+\frac {\sinh ^7(x)}{7 a}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\sinh ^7(x)}{7 a}+\frac {\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)+\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)}{a}\) |
Sinh[x]^7/(7*a) + (-1/6*(Cosh[x]*Sinh[x]^5) + (5*((Cosh[x]*Sinh[x]^3)/4 + (3*(x/2 - (Cosh[x]*Sinh[x])/2))/4))/6)/a
3.2.52.3.1 Defintions of rubi rules used
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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[g*((g*Cos[e + f*x])^(p - 1)/(b*f*(p - 1))), x] + Si mp[g^2/a Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g}, x ] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*p]
Leaf count of result is larger than twice the leaf count of optimal. \(131\) vs. \(2(47)=94\).
Time = 304.86 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.32
method | result | size |
risch | \(\frac {5 x}{16 a}+\frac {{\mathrm e}^{7 x}}{896 a}-\frac {{\mathrm e}^{6 x}}{384 a}-\frac {{\mathrm e}^{5 x}}{128 a}+\frac {3 \,{\mathrm e}^{4 x}}{128 a}+\frac {3 \,{\mathrm e}^{3 x}}{128 a}-\frac {15 \,{\mathrm e}^{2 x}}{128 a}-\frac {5 \,{\mathrm e}^{x}}{128 a}+\frac {5 \,{\mathrm e}^{-x}}{128 a}+\frac {15 \,{\mathrm e}^{-2 x}}{128 a}-\frac {3 \,{\mathrm e}^{-3 x}}{128 a}-\frac {3 \,{\mathrm e}^{-4 x}}{128 a}+\frac {{\mathrm e}^{-5 x}}{128 a}+\frac {{\mathrm e}^{-6 x}}{384 a}-\frac {{\mathrm e}^{-7 x}}{896 a}\) | \(132\) |
default | \(\frac {-\frac {1}{7 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{7}}-\frac {2}{3 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{6}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {11}{24 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {1}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {5}{16 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {5 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{16}-\frac {1}{7 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{7}}+\frac {2}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{6}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}+\frac {11}{24 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {5}{16 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {5 \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{16}}{a}\) | \(165\) |
5/16*x/a+1/896/a*exp(7*x)-1/384/a*exp(6*x)-1/128/a*exp(5*x)+3/128/a*exp(4* x)+3/128/a*exp(3*x)-15/128/a*exp(2*x)-5/128/a*exp(x)+5/128/a*exp(-x)+15/12 8/a*exp(-2*x)-3/128/a*exp(-3*x)-3/128/a*exp(-4*x)+1/128/a*exp(-5*x)+1/384/ a*exp(-6*x)-1/896/a*exp(-7*x)
Leaf count of result is larger than twice the leaf count of optimal. 101 vs. \(2 (47) = 94\).
Time = 0.26 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.77 \[ \int \frac {\sinh ^8(x)}{a+a \cosh (x)} \, dx=\frac {3 \, \sinh \left (x\right )^{7} + 21 \, {\left (3 \, \cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{5} + 7 \, {\left (15 \, \cosh \left (x\right )^{4} - 20 \, \cosh \left (x\right )^{3} - 30 \, \cosh \left (x\right )^{2} + 36 \, \cosh \left (x\right ) + 9\right )} \sinh \left (x\right )^{3} + 21 \, {\left (\cosh \left (x\right )^{6} - 2 \, \cosh \left (x\right )^{5} - 5 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )^{2} - 30 \, \cosh \left (x\right ) - 5\right )} \sinh \left (x\right ) + 420 \, x}{1344 \, a} \]
1/1344*(3*sinh(x)^7 + 21*(3*cosh(x)^2 - 2*cosh(x) - 1)*sinh(x)^5 + 7*(15*c osh(x)^4 - 20*cosh(x)^3 - 30*cosh(x)^2 + 36*cosh(x) + 9)*sinh(x)^3 + 21*(c osh(x)^6 - 2*cosh(x)^5 - 5*cosh(x)^4 + 12*cosh(x)^3 + 9*cosh(x)^2 - 30*cos h(x) - 5)*sinh(x) + 420*x)/a
Leaf count of result is larger than twice the leaf count of optimal. 1253 vs. \(2 (51) = 102\).
Time = 2.83 (sec) , antiderivative size = 1253, normalized size of antiderivative = 21.98 \[ \int \frac {\sinh ^8(x)}{a+a \cosh (x)} \, dx=\text {Too large to display} \]
105*x*tanh(x/2)**14/(336*a*tanh(x/2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*t anh(x/2)**10 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x /2)**4 + 2352*a*tanh(x/2)**2 - 336*a) - 735*x*tanh(x/2)**12/(336*a*tanh(x/ 2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)** 8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)**2 - 336 *a) + 2205*x*tanh(x/2)**10/(336*a*tanh(x/2)**14 - 2352*a*tanh(x/2)**12 + 7 056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a *tanh(x/2)**4 + 2352*a*tanh(x/2)**2 - 336*a) - 3675*x*tanh(x/2)**8/(336*a* tanh(x/2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh (x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)** 2 - 336*a) + 3675*x*tanh(x/2)**6/(336*a*tanh(x/2)**14 - 2352*a*tanh(x/2)** 12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)**2 - 336*a) - 2205*x*tanh(x/2)**4/( 336*a*tanh(x/2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760* a*tanh(x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh( x/2)**2 - 336*a) + 735*x*tanh(x/2)**2/(336*a*tanh(x/2)**14 - 2352*a*tanh(x /2)**12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/2)* *6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)**2 - 336*a) - 105*x/(336*a*tan h(x/2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/ 2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)**...
Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (47) = 94\).
Time = 0.18 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.79 \[ \int \frac {\sinh ^8(x)}{a+a \cosh (x)} \, dx=-\frac {{\left (7 \, e^{\left (-x\right )} + 21 \, e^{\left (-2 \, x\right )} - 63 \, e^{\left (-3 \, x\right )} - 63 \, e^{\left (-4 \, x\right )} + 315 \, e^{\left (-5 \, x\right )} + 105 \, e^{\left (-6 \, x\right )} - 3\right )} e^{\left (7 \, x\right )}}{2688 \, a} + \frac {5 \, x}{16 \, a} + \frac {105 \, e^{\left (-x\right )} + 315 \, e^{\left (-2 \, x\right )} - 63 \, e^{\left (-3 \, x\right )} - 63 \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} + 7 \, e^{\left (-6 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}}{2688 \, a} \]
-1/2688*(7*e^(-x) + 21*e^(-2*x) - 63*e^(-3*x) - 63*e^(-4*x) + 315*e^(-5*x) + 105*e^(-6*x) - 3)*e^(7*x)/a + 5/16*x/a + 1/2688*(105*e^(-x) + 315*e^(-2 *x) - 63*e^(-3*x) - 63*e^(-4*x) + 21*e^(-5*x) + 7*e^(-6*x) - 3*e^(-7*x))/a
Time = 0.25 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.58 \[ \int \frac {\sinh ^8(x)}{a+a \cosh (x)} \, dx=\frac {{\left (105 \, e^{\left (6 \, x\right )} + 315 \, e^{\left (5 \, x\right )} - 63 \, e^{\left (4 \, x\right )} - 63 \, e^{\left (3 \, x\right )} + 21 \, e^{\left (2 \, x\right )} + 7 \, e^{x} - 3\right )} e^{\left (-7 \, x\right )} + 840 \, x + 3 \, e^{\left (7 \, x\right )} - 7 \, e^{\left (6 \, x\right )} - 21 \, e^{\left (5 \, x\right )} + 63 \, e^{\left (4 \, x\right )} + 63 \, e^{\left (3 \, x\right )} - 315 \, e^{\left (2 \, x\right )} - 105 \, e^{x}}{2688 \, a} \]
1/2688*((105*e^(6*x) + 315*e^(5*x) - 63*e^(4*x) - 63*e^(3*x) + 21*e^(2*x) + 7*e^x - 3)*e^(-7*x) + 840*x + 3*e^(7*x) - 7*e^(6*x) - 21*e^(5*x) + 63*e^ (4*x) + 63*e^(3*x) - 315*e^(2*x) - 105*e^x)/a
Time = 1.98 (sec) , antiderivative size = 131, normalized size of antiderivative = 2.30 \[ \int \frac {\sinh ^8(x)}{a+a \cosh (x)} \, dx=\frac {5\,{\mathrm {e}}^{-x}}{128\,a}+\frac {15\,{\mathrm {e}}^{-2\,x}}{128\,a}-\frac {15\,{\mathrm {e}}^{2\,x}}{128\,a}-\frac {3\,{\mathrm {e}}^{-3\,x}}{128\,a}+\frac {3\,{\mathrm {e}}^{3\,x}}{128\,a}-\frac {3\,{\mathrm {e}}^{-4\,x}}{128\,a}+\frac {3\,{\mathrm {e}}^{4\,x}}{128\,a}+\frac {{\mathrm {e}}^{-5\,x}}{128\,a}-\frac {{\mathrm {e}}^{5\,x}}{128\,a}+\frac {{\mathrm {e}}^{-6\,x}}{384\,a}-\frac {{\mathrm {e}}^{6\,x}}{384\,a}-\frac {{\mathrm {e}}^{-7\,x}}{896\,a}+\frac {{\mathrm {e}}^{7\,x}}{896\,a}+\frac {5\,x}{16\,a}-\frac {5\,{\mathrm {e}}^x}{128\,a} \]
(5*exp(-x))/(128*a) + (15*exp(-2*x))/(128*a) - (15*exp(2*x))/(128*a) - (3* exp(-3*x))/(128*a) + (3*exp(3*x))/(128*a) - (3*exp(-4*x))/(128*a) + (3*exp (4*x))/(128*a) + exp(-5*x)/(128*a) - exp(5*x)/(128*a) + exp(-6*x)/(384*a) - exp(6*x)/(384*a) - exp(-7*x)/(896*a) + exp(7*x)/(896*a) + (5*x)/(16*a) - (5*exp(x))/(128*a)