Integrand size = 12, antiderivative size = 98 \[ \int \frac {1}{(3+5 \cosh (c+d x))^4} \, dx=-\frac {279 \arctan \left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{16384 d}+\frac {5 \sinh (c+d x)}{48 d (3+5 \cosh (c+d x))^3}-\frac {25 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))^2}+\frac {995 \sinh (c+d x)}{24576 d (3+5 \cosh (c+d x))} \] Output:
-279/16384*arctan(1/2*tanh(1/2*d*x+1/2*c))/d+5/48*sinh(d*x+c)/d/(3+5*cosh( d*x+c))^3-25/512*sinh(d*x+c)/d/(3+5*cosh(d*x+c))^2+995/24576*sinh(d*x+c)/d /(3+5*cosh(d*x+c))
Time = 0.17 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(3+5 \cosh (c+d x))^4} \, dx=\frac {837 \arctan \left (2 \coth \left (\frac {1}{2} (c+d x)\right )\right )+\frac {5 (8141+9540 \cosh (c+d x)+4975 \cosh (2 (c+d x))) \sinh (c+d x)}{(3+5 \cosh (c+d x))^3}}{49152 d} \] Input:
Integrate[(3 + 5*Cosh[c + d*x])^(-4),x]
Output:
(837*ArcTan[2*Coth[(c + d*x)/2]] + (5*(8141 + 9540*Cosh[c + d*x] + 4975*Co sh[2*(c + d*x)])*Sinh[c + d*x])/(3 + 5*Cosh[c + d*x])^3)/(49152*d)
Time = 0.53 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3143, 25, 3042, 3233, 25, 3042, 3233, 27, 3042, 3138, 219}
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.
| \(\displaystyle \int \frac {1}{(5 \cosh (c+d x)+3)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (3+5 \sin \left (i c+i d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3143 |
| \(\displaystyle \frac {1}{48} \int -\frac {9-10 \cosh (c+d x)}{(5 \cosh (c+d x)+3)^3}dx+\frac {5 \sinh (c+d x)}{48 d (5 \cosh (c+d x)+3)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5 \sinh (c+d x)}{48 d (5 \cosh (c+d x)+3)^3}-\frac {1}{48} \int \frac {9-10 \cosh (c+d x)}{(5 \cosh (c+d x)+3)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \sinh (c+d x)}{48 d (5 \cosh (c+d x)+3)^3}-\frac {1}{48} \int \frac {9-10 \sin \left (i c+i d x+\frac {\pi }{2}\right )}{\left (5 \sin \left (i c+i d x+\frac {\pi }{2}\right )+3\right )^3}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{48} \left (-\frac {1}{32} \int -\frac {154-75 \cosh (c+d x)}{(5 \cosh (c+d x)+3)^2}dx-\frac {75 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}\right )+\frac {5 \sinh (c+d x)}{48 d (5 \cosh (c+d x)+3)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \int \frac {154-75 \cosh (c+d x)}{(5 \cosh (c+d x)+3)^2}dx-\frac {75 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}\right )+\frac {5 \sinh (c+d x)}{48 d (5 \cosh (c+d x)+3)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \sinh (c+d x)}{48 d (5 \cosh (c+d x)+3)^3}+\frac {1}{48} \left (-\frac {75 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}+\frac {1}{32} \int \frac {154-75 \sin \left (i c+i d x+\frac {\pi }{2}\right )}{\left (5 \sin \left (i c+i d x+\frac {\pi }{2}\right )+3\right )^2}dx\right )\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {1}{16} \int -\frac {837}{5 \cosh (c+d x)+3}dx+\frac {995 \sinh (c+d x)}{16 d (5 \cosh (c+d x)+3)}\right )-\frac {75 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}\right )+\frac {5 \sinh (c+d x)}{48 d (5 \cosh (c+d x)+3)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {995 \sinh (c+d x)}{16 d (5 \cosh (c+d x)+3)}-\frac {837}{16} \int \frac {1}{5 \cosh (c+d x)+3}dx\right )-\frac {75 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}\right )+\frac {5 \sinh (c+d x)}{48 d (5 \cosh (c+d x)+3)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5 \sinh (c+d x)}{48 d (5 \cosh (c+d x)+3)^3}+\frac {1}{48} \left (-\frac {75 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}+\frac {1}{32} \left (\frac {995 \sinh (c+d x)}{16 d (5 \cosh (c+d x)+3)}-\frac {837}{16} \int \frac {1}{5 \sin \left (i c+i d x+\frac {\pi }{2}\right )+3}dx\right )\right )\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {5 \sinh (c+d x)}{48 d (5 \cosh (c+d x)+3)^3}+\frac {1}{48} \left (-\frac {75 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}+\frac {1}{32} \left (\frac {995 \sinh (c+d x)}{16 d (5 \cosh (c+d x)+3)}+\frac {837 i \int \frac {1}{2 \tanh ^2\left (\frac {1}{2} (c+d x)\right )+8}d\left (i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}\right )\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{48} \left (\frac {1}{32} \left (\frac {995 \sinh (c+d x)}{16 d (5 \cosh (c+d x)+3)}-\frac {837 \arctan \left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{32 d}\right )-\frac {75 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}\right )+\frac {5 \sinh (c+d x)}{48 d (5 \cosh (c+d x)+3)^3}\) |
Input:
Int[(3 + 5*Cosh[c + d*x])^(-4),x]
Output:
(5*Sinh[c + d*x])/(48*d*(3 + 5*Cosh[c + d*x])^3) + ((-75*Sinh[c + d*x])/(3 2*d*(3 + 5*Cosh[c + d*x])^2) + ((-837*ArcTan[Tanh[(c + d*x)/2]/2])/(32*d) + (995*Sinh[c + d*x])/(16*d*(3 + 5*Cosh[c + d*x])))/32)/48
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_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0]
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 = 0.81 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.77
| method | result | size |
| derivativedivides | \(\frac {-\frac {-\frac {745 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{1024}-\frac {265 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{96}-\frac {295 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}}{8 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4\right )^{3}}-\frac {279 \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{16384}}{d}\) | \(75\) |
| default | \(\frac {-\frac {-\frac {745 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{1024}-\frac {265 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{96}-\frac {295 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}}{8 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4\right )^{3}}-\frac {279 \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{16384}}{d}\) | \(75\) |
| risch | \(-\frac {20925 \,{\mathrm e}^{5 d x +5 c}+62775 \,{\mathrm e}^{4 d x +4 c}+111042 \,{\mathrm e}^{3 d x +3 c}+119310 \,{\mathrm e}^{2 d x +2 c}+68625 \,{\mathrm e}^{d x +c}+24875}{12288 d \left (5 \,{\mathrm e}^{2 d x +2 c}+6 \,{\mathrm e}^{d x +c}+5\right )^{3}}+\frac {279 i \ln \left ({\mathrm e}^{d x +c}+\frac {3}{5}-\frac {4 i}{5}\right )}{32768 d}-\frac {279 i \ln \left ({\mathrm e}^{d x +c}+\frac {3}{5}+\frac {4 i}{5}\right )}{32768 d}\) | \(118\) |
| parallelrisch | \(\frac {837 i \left (125 \cosh \left (3 d x +3 c \right )+915 \cosh \left (d x +c \right )+450 \cosh \left (2 d x +2 c \right )+558\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 i\right )+837 i \left (-125 \cosh \left (3 d x +3 c \right )-915 \cosh \left (d x +c \right )-450 \cosh \left (2 d x +2 c \right )-558\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 i\right )+226140 \sinh \left (d x +c \right )+190800 \sinh \left (2 d x +2 c \right )+99500 \sinh \left (3 d x +3 c \right )}{98304 d \left (125 \cosh \left (3 d x +3 c \right )+915 \cosh \left (d x +c \right )+450 \cosh \left (2 d x +2 c \right )+558\right )}\) | \(167\) |
Input:
int(1/(3+5*cosh(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/8*(-745/1024*tanh(1/2*d*x+1/2*c)^5-265/96*tanh(1/2*d*x+1/2*c)^3-29 5/64*tanh(1/2*d*x+1/2*c))/(tanh(1/2*d*x+1/2*c)^2+4)^3-279/16384*arctan(1/2 *tanh(1/2*d*x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 793 vs. \(2 (87) = 174\).
Time = 0.09 (sec) , antiderivative size = 793, normalized size of antiderivative = 8.09 \[ \int \frac {1}{(3+5 \cosh (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(3+5*cosh(d*x+c))^4,x, algorithm="fricas")
Output:
-1/49152*(83700*cosh(d*x + c)^5 + 83700*(5*cosh(d*x + c) + 3)*sinh(d*x + c )^4 + 83700*sinh(d*x + c)^5 + 251100*cosh(d*x + c)^4 + 2232*(375*cosh(d*x + c)^2 + 450*cosh(d*x + c) + 199)*sinh(d*x + c)^3 + 444168*cosh(d*x + c)^3 + 24*(34875*cosh(d*x + c)^3 + 62775*cosh(d*x + c)^2 + 55521*cosh(d*x + c) + 19885)*sinh(d*x + c)^2 + 837*(125*cosh(d*x + c)^6 + 150*(5*cosh(d*x + c ) + 3)*sinh(d*x + c)^5 + 125*sinh(d*x + c)^6 + 450*cosh(d*x + c)^5 + 15*(1 25*cosh(d*x + c)^2 + 150*cosh(d*x + c) + 61)*sinh(d*x + c)^4 + 915*cosh(d* x + c)^4 + 4*(625*cosh(d*x + c)^3 + 1125*cosh(d*x + c)^2 + 915*cosh(d*x + c) + 279)*sinh(d*x + c)^3 + 1116*cosh(d*x + c)^3 + 3*(625*cosh(d*x + c)^4 + 1500*cosh(d*x + c)^3 + 1830*cosh(d*x + c)^2 + 1116*cosh(d*x + c) + 305)* sinh(d*x + c)^2 + 915*cosh(d*x + c)^2 + 6*(125*cosh(d*x + c)^5 + 375*cosh( d*x + c)^4 + 610*cosh(d*x + c)^3 + 558*cosh(d*x + c)^2 + 305*cosh(d*x + c) + 75)*sinh(d*x + c) + 450*cosh(d*x + c) + 125)*arctan(5/4*cosh(d*x + c) + 5/4*sinh(d*x + c) + 3/4) + 477240*cosh(d*x + c)^2 + 12*(34875*cosh(d*x + c)^4 + 83700*cosh(d*x + c)^3 + 111042*cosh(d*x + c)^2 + 79540*cosh(d*x + c ) + 22875)*sinh(d*x + c) + 274500*cosh(d*x + c) + 99500)/(125*d*cosh(d*x + c)^6 + 125*d*sinh(d*x + c)^6 + 450*d*cosh(d*x + c)^5 + 150*(5*d*cosh(d*x + c) + 3*d)*sinh(d*x + c)^5 + 915*d*cosh(d*x + c)^4 + 15*(125*d*cosh(d*x + c)^2 + 150*d*cosh(d*x + c) + 61*d)*sinh(d*x + c)^4 + 1116*d*cosh(d*x + c) ^3 + 4*(625*d*cosh(d*x + c)^3 + 1125*d*cosh(d*x + c)^2 + 915*d*cosh(d*x...
Result contains complex when optimal does not.
Time = 4.56 (sec) , antiderivative size = 784, normalized size of antiderivative = 8.00 \[ \int \frac {1}{(3+5 \cosh (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(3+5*cosh(d*x+c))**4,x)
Output:
Piecewise((-log(-3*exp(-d*x) - 4*I*exp(-d*x))/(625*d*cosh(d*x + log(-3*exp (-d*x) - 4*I*exp(-d*x)) - log(5))**4 + 1500*d*cosh(d*x + log(-3*exp(-d*x) - 4*I*exp(-d*x)) - log(5))**3 + 1350*d*cosh(d*x + log(-3*exp(-d*x) - 4*I*e xp(-d*x)) - log(5))**2 + 540*d*cosh(d*x + log(-3*exp(-d*x) - 4*I*exp(-d*x) ) - log(5)) + 81*d), Eq(c, log((-3 - 4*I)*exp(-d*x)) - log(5))), (x/(625*c osh(d*x + log(-3*exp(-d*x) + 4*I*exp(-d*x)) - log(5))**4 + 1500*cosh(d*x + log(-3*exp(-d*x) + 4*I*exp(-d*x)) - log(5))**3 + 1350*cosh(d*x + log(-3*e xp(-d*x) + 4*I*exp(-d*x)) - log(5))**2 + 540*cosh(d*x + log(-3*exp(-d*x) + 4*I*exp(-d*x)) - log(5)) + 81), Eq(c, log((-3 + 4*I)*exp(-d*x)) - log(5)) ), (x/(5*cosh(c) + 3)**4, Eq(d, 0)), (-837*tanh(c/2 + d*x/2)**6*atan(tanh( c/2 + d*x/2)/2)/(49152*d*tanh(c/2 + d*x/2)**6 + 589824*d*tanh(c/2 + d*x/2) **4 + 2359296*d*tanh(c/2 + d*x/2)**2 + 3145728*d) + 4470*tanh(c/2 + d*x/2) **5/(49152*d*tanh(c/2 + d*x/2)**6 + 589824*d*tanh(c/2 + d*x/2)**4 + 235929 6*d*tanh(c/2 + d*x/2)**2 + 3145728*d) - 10044*tanh(c/2 + d*x/2)**4*atan(ta nh(c/2 + d*x/2)/2)/(49152*d*tanh(c/2 + d*x/2)**6 + 589824*d*tanh(c/2 + d*x /2)**4 + 2359296*d*tanh(c/2 + d*x/2)**2 + 3145728*d) + 16960*tanh(c/2 + d* x/2)**3/(49152*d*tanh(c/2 + d*x/2)**6 + 589824*d*tanh(c/2 + d*x/2)**4 + 23 59296*d*tanh(c/2 + d*x/2)**2 + 3145728*d) - 40176*tanh(c/2 + d*x/2)**2*ata n(tanh(c/2 + d*x/2)/2)/(49152*d*tanh(c/2 + d*x/2)**6 + 589824*d*tanh(c/2 + d*x/2)**4 + 2359296*d*tanh(c/2 + d*x/2)**2 + 3145728*d) + 28320*tanh(c...
Time = 0.13 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.55 \[ \int \frac {1}{(3+5 \cosh (c+d x))^4} \, dx=\frac {279 \, \arctan \left (\frac {5}{4} \, e^{\left (-d x - c\right )} + \frac {3}{4}\right )}{16384 \, d} + \frac {68625 \, e^{\left (-d x - c\right )} + 119310 \, e^{\left (-2 \, d x - 2 \, c\right )} + 111042 \, e^{\left (-3 \, d x - 3 \, c\right )} + 62775 \, e^{\left (-4 \, d x - 4 \, c\right )} + 20925 \, e^{\left (-5 \, d x - 5 \, c\right )} + 24875}{12288 \, d {\left (450 \, e^{\left (-d x - c\right )} + 915 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1116 \, e^{\left (-3 \, d x - 3 \, c\right )} + 915 \, e^{\left (-4 \, d x - 4 \, c\right )} + 450 \, e^{\left (-5 \, d x - 5 \, c\right )} + 125 \, e^{\left (-6 \, d x - 6 \, c\right )} + 125\right )}} \] Input:
integrate(1/(3+5*cosh(d*x+c))^4,x, algorithm="maxima")
Output:
279/16384*arctan(5/4*e^(-d*x - c) + 3/4)/d + 1/12288*(68625*e^(-d*x - c) + 119310*e^(-2*d*x - 2*c) + 111042*e^(-3*d*x - 3*c) + 62775*e^(-4*d*x - 4*c ) + 20925*e^(-5*d*x - 5*c) + 24875)/(d*(450*e^(-d*x - c) + 915*e^(-2*d*x - 2*c) + 1116*e^(-3*d*x - 3*c) + 915*e^(-4*d*x - 4*c) + 450*e^(-5*d*x - 5*c ) + 125*e^(-6*d*x - 6*c) + 125))
Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(3+5 \cosh (c+d x))^4} \, dx=-\frac {\frac {4 \, {\left (20925 \, e^{\left (5 \, d x + 5 \, c\right )} + 62775 \, e^{\left (4 \, d x + 4 \, c\right )} + 111042 \, e^{\left (3 \, d x + 3 \, c\right )} + 119310 \, e^{\left (2 \, d x + 2 \, c\right )} + 68625 \, e^{\left (d x + c\right )} + 24875\right )}}{{\left (5 \, e^{\left (2 \, d x + 2 \, c\right )} + 6 \, e^{\left (d x + c\right )} + 5\right )}^{3}} + 837 \, \arctan \left (\frac {5}{4} \, e^{\left (d x + c\right )} + \frac {3}{4}\right )}{49152 \, d} \] Input:
integrate(1/(3+5*cosh(d*x+c))^4,x, algorithm="giac")
Output:
-1/49152*(4*(20925*e^(5*d*x + 5*c) + 62775*e^(4*d*x + 4*c) + 111042*e^(3*d *x + 3*c) + 119310*e^(2*d*x + 2*c) + 68625*e^(d*x + c) + 24875)/(5*e^(2*d* x + 2*c) + 6*e^(d*x + c) + 5)^3 + 837*arctan(5/4*e^(d*x + c) + 3/4))/d
Time = 2.04 (sec) , antiderivative size = 223, normalized size of antiderivative = 2.28 \[ \int \frac {1}{(3+5 \cosh (c+d x))^4} \, dx=\frac {\frac {39\,{\mathrm {e}}^{c+d\,x}}{50\,d}+\frac {7}{30\,d}}{450\,{\mathrm {e}}^{c+d\,x}+915\,{\mathrm {e}}^{2\,c+2\,d\,x}+1116\,{\mathrm {e}}^{3\,c+3\,d\,x}+915\,{\mathrm {e}}^{4\,c+4\,d\,x}+450\,{\mathrm {e}}^{5\,c+5\,d\,x}+125\,{\mathrm {e}}^{6\,c+6\,d\,x}+125}-\frac {\frac {93\,{\mathrm {e}}^{c+d\,x}}{640\,d}+\frac {791}{3200\,d}}{60\,{\mathrm {e}}^{c+d\,x}+86\,{\mathrm {e}}^{2\,c+2\,d\,x}+60\,{\mathrm {e}}^{3\,c+3\,d\,x}+25\,{\mathrm {e}}^{4\,c+4\,d\,x}+25}-\frac {279\,\mathrm {atan}\left (\left (\frac {3}{4\,d}+\frac {5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{4\,d}\right )\,\sqrt {d^2}\right )}{16384\,\sqrt {d^2}}-\frac {\frac {279\,{\mathrm {e}}^{c+d\,x}}{4096\,d}+\frac {837}{20480\,d}}{6\,{\mathrm {e}}^{c+d\,x}+5\,{\mathrm {e}}^{2\,c+2\,d\,x}+5} \] Input:
int(1/(5*cosh(c + d*x) + 3)^4,x)
Output:
((39*exp(c + d*x))/(50*d) + 7/(30*d))/(450*exp(c + d*x) + 915*exp(2*c + 2* d*x) + 1116*exp(3*c + 3*d*x) + 915*exp(4*c + 4*d*x) + 450*exp(5*c + 5*d*x) + 125*exp(6*c + 6*d*x) + 125) - ((93*exp(c + d*x))/(640*d) + 791/(3200*d) )/(60*exp(c + d*x) + 86*exp(2*c + 2*d*x) + 60*exp(3*c + 3*d*x) + 25*exp(4* c + 4*d*x) + 25) - (279*atan((3/(4*d) + (5*exp(d*x)*exp(c))/(4*d))*(d^2)^( 1/2)))/(16384*(d^2)^(1/2)) - ((279*exp(c + d*x))/(4096*d) + 837/(20480*d)) /(6*exp(c + d*x) + 5*exp(2*c + 2*d*x) + 5)
Time = 0.21 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.98 \[ \int \frac {1}{(3+5 \cosh (c+d x))^4} \, dx=\frac {-104625 e^{6 d x +6 c} \mathit {atan} \left (\frac {5 e^{d x +c}}{4}+\frac {3}{4}\right )-376650 e^{5 d x +5 c} \mathit {atan} \left (\frac {5 e^{d x +c}}{4}+\frac {3}{4}\right )-765855 e^{4 d x +4 c} \mathit {atan} \left (\frac {5 e^{d x +c}}{4}+\frac {3}{4}\right )-934092 e^{3 d x +3 c} \mathit {atan} \left (\frac {5 e^{d x +c}}{4}+\frac {3}{4}\right )-765855 e^{2 d x +2 c} \mathit {atan} \left (\frac {5 e^{d x +c}}{4}+\frac {3}{4}\right )-376650 e^{d x +c} \mathit {atan} \left (\frac {5 e^{d x +c}}{4}+\frac {3}{4}\right )-104625 \mathit {atan} \left (\frac {5 e^{d x +c}}{4}+\frac {3}{4}\right )+23250 e^{6 d x +6 c}-80910 e^{4 d x +4 c}-236592 e^{3 d x +3 c}-307050 e^{2 d x +2 c}-190800 e^{d x +c}-76250}{49152 d \left (125 e^{6 d x +6 c}+450 e^{5 d x +5 c}+915 e^{4 d x +4 c}+1116 e^{3 d x +3 c}+915 e^{2 d x +2 c}+450 e^{d x +c}+125\right )} \] Input:
int(1/(3+5*cosh(d*x+c))^4,x)
Output:
( - 104625*e**(6*c + 6*d*x)*atan((5*e**(c + d*x) + 3)/4) - 376650*e**(5*c + 5*d*x)*atan((5*e**(c + d*x) + 3)/4) - 765855*e**(4*c + 4*d*x)*atan((5*e* *(c + d*x) + 3)/4) - 934092*e**(3*c + 3*d*x)*atan((5*e**(c + d*x) + 3)/4) - 765855*e**(2*c + 2*d*x)*atan((5*e**(c + d*x) + 3)/4) - 376650*e**(c + d* x)*atan((5*e**(c + d*x) + 3)/4) - 104625*atan((5*e**(c + d*x) + 3)/4) + 23 250*e**(6*c + 6*d*x) - 80910*e**(4*c + 4*d*x) - 236592*e**(3*c + 3*d*x) - 307050*e**(2*c + 2*d*x) - 190800*e**(c + d*x) - 76250)/(49152*d*(125*e**(6 *c + 6*d*x) + 450*e**(5*c + 5*d*x) + 915*e**(4*c + 4*d*x) + 1116*e**(3*c + 3*d*x) + 915*e**(2*c + 2*d*x) + 450*e**(c + d*x) + 125))