Integrand size = 13, antiderivative size = 44 \[ \int \frac {\sinh ^6(x)}{a+a \cosh (x)} \, dx=-\frac {3 x}{8 a}+\frac {3 \cosh (x) \sinh (x)}{8 a}-\frac {\cosh (x) \sinh ^3(x)}{4 a}+\frac {\sinh ^5(x)}{5 a} \] Output:
-3/8*x/a+3/8*cosh(x)*sinh(x)/a-1/4*cosh(x)*sinh(x)^3/a+1/5*sinh(x)^5/a
Time = 0.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.89 \[ \int \frac {\sinh ^6(x)}{a+a \cosh (x)} \, dx=\frac {-60 x+20 \sinh (x)+40 \sinh (2 x)-10 \sinh (3 x)-5 \sinh (4 x)+2 \sinh (5 x)}{160 a} \] Input:
Integrate[Sinh[x]^6/(a + a*Cosh[x]),x]
Output:
(-60*x + 20*Sinh[x] + 40*Sinh[2*x] - 10*Sinh[3*x] - 5*Sinh[4*x] + 2*Sinh[5 *x])/(160*a)
Time = 0.33 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.769, Rules used = {3042, 25, 3161, 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.
\(\displaystyle \int \frac {\sinh ^6(x)}{a \cosh (x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\cos \left (-\frac {\pi }{2}+i x\right )^6}{a-a \sin \left (-\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cos \left (i x-\frac {\pi }{2}\right )^6}{a-a \sin \left (i x-\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3161 |
\(\displaystyle \frac {\sinh ^5(x)}{5 a}-\frac {\int \sinh ^4(x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh ^5(x)}{5 a}-\frac {\int \sin (i x)^4dx}{a}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\sinh ^5(x)}{5 a}-\frac {\frac {3}{4} \int -\sinh ^2(x)dx+\frac {1}{4} \sinh ^3(x) \cosh (x)}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sinh ^5(x)}{5 a}-\frac {\frac {1}{4} \sinh ^3(x) \cosh (x)-\frac {3}{4} \int \sinh ^2(x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sinh ^5(x)}{5 a}-\frac {\frac {1}{4} \sinh ^3(x) \cosh (x)-\frac {3}{4} \int -\sin (i x)^2dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\sinh ^5(x)}{5 a}-\frac {\frac {1}{4} \sinh ^3(x) \cosh (x)+\frac {3}{4} \int \sin (i x)^2dx}{a}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {\sinh ^5(x)}{5 a}-\frac {\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {1}{4} \sinh ^3(x) \cosh (x)}{a}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {\sinh ^5(x)}{5 a}-\frac {\frac {1}{4} \sinh ^3(x) \cosh (x)+\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )}{a}\) |
Input:
Int[Sinh[x]^6/(a + a*Cosh[x]),x]
Output:
Sinh[x]^5/(5*a) - ((Cosh[x]*Sinh[x]^3)/4 + (3*(x/2 - (Cosh[x]*Sinh[x])/2)) /4)/a
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. \(95\) vs. \(2(36)=72\).
Time = 61.89 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.18
method | result | size |
risch | \(-\frac {3 x}{8 a}+\frac {{\mathrm e}^{5 x}}{160 a}-\frac {{\mathrm e}^{4 x}}{64 a}-\frac {{\mathrm e}^{3 x}}{32 a}+\frac {{\mathrm e}^{2 x}}{8 a}+\frac {{\mathrm e}^{x}}{16 a}-\frac {{\mathrm e}^{-x}}{16 a}-\frac {{\mathrm e}^{-2 x}}{8 a}+\frac {{\mathrm e}^{-3 x}}{32 a}+\frac {{\mathrm e}^{-4 x}}{64 a}-\frac {{\mathrm e}^{-5 x}}{160 a}\) | \(96\) |
default | \(\frac {-\frac {1}{5 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {3}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {3}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8}-\frac {1}{5 \left (1+\tanh \left (\frac {x}{2}\right )\right )^{5}}+\frac {3}{4 \left (1+\tanh \left (\frac {x}{2}\right )\right )^{4}}-\frac {3}{4 \left (1+\tanh \left (\frac {x}{2}\right )\right )^{3}}-\frac {1}{4 \left (1+\tanh \left (\frac {x}{2}\right )\right )^{2}}+\frac {3}{8 \left (1+\tanh \left (\frac {x}{2}\right )\right )}-\frac {3 \ln \left (1+\tanh \left (\frac {x}{2}\right )\right )}{8}}{a}\) | \(125\) |
Input:
int(sinh(x)^6/(a+cosh(x)*a),x,method=_RETURNVERBOSE)
Output:
-3/8*x/a+1/160/a*exp(5*x)-1/64/a*exp(4*x)-1/32/a*exp(3*x)+1/8/a*exp(2*x)+1 /16/a*exp(x)-1/16/a*exp(-x)-1/8/a*exp(-2*x)+1/32/a*exp(-3*x)+1/64/a*exp(-4 *x)-1/160/a*exp(-5*x)
Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.30 \[ \int \frac {\sinh ^6(x)}{a+a \cosh (x)} \, dx=\frac {\sinh \left (x\right )^{5} + 5 \, {\left (2 \, \cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + 5 \, {\left (\cosh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} + 8 \, \cosh \left (x\right ) + 2\right )} \sinh \left (x\right ) - 30 \, x}{80 \, a} \] Input:
integrate(sinh(x)^6/(a+a*cosh(x)),x, algorithm="fricas")
Output:
1/80*(sinh(x)^5 + 5*(2*cosh(x)^2 - 2*cosh(x) - 1)*sinh(x)^3 + 5*(cosh(x)^4 - 2*cosh(x)^3 - 3*cosh(x)^2 + 8*cosh(x) + 2)*sinh(x) - 30*x)/a
Leaf count of result is larger than twice the leaf count of optimal. 692 vs. \(2 (37) = 74\).
Time = 1.25 (sec) , antiderivative size = 692, normalized size of antiderivative = 15.73 \[ \int \frac {\sinh ^6(x)}{a+a \cosh (x)} \, dx=\text {Too large to display} \] Input:
integrate(sinh(x)**6/(a+a*cosh(x)),x)
Output:
-15*x*tanh(x/2)**10/(40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**8 + 400*a*tanh( x/2)**6 - 400*a*tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a) + 75*x*tanh(x/2) **8/(40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*a* tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a) - 150*x*tanh(x/2)**6/(40*a*tanh( x/2)**10 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a) + 150*x*tanh(x/2)**4/(40*a*tanh(x/2)**10 - 200* a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 200*a*tanh(x/2) **2 - 40*a) - 75*x*tanh(x/2)**2/(40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a) + 15 *x/(40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*a*t anh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a) + 30*tanh(x/2)**9/(40*a*tanh(x/2) **10 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 200* a*tanh(x/2)**2 - 40*a) - 140*tanh(x/2)**7/(40*a*tanh(x/2)**10 - 200*a*tanh (x/2)**8 + 400*a*tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a) - 256*tanh(x/2)**5/(40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**8 + 400*a* tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a) + 140*tanh( x/2)**3/(40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 40 0*a*tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a) - 30*tanh(x/2)/(40*a*tanh(x/ 2)**10 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 20 0*a*tanh(x/2)**2 - 40*a)
Leaf count of result is larger than twice the leaf count of optimal. 78 vs. \(2 (36) = 72\).
Time = 0.03 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.77 \[ \int \frac {\sinh ^6(x)}{a+a \cosh (x)} \, dx=-\frac {{\left (5 \, e^{\left (-x\right )} + 10 \, e^{\left (-2 \, x\right )} - 40 \, e^{\left (-3 \, x\right )} - 20 \, e^{\left (-4 \, x\right )} - 2\right )} e^{\left (5 \, x\right )}}{320 \, a} - \frac {3 \, x}{8 \, a} - \frac {20 \, e^{\left (-x\right )} + 40 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-3 \, x\right )} - 5 \, e^{\left (-4 \, x\right )} + 2 \, e^{\left (-5 \, x\right )}}{320 \, a} \] Input:
integrate(sinh(x)^6/(a+a*cosh(x)),x, algorithm="maxima")
Output:
-1/320*(5*e^(-x) + 10*e^(-2*x) - 40*e^(-3*x) - 20*e^(-4*x) - 2)*e^(5*x)/a - 3/8*x/a - 1/320*(20*e^(-x) + 40*e^(-2*x) - 10*e^(-3*x) - 5*e^(-4*x) + 2* e^(-5*x))/a
Time = 0.11 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.50 \[ \int \frac {\sinh ^6(x)}{a+a \cosh (x)} \, dx=-\frac {{\left (20 \, e^{\left (4 \, x\right )} + 40 \, e^{\left (3 \, x\right )} - 10 \, e^{\left (2 \, x\right )} - 5 \, e^{x} + 2\right )} e^{\left (-5 \, x\right )} + 120 \, x - 2 \, e^{\left (5 \, x\right )} + 5 \, e^{\left (4 \, x\right )} + 10 \, e^{\left (3 \, x\right )} - 40 \, e^{\left (2 \, x\right )} - 20 \, e^{x}}{320 \, a} \] Input:
integrate(sinh(x)^6/(a+a*cosh(x)),x, algorithm="giac")
Output:
-1/320*((20*e^(4*x) + 40*e^(3*x) - 10*e^(2*x) - 5*e^x + 2)*e^(-5*x) + 120* x - 2*e^(5*x) + 5*e^(4*x) + 10*e^(3*x) - 40*e^(2*x) - 20*e^x)/a
Time = 2.07 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.16 \[ \int \frac {\sinh ^6(x)}{a+a \cosh (x)} \, dx=\frac {{\mathrm {e}}^{2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {{\mathrm {e}}^{-x}}{16\,a}+\frac {{\mathrm {e}}^{-3\,x}}{32\,a}-\frac {{\mathrm {e}}^{3\,x}}{32\,a}+\frac {{\mathrm {e}}^{-4\,x}}{64\,a}-\frac {{\mathrm {e}}^{4\,x}}{64\,a}-\frac {{\mathrm {e}}^{-5\,x}}{160\,a}+\frac {{\mathrm {e}}^{5\,x}}{160\,a}-\frac {3\,x}{8\,a}+\frac {{\mathrm {e}}^x}{16\,a} \] Input:
int(sinh(x)^6/(a + a*cosh(x)),x)
Output:
exp(2*x)/(8*a) - exp(-2*x)/(8*a) - exp(-x)/(16*a) + exp(-3*x)/(32*a) - exp (3*x)/(32*a) + exp(-4*x)/(64*a) - exp(4*x)/(64*a) - exp(-5*x)/(160*a) + ex p(5*x)/(160*a) - (3*x)/(8*a) + exp(x)/(16*a)
Time = 0.27 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.89 \[ \int \frac {\sinh ^6(x)}{a+a \cosh (x)} \, dx=\frac {2 e^{10 x}-5 e^{9 x}-10 e^{8 x}+40 e^{7 x}+20 e^{6 x}-120 e^{5 x} x -20 e^{4 x}-40 e^{3 x}+10 e^{2 x}+5 e^{x}-2}{320 e^{5 x} a} \] Input:
int(sinh(x)^6/(a+a*cosh(x)),x)
Output:
(2*e**(10*x) - 5*e**(9*x) - 10*e**(8*x) + 40*e**(7*x) + 20*e**(6*x) - 120* e**(5*x)*x - 20*e**(4*x) - 40*e**(3*x) + 10*e**(2*x) + 5*e**x - 2)/(320*e* *(5*x)*a)