Integrand size = 12, antiderivative size = 106 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=\frac {385 x}{32768}-\frac {385 \text {arctanh}\left (\frac {\sinh (c+d x)}{3+\cosh (c+d x)}\right )}{16384 d}-\frac {\sinh (c+d x)}{16 d (5+3 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))^2}-\frac {311 \sinh (c+d x)}{8192 d (5+3 \cosh (c+d x))} \] Output:
385/32768*x-385/16384*arctanh(sinh(d*x+c)/(3+cosh(d*x+c)))/d-1/16*sinh(d*x +c)/d/(5+3*cosh(d*x+c))^3-25/512*sinh(d*x+c)/d/(5+3*cosh(d*x+c))^2-311/819 2*sinh(d*x+c)/d/(5+3*cosh(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(296\) vs. \(2(106)=212\).
Time = 0.25 (sec) , antiderivative size = 296, normalized size of antiderivative = 2.79 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=-\frac {296450 \log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )\right )+10395 \cosh (3 (c+d x)) \log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )\right )+377685 \cosh (c+d x) \left (\log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )+103950 \cosh (2 (c+d x)) \left (\log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )\right )-296450 \log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )-10395 \cosh (3 (c+d x)) \log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )+175788 \sinh (c+d x)+84240 \sinh (2 (c+d x))+11196 \sinh (3 (c+d x))}{131072 d (5+3 \cosh (c+d x))^3} \] Input:
Integrate[(5 + 3*Cosh[c + d*x])^(-4),x]
Output:
-1/131072*(296450*Log[2*Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2]] + 10395*Cos h[3*(c + d*x)]*Log[2*Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2]] + 377685*Cosh[ c + d*x]*(Log[2*Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2]] - Log[2*Cosh[(c + d *x)/2] + Sinh[(c + d*x)/2]]) + 103950*Cosh[2*(c + d*x)]*(Log[2*Cosh[(c + d *x)/2] - Sinh[(c + d*x)/2]] - Log[2*Cosh[(c + d*x)/2] + Sinh[(c + d*x)/2]] ) - 296450*Log[2*Cosh[(c + d*x)/2] + Sinh[(c + d*x)/2]] - 10395*Cosh[3*(c + d*x)]*Log[2*Cosh[(c + d*x)/2] + Sinh[(c + d*x)/2]] + 175788*Sinh[c + d*x ] + 84240*Sinh[2*(c + d*x)] + 11196*Sinh[3*(c + d*x)])/(d*(5 + 3*Cosh[c + d*x])^3)
Time = 0.54 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {3042, 3143, 27, 3042, 3233, 25, 3042, 3233, 27, 3042, 3136}
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.
| \(\displaystyle \int \frac {1}{(3 \cosh (c+d x)+5)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (5+3 \sin \left (i c+i d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3143 |
| \(\displaystyle -\frac {1}{48} \int -\frac {3 (5-2 \cosh (c+d x))}{(3 \cosh (c+d x)+5)^3}dx-\frac {\sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \int \frac {5-2 \cosh (c+d x)}{(3 \cosh (c+d x)+5)^3}dx-\frac {\sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)^3}+\frac {1}{16} \int \frac {5-2 \sin \left (i c+i d x+\frac {\pi }{2}\right )}{\left (3 \sin \left (i c+i d x+\frac {\pi }{2}\right )+5\right )^3}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{16} \left (-\frac {1}{32} \int -\frac {62-25 \cosh (c+d x)}{(3 \cosh (c+d x)+5)^2}dx-\frac {25 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}\right )-\frac {\sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \int \frac {62-25 \cosh (c+d x)}{(3 \cosh (c+d x)+5)^2}dx-\frac {25 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}\right )-\frac {\sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)^3}+\frac {1}{16} \left (-\frac {25 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}+\frac {1}{32} \int \frac {62-25 \sin \left (i c+i d x+\frac {\pi }{2}\right )}{\left (3 \sin \left (i c+i d x+\frac {\pi }{2}\right )+5\right )^2}dx\right )\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (-\frac {1}{16} \int -\frac {385}{3 \cosh (c+d x)+5}dx-\frac {311 \sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)}\right )-\frac {25 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}\right )-\frac {\sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {385}{16} \int \frac {1}{3 \cosh (c+d x)+5}dx-\frac {311 \sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)}\right )-\frac {25 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}\right )-\frac {\sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)^3}+\frac {1}{16} \left (-\frac {25 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}+\frac {1}{32} \left (-\frac {311 \sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)}+\frac {385}{16} \int \frac {1}{3 \sin \left (i c+i d x+\frac {\pi }{2}\right )+5}dx\right )\right )\) |
| \(\Big \downarrow \) 3136 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {385}{16} \left (\frac {x}{4}-\frac {\text {arctanh}\left (\frac {\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{2 d}\right )-\frac {311 \sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)}\right )-\frac {25 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}\right )-\frac {\sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)^3}\) |
Input:
Int[(5 + 3*Cosh[c + d*x])^(-4),x]
Output:
-1/16*Sinh[c + d*x]/(d*(5 + 3*Cosh[c + d*x])^3) + ((-25*Sinh[c + d*x])/(32 *d*(5 + 3*Cosh[c + d*x])^2) + ((385*(x/4 - ArcTanh[Sinh[c + d*x]/(3 + Cosh [c + d*x])]/(2*d)))/16 - (311*Sinh[c + d*x])/(16*d*(5 + 3*Cosh[c + d*x]))) /32)/16
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[ a^2 - b^2, 2]}, Simp[x/q, x] + Simp[(2/(d*q))*ArcTan[b*(Cos[c + d*x]/(a + q + b*Sin[c + d*x]))], x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2, 0] && PosQ[a]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n + 1)/(d*(n + 1)*(a^2 - b^2))), x] + Simp [1/((n + 1)*(a^2 - b^2)) Int[(a + b*Sin[c + d*x])^(n + 1)*Simp[a*(n + 1) - b*(n + 2)*Sin[c + d*x], x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Time = 0.75 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.06
| method | result | size |
| risch | \(\frac {10395 \,{\mathrm e}^{5 d x +5 c}+86625 \,{\mathrm e}^{4 d x +4 c}+239470 \,{\mathrm e}^{3 d x +3 c}+218466 \,{\mathrm e}^{2 d x +2 c}+73575 \,{\mathrm e}^{d x +c}+8397}{12288 d \left (3 \,{\mathrm e}^{2 d x +2 c}+10 \,{\mathrm e}^{d x +c}+3\right )^{3}}-\frac {385 \ln \left (3+{\mathrm e}^{d x +c}\right )}{32768 d}+\frac {385 \ln \left ({\mathrm e}^{d x +c}+\frac {1}{3}\right )}{32768 d}\) | \(112\) |
| derivativedivides | \(\frac {\frac {9}{2048 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}-\frac {81}{4096 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}+\frac {639}{16384 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}+\frac {385 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768}+\frac {9}{2048 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}+\frac {81}{4096 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}+\frac {639}{16384 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}-\frac {385 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768}}{d}\) | \(124\) |
| default | \(\frac {\frac {9}{2048 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}-\frac {81}{4096 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}+\frac {639}{16384 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}+\frac {385 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{32768}+\frac {9}{2048 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}+\frac {81}{4096 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}+\frac {639}{16384 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}-\frac {385 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{32768}}{d}\) | \(124\) |
| parallelrisch | \(\frac {\left (-377685 \cosh \left (d x +c \right )-103950 \cosh \left (2 d x +2 c \right )-10395 \cosh \left (3 d x +3 c \right )-296450\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )+\left (377685 \cosh \left (d x +c \right )+103950 \cosh \left (2 d x +2 c \right )+10395 \cosh \left (3 d x +3 c \right )+296450\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )-175788 \sinh \left (d x +c \right )-84240 \sinh \left (2 d x +2 c \right )-11196 \sinh \left (3 d x +3 c \right )}{32768 d \left (770+27 \cosh \left (3 d x +3 c \right )+981 \cosh \left (d x +c \right )+270 \cosh \left (2 d x +2 c \right )\right )}\) | \(161\) |
Input:
int(1/(5+3*cosh(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/12288*(10395*exp(5*d*x+5*c)+86625*exp(4*d*x+4*c)+239470*exp(3*d*x+3*c)+2 18466*exp(2*d*x+2*c)+73575*exp(d*x+c)+8397)/d/(3*exp(2*d*x+2*c)+10*exp(d*x +c)+3)^3-385/32768/d*ln(3+exp(d*x+c))+385/32768/d*ln(exp(d*x+c)+1/3)
Leaf count of result is larger than twice the leaf count of optimal. 1078 vs. \(2 (96) = 192\).
Time = 0.09 (sec) , antiderivative size = 1078, normalized size of antiderivative = 10.17 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(5+3*cosh(d*x+c))^4,x, algorithm="fricas")
Output:
1/98304*(83160*cosh(d*x + c)^5 + 138600*(3*cosh(d*x + c) + 5)*sinh(d*x + c )^4 + 83160*sinh(d*x + c)^5 + 693000*cosh(d*x + c)^4 + 6160*(135*cosh(d*x + c)^2 + 450*cosh(d*x + c) + 311)*sinh(d*x + c)^3 + 1915760*cosh(d*x + c)^ 3 + 48*(17325*cosh(d*x + c)^3 + 86625*cosh(d*x + c)^2 + 119735*cosh(d*x + c) + 36411)*sinh(d*x + c)^2 + 1747728*cosh(d*x + c)^2 + 1155*(27*cosh(d*x + c)^6 + 54*(3*cosh(d*x + c) + 5)*sinh(d*x + c)^5 + 27*sinh(d*x + c)^6 + 2 70*cosh(d*x + c)^5 + 9*(45*cosh(d*x + c)^2 + 150*cosh(d*x + c) + 109)*sinh (d*x + c)^4 + 981*cosh(d*x + c)^4 + 4*(135*cosh(d*x + c)^3 + 675*cosh(d*x + c)^2 + 981*cosh(d*x + c) + 385)*sinh(d*x + c)^3 + 1540*cosh(d*x + c)^3 + 3*(135*cosh(d*x + c)^4 + 900*cosh(d*x + c)^3 + 1962*cosh(d*x + c)^2 + 154 0*cosh(d*x + c) + 327)*sinh(d*x + c)^2 + 981*cosh(d*x + c)^2 + 6*(27*cosh( d*x + c)^5 + 225*cosh(d*x + c)^4 + 654*cosh(d*x + c)^3 + 770*cosh(d*x + c) ^2 + 327*cosh(d*x + c) + 45)*sinh(d*x + c) + 270*cosh(d*x + c) + 27)*log(3 *cosh(d*x + c) + 3*sinh(d*x + c) + 1) - 1155*(27*cosh(d*x + c)^6 + 54*(3*c osh(d*x + c) + 5)*sinh(d*x + c)^5 + 27*sinh(d*x + c)^6 + 270*cosh(d*x + c) ^5 + 9*(45*cosh(d*x + c)^2 + 150*cosh(d*x + c) + 109)*sinh(d*x + c)^4 + 98 1*cosh(d*x + c)^4 + 4*(135*cosh(d*x + c)^3 + 675*cosh(d*x + c)^2 + 981*cos h(d*x + c) + 385)*sinh(d*x + c)^3 + 1540*cosh(d*x + c)^3 + 3*(135*cosh(d*x + c)^4 + 900*cosh(d*x + c)^3 + 1962*cosh(d*x + c)^2 + 1540*cosh(d*x + c) + 327)*sinh(d*x + c)^2 + 981*cosh(d*x + c)^2 + 6*(27*cosh(d*x + c)^5 + ...
Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (94) = 188\).
Time = 3.23 (sec) , antiderivative size = 784, normalized size of antiderivative = 7.40 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(5+3*cosh(d*x+c))**4,x)
Output:
Piecewise((-385*log(tanh(c/2 + d*x/2) - 2)*tanh(c/2 + d*x/2)**6/(32768*d*t anh(c/2 + d*x/2)**6 - 393216*d*tanh(c/2 + d*x/2)**4 + 1572864*d*tanh(c/2 + d*x/2)**2 - 2097152*d) + 4620*log(tanh(c/2 + d*x/2) - 2)*tanh(c/2 + d*x/2 )**4/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tanh(c/2 + d*x/2)**4 + 15728 64*d*tanh(c/2 + d*x/2)**2 - 2097152*d) - 18480*log(tanh(c/2 + d*x/2) - 2)* tanh(c/2 + d*x/2)**2/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tanh(c/2 + d *x/2)**4 + 1572864*d*tanh(c/2 + d*x/2)**2 - 2097152*d) + 24640*log(tanh(c/ 2 + d*x/2) - 2)/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tanh(c/2 + d*x/2) **4 + 1572864*d*tanh(c/2 + d*x/2)**2 - 2097152*d) + 385*log(tanh(c/2 + d*x /2) + 2)*tanh(c/2 + d*x/2)**6/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tan h(c/2 + d*x/2)**4 + 1572864*d*tanh(c/2 + d*x/2)**2 - 2097152*d) - 4620*log (tanh(c/2 + d*x/2) + 2)*tanh(c/2 + d*x/2)**4/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tanh(c/2 + d*x/2)**4 + 1572864*d*tanh(c/2 + d*x/2)**2 - 209715 2*d) + 18480*log(tanh(c/2 + d*x/2) + 2)*tanh(c/2 + d*x/2)**2/(32768*d*tanh (c/2 + d*x/2)**6 - 393216*d*tanh(c/2 + d*x/2)**4 + 1572864*d*tanh(c/2 + d* x/2)**2 - 2097152*d) - 24640*log(tanh(c/2 + d*x/2) + 2)/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tanh(c/2 + d*x/2)**4 + 1572864*d*tanh(c/2 + d*x/2)* *2 - 2097152*d) + 2556*tanh(c/2 + d*x/2)**5/(32768*d*tanh(c/2 + d*x/2)**6 - 393216*d*tanh(c/2 + d*x/2)**4 + 1572864*d*tanh(c/2 + d*x/2)**2 - 2097152 *d) - 14976*tanh(c/2 + d*x/2)**3/(32768*d*tanh(c/2 + d*x/2)**6 - 393216...
Time = 0.05 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.59 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=-\frac {385 \, \log \left (3 \, e^{\left (-d x - c\right )} + 1\right )}{32768 \, d} + \frac {385 \, \log \left (e^{\left (-d x - c\right )} + 3\right )}{32768 \, d} - \frac {73575 \, e^{\left (-d x - c\right )} + 218466 \, e^{\left (-2 \, d x - 2 \, c\right )} + 239470 \, e^{\left (-3 \, d x - 3 \, c\right )} + 86625 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10395 \, e^{\left (-5 \, d x - 5 \, c\right )} + 8397}{12288 \, d {\left (270 \, e^{\left (-d x - c\right )} + 981 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1540 \, e^{\left (-3 \, d x - 3 \, c\right )} + 981 \, e^{\left (-4 \, d x - 4 \, c\right )} + 270 \, e^{\left (-5 \, d x - 5 \, c\right )} + 27 \, e^{\left (-6 \, d x - 6 \, c\right )} + 27\right )}} \] Input:
integrate(1/(5+3*cosh(d*x+c))^4,x, algorithm="maxima")
Output:
-385/32768*log(3*e^(-d*x - c) + 1)/d + 385/32768*log(e^(-d*x - c) + 3)/d - 1/12288*(73575*e^(-d*x - c) + 218466*e^(-2*d*x - 2*c) + 239470*e^(-3*d*x - 3*c) + 86625*e^(-4*d*x - 4*c) + 10395*e^(-5*d*x - 5*c) + 8397)/(d*(270*e ^(-d*x - c) + 981*e^(-2*d*x - 2*c) + 1540*e^(-3*d*x - 3*c) + 981*e^(-4*d*x - 4*c) + 270*e^(-5*d*x - 5*c) + 27*e^(-6*d*x - 6*c) + 27))
Time = 0.11 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.03 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=\frac {\frac {8 \, {\left (10395 \, e^{\left (5 \, d x + 5 \, c\right )} + 86625 \, e^{\left (4 \, d x + 4 \, c\right )} + 239470 \, e^{\left (3 \, d x + 3 \, c\right )} + 218466 \, e^{\left (2 \, d x + 2 \, c\right )} + 73575 \, e^{\left (d x + c\right )} + 8397\right )}}{{\left (3 \, e^{\left (2 \, d x + 2 \, c\right )} + 10 \, e^{\left (d x + c\right )} + 3\right )}^{3}} + 1155 \, \log \left (3 \, e^{\left (d x + c\right )} + 1\right ) - 1155 \, \log \left (e^{\left (d x + c\right )} + 3\right )}{98304 \, d} \] Input:
integrate(1/(5+3*cosh(d*x+c))^4,x, algorithm="giac")
Output:
1/98304*(8*(10395*e^(5*d*x + 5*c) + 86625*e^(4*d*x + 4*c) + 239470*e^(3*d* x + 3*c) + 218466*e^(2*d*x + 2*c) + 73575*e^(d*x + c) + 8397)/(3*e^(2*d*x + 2*c) + 10*e^(d*x + c) + 3)^3 + 1155*log(3*e^(d*x + c) + 1) - 1155*log(e^ (d*x + c) + 3))/d
Time = 2.04 (sec) , antiderivative size = 226, normalized size of antiderivative = 2.13 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=\frac {\frac {385\,{\mathrm {e}}^{c+d\,x}}{4096\,d}+\frac {1925}{12288\,d}}{10\,{\mathrm {e}}^{c+d\,x}+3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3}-\frac {385\,\mathrm {atan}\left (\left (\frac {5}{4\,d}+\frac {3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{4\,d}\right )\,\sqrt {-d^2}\right )}{16384\,\sqrt {-d^2}}-\frac {\frac {385\,{\mathrm {e}}^{c+d\,x}}{1152\,d}+\frac {3461}{3456\,d}}{60\,{\mathrm {e}}^{c+d\,x}+118\,{\mathrm {e}}^{2\,c+2\,d\,x}+60\,{\mathrm {e}}^{3\,c+3\,d\,x}+9\,{\mathrm {e}}^{4\,c+4\,d\,x}+9}+\frac {\frac {365\,{\mathrm {e}}^{c+d\,x}}{54\,d}+\frac {41}{18\,d}}{270\,{\mathrm {e}}^{c+d\,x}+981\,{\mathrm {e}}^{2\,c+2\,d\,x}+1540\,{\mathrm {e}}^{3\,c+3\,d\,x}+981\,{\mathrm {e}}^{4\,c+4\,d\,x}+270\,{\mathrm {e}}^{5\,c+5\,d\,x}+27\,{\mathrm {e}}^{6\,c+6\,d\,x}+27} \] Input:
int(1/(3*cosh(c + d*x) + 5)^4,x)
Output:
((385*exp(c + d*x))/(4096*d) + 1925/(12288*d))/(10*exp(c + d*x) + 3*exp(2* c + 2*d*x) + 3) - (385*atan((5/(4*d) + (3*exp(d*x)*exp(c))/(4*d))*(-d^2)^( 1/2)))/(16384*(-d^2)^(1/2)) - ((385*exp(c + d*x))/(1152*d) + 3461/(3456*d) )/(60*exp(c + d*x) + 118*exp(2*c + 2*d*x) + 60*exp(3*c + 3*d*x) + 9*exp(4* c + 4*d*x) + 9) + ((365*exp(c + d*x))/(54*d) + 41/(18*d))/(270*exp(c + d*x ) + 981*exp(2*c + 2*d*x) + 1540*exp(3*c + 3*d*x) + 981*exp(4*c + 4*d*x) + 270*exp(5*c + 5*d*x) + 27*exp(6*c + 6*d*x) + 27)
Time = 0.21 (sec) , antiderivative size = 433, normalized size of antiderivative = 4.08 \[ \int \frac {1}{(5+3 \cosh (c+d x))^4} \, dx=\frac {-10395 e^{6 d x +6 c} \mathrm {log}\left (e^{d x +c}+3\right )+10395 e^{6 d x +6 c} \mathrm {log}\left (3 e^{d x +c}+1\right )-2772 e^{6 d x +6 c}-103950 e^{5 d x +5 c} \mathrm {log}\left (e^{d x +c}+3\right )+103950 e^{5 d x +5 c} \mathrm {log}\left (3 e^{d x +c}+1\right )-377685 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+3\right )+377685 e^{4 d x +4 c} \mathrm {log}\left (3 e^{d x +c}+1\right )+130284 e^{4 d x +4 c}-592900 e^{3 d x +3 c} \mathrm {log}\left (e^{d x +c}+3\right )+592900 e^{3 d x +3 c} \mathrm {log}\left (3 e^{d x +c}+1\right )+480480 e^{3 d x +3 c}-377685 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+3\right )+377685 e^{2 d x +2 c} \mathrm {log}\left (3 e^{d x +c}+1\right )+481860 e^{2 d x +2 c}-103950 e^{d x +c} \mathrm {log}\left (e^{d x +c}+3\right )+103950 e^{d x +c} \mathrm {log}\left (3 e^{d x +c}+1\right )+168480 e^{d x +c}-10395 \,\mathrm {log}\left (e^{d x +c}+3\right )+10395 \,\mathrm {log}\left (3 e^{d x +c}+1\right )+19620}{32768 d \left (27 e^{6 d x +6 c}+270 e^{5 d x +5 c}+981 e^{4 d x +4 c}+1540 e^{3 d x +3 c}+981 e^{2 d x +2 c}+270 e^{d x +c}+27\right )} \] Input:
int(1/(5+3*cosh(d*x+c))^4,x)
Output:
( - 10395*e**(6*c + 6*d*x)*log(e**(c + d*x) + 3) + 10395*e**(6*c + 6*d*x)* log(3*e**(c + d*x) + 1) - 2772*e**(6*c + 6*d*x) - 103950*e**(5*c + 5*d*x)* log(e**(c + d*x) + 3) + 103950*e**(5*c + 5*d*x)*log(3*e**(c + d*x) + 1) - 377685*e**(4*c + 4*d*x)*log(e**(c + d*x) + 3) + 377685*e**(4*c + 4*d*x)*lo g(3*e**(c + d*x) + 1) + 130284*e**(4*c + 4*d*x) - 592900*e**(3*c + 3*d*x)* log(e**(c + d*x) + 3) + 592900*e**(3*c + 3*d*x)*log(3*e**(c + d*x) + 1) + 480480*e**(3*c + 3*d*x) - 377685*e**(2*c + 2*d*x)*log(e**(c + d*x) + 3) + 377685*e**(2*c + 2*d*x)*log(3*e**(c + d*x) + 1) + 481860*e**(2*c + 2*d*x) - 103950*e**(c + d*x)*log(e**(c + d*x) + 3) + 103950*e**(c + d*x)*log(3*e* *(c + d*x) + 1) + 168480*e**(c + d*x) - 10395*log(e**(c + d*x) + 3) + 1039 5*log(3*e**(c + d*x) + 1) + 19620)/(32768*d*(27*e**(6*c + 6*d*x) + 270*e** (5*c + 5*d*x) + 981*e**(4*c + 4*d*x) + 1540*e**(3*c + 3*d*x) + 981*e**(2*c + 2*d*x) + 270*e**(c + d*x) + 27))