Integrand size = 12, antiderivative size = 106 \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=\frac {385 x}{32768}-\frac {385 \arctan \left (\frac {\sin (c+d x)}{3+\cos (c+d x)}\right )}{16384 d}-\frac {\sin (c+d x)}{16 d (5+3 \cos (c+d x))^3}-\frac {25 \sin (c+d x)}{512 d (5+3 \cos (c+d x))^2}-\frac {311 \sin (c+d x)}{8192 d (5+3 \cos (c+d x))} \]
385/32768*x-385/16384*arctan(sin(d*x+c)/(3+cos(d*x+c)))/d-1/16*sin(d*x+c)/ d/(5+3*cos(d*x+c))^3-25/512*sin(d*x+c)/d/(5+3*cos(d*x+c))^2-311/8192*sin(d *x+c)/d/(5+3*cos(d*x+c))
Time = 0.01 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.62 \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=-\frac {770 \arctan \left (2 \cot \left (\frac {1}{2} (c+d x)\right )\right )+\frac {9 (4883 \sin (c+d x)+2340 \sin (2 (c+d x))+311 \sin (3 (c+d x)))}{(5+3 \cos (c+d x))^3}}{32768 d} \]
-1/32768*(770*ArcTan[2*Cot[(c + d*x)/2]] + (9*(4883*Sin[c + d*x] + 2340*Si n[2*(c + d*x)] + 311*Sin[3*(c + d*x)]))/(5 + 3*Cos[c + d*x])^3)/d
Time = 0.49 (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.
| \(\displaystyle \int \frac {1}{(-3 \cos (c+d x)-5)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (-3 \sin \left (c+d x+\frac {\pi }{2}\right )-5\right )^4}dx\) |
| \(\Big \downarrow \) 3143 |
| \(\displaystyle -\frac {1}{48} \int -\frac {3 (5-2 \cos (c+d x))}{(3 \cos (c+d x)+5)^3}dx-\frac {\sin (c+d x)}{16 d (3 \cos (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \int \frac {5-2 \cos (c+d x)}{(3 \cos (c+d x)+5)^3}dx-\frac {\sin (c+d x)}{16 d (3 \cos (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{16} \int \frac {5-2 \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (3 \sin \left (c+d x+\frac {\pi }{2}\right )+5\right )^3}dx-\frac {\sin (c+d x)}{16 d (3 \cos (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{16} \left (-\frac {1}{32} \int -\frac {62-25 \cos (c+d x)}{(3 \cos (c+d x)+5)^2}dx-\frac {25 \sin (c+d x)}{32 d (3 \cos (c+d x)+5)^2}\right )-\frac {\sin (c+d x)}{16 d (3 \cos (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \int \frac {62-25 \cos (c+d x)}{(3 \cos (c+d x)+5)^2}dx-\frac {25 \sin (c+d x)}{32 d (3 \cos (c+d x)+5)^2}\right )-\frac {\sin (c+d x)}{16 d (3 \cos (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \int \frac {62-25 \sin \left (c+d x+\frac {\pi }{2}\right )}{\left (3 \sin \left (c+d x+\frac {\pi }{2}\right )+5\right )^2}dx-\frac {25 \sin (c+d x)}{32 d (3 \cos (c+d x)+5)^2}\right )-\frac {\sin (c+d x)}{16 d (3 \cos (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (-\frac {1}{16} \int -\frac {385}{3 \cos (c+d x)+5}dx-\frac {311 \sin (c+d x)}{16 d (3 \cos (c+d x)+5)}\right )-\frac {25 \sin (c+d x)}{32 d (3 \cos (c+d x)+5)^2}\right )-\frac {\sin (c+d x)}{16 d (3 \cos (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {385}{16} \int \frac {1}{3 \cos (c+d x)+5}dx-\frac {311 \sin (c+d x)}{16 d (3 \cos (c+d x)+5)}\right )-\frac {25 \sin (c+d x)}{32 d (3 \cos (c+d x)+5)^2}\right )-\frac {\sin (c+d x)}{16 d (3 \cos (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {385}{16} \int \frac {1}{3 \sin \left (c+d x+\frac {\pi }{2}\right )+5}dx-\frac {311 \sin (c+d x)}{16 d (3 \cos (c+d x)+5)}\right )-\frac {25 \sin (c+d x)}{32 d (3 \cos (c+d x)+5)^2}\right )-\frac {\sin (c+d x)}{16 d (3 \cos (c+d x)+5)^3}\) |
| \(\Big \downarrow \) 3136 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {385}{16} \left (\frac {x}{4}-\frac {\arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)+3}\right )}{2 d}\right )-\frac {311 \sin (c+d x)}{16 d (3 \cos (c+d x)+5)}\right )-\frac {25 \sin (c+d x)}{32 d (3 \cos (c+d x)+5)^2}\right )-\frac {\sin (c+d x)}{16 d (3 \cos (c+d x)+5)^3}\) |
-1/16*Sin[c + d*x]/(d*(5 + 3*Cos[c + d*x])^3) + ((-25*Sin[c + d*x])/(32*d* (5 + 3*Cos[c + d*x])^2) + ((385*(x/4 - ArcTan[Sin[c + d*x]/(3 + Cos[c + d* x])]/(2*d)))/16 - (311*Sin[c + d*x])/(16*d*(5 + 3*Cos[c + d*x])))/32)/16
3.1.33.3.1 Defintions of rubi rules used
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 = 0.82 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.71
| method | result | size |
| derivativedivides | \(\frac {\frac {-\frac {639 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {117 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {369 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}}{8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{3}}+\frac {385 \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{16384}}{d}\) | \(75\) |
| default | \(\frac {\frac {-\frac {639 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1024}-\frac {117 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}-\frac {369 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64}}{8 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+4\right )^{3}}+\frac {385 \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{16384}}{d}\) | \(75\) |
| risch | \(-\frac {i \left (10395 \,{\mathrm e}^{5 i \left (d x +c \right )}+86625 \,{\mathrm e}^{4 i \left (d x +c \right )}+239470 \,{\mathrm e}^{3 i \left (d x +c \right )}+218466 \,{\mathrm e}^{2 i \left (d x +c \right )}+73575 \,{\mathrm e}^{i \left (d x +c \right )}+8397\right )}{12288 d \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+10 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )^{3}}+\frac {385 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+3\right )}{32768 d}-\frac {385 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {1}{3}\right )}{32768 d}\) | \(127\) |
| parallelrisch | \(\frac {385 i \left (-770-27 \cos \left (3 d x +3 c \right )-981 \cos \left (d x +c \right )-270 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 i\right )+385 i \left (770+27 \cos \left (3 d x +3 c \right )+981 \cos \left (d x +c \right )+270 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 i\right )-175788 \sin \left (d x +c \right )-84240 \sin \left (2 d x +2 c \right )-11196 \sin \left (3 d x +3 c \right )}{32768 d \left (770+27 \cos \left (3 d x +3 c \right )+981 \cos \left (d x +c \right )+270 \cos \left (2 d x +2 c \right )\right )}\) | \(167\) |
1/d*(1/8*(-639/1024*tan(1/2*d*x+1/2*c)^5-117/32*tan(1/2*d*x+1/2*c)^3-369/6 4*tan(1/2*d*x+1/2*c))/(tan(1/2*d*x+1/2*c)^2+4)^3+385/16384*arctan(1/2*tan( 1/2*d*x+1/2*c)))
Time = 0.26 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=-\frac {385 \, {\left (27 \, \cos \left (d x + c\right )^{3} + 135 \, \cos \left (d x + c\right )^{2} + 225 \, \cos \left (d x + c\right ) + 125\right )} \arctan \left (\frac {5 \, \cos \left (d x + c\right ) + 3}{4 \, \sin \left (d x + c\right )}\right ) + 36 \, {\left (311 \, \cos \left (d x + c\right )^{2} + 1170 \, \cos \left (d x + c\right ) + 1143\right )} \sin \left (d x + c\right )}{32768 \, {\left (27 \, d \cos \left (d x + c\right )^{3} + 135 \, d \cos \left (d x + c\right )^{2} + 225 \, d \cos \left (d x + c\right ) + 125 \, d\right )}} \]
-1/32768*(385*(27*cos(d*x + c)^3 + 135*cos(d*x + c)^2 + 225*cos(d*x + c) + 125)*arctan(1/4*(5*cos(d*x + c) + 3)/sin(d*x + c)) + 36*(311*cos(d*x + c) ^2 + 1170*cos(d*x + c) + 1143)*sin(d*x + c))/(27*d*cos(d*x + c)^3 + 135*d* cos(d*x + c)^2 + 225*d*cos(d*x + c) + 125*d)
Result contains complex when optimal does not.
Time = 3.15 (sec) , antiderivative size = 595, normalized size of antiderivative = 5.61 \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=\begin {cases} \frac {x}{\left (-5 - 3 \cosh {\left (2 \operatorname {atanh}{\left (2 \right )} \right )}\right )^{4}} & \text {for}\: c = - d x - 2 i \operatorname {atanh}{\left (2 \right )} \vee c = - d x + 2 i \operatorname {atanh}{\left (2 \right )} \\\frac {x}{\left (- 3 \cos {\left (c \right )} - 5\right )^{4}} & \text {for}\: d = 0 \\\frac {385 \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16384 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1048576 d} + \frac {4620 \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16384 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1048576 d} + \frac {18480 \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right ) \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16384 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1048576 d} + \frac {24640 \left (\operatorname {atan}{\left (\frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{16384 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1048576 d} - \frac {1278 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16384 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1048576 d} - \frac {7488 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16384 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1048576 d} - \frac {11808 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{16384 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 1048576 d} & \text {otherwise} \end {cases} \]
Piecewise((x/(-5 - 3*cosh(2*atanh(2)))**4, Eq(c, -d*x - 2*I*atanh(2)) | Eq (c, -d*x + 2*I*atanh(2))), (x/(-3*cos(c) - 5)**4, Eq(d, 0)), (385*(atan(ta n(c/2 + d*x/2)/2) + pi*floor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)**6 /(16384*d*tan(c/2 + d*x/2)**6 + 196608*d*tan(c/2 + d*x/2)**4 + 786432*d*ta n(c/2 + d*x/2)**2 + 1048576*d) + 4620*(atan(tan(c/2 + d*x/2)/2) + pi*floor ((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)**4/(16384*d*tan(c/2 + d*x/2)** 6 + 196608*d*tan(c/2 + d*x/2)**4 + 786432*d*tan(c/2 + d*x/2)**2 + 1048576* d) + 18480*(atan(tan(c/2 + d*x/2)/2) + pi*floor((c/2 + d*x/2 - pi/2)/pi))* tan(c/2 + d*x/2)**2/(16384*d*tan(c/2 + d*x/2)**6 + 196608*d*tan(c/2 + d*x/ 2)**4 + 786432*d*tan(c/2 + d*x/2)**2 + 1048576*d) + 24640*(atan(tan(c/2 + d*x/2)/2) + pi*floor((c/2 + d*x/2 - pi/2)/pi))/(16384*d*tan(c/2 + d*x/2)** 6 + 196608*d*tan(c/2 + d*x/2)**4 + 786432*d*tan(c/2 + d*x/2)**2 + 1048576* d) - 1278*tan(c/2 + d*x/2)**5/(16384*d*tan(c/2 + d*x/2)**6 + 196608*d*tan( c/2 + d*x/2)**4 + 786432*d*tan(c/2 + d*x/2)**2 + 1048576*d) - 7488*tan(c/2 + d*x/2)**3/(16384*d*tan(c/2 + d*x/2)**6 + 196608*d*tan(c/2 + d*x/2)**4 + 786432*d*tan(c/2 + d*x/2)**2 + 1048576*d) - 11808*tan(c/2 + d*x/2)/(16384 *d*tan(c/2 + d*x/2)**6 + 196608*d*tan(c/2 + d*x/2)**4 + 786432*d*tan(c/2 + d*x/2)**2 + 1048576*d), True))
Time = 0.33 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.42 \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=-\frac {\frac {18 \, {\left (\frac {656 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {416 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {71 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {12 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 64} - 385 \, \arctan \left (\frac {\sin \left (d x + c\right )}{2 \, {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}{16384 \, d} \]
-1/16384*(18*(656*sin(d*x + c)/(cos(d*x + c) + 1) + 416*sin(d*x + c)^3/(co s(d*x + c) + 1)^3 + 71*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(48*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 12*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + sin(d *x + c)^6/(cos(d*x + c) + 1)^6 + 64) - 385*arctan(1/2*sin(d*x + c)/(cos(d* x + c) + 1)))/d
Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=\frac {385 \, d x + 385 \, c - \frac {36 \, {\left (71 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 416 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 656 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4\right )}^{3}} - 770 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 3}\right )}{32768 \, d} \]
1/32768*(385*d*x + 385*c - 36*(71*tan(1/2*d*x + 1/2*c)^5 + 416*tan(1/2*d*x + 1/2*c)^3 + 656*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 4)^3 - 7 70*arctan(sin(d*x + c)/(cos(d*x + c) + 3)))/d
Time = 0.00 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(-5-3 \cos (c+d x))^4} \, dx=\frac {385\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{16384\,d}-\frac {385\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{16384\,d}-\frac {\frac {639\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8192}+\frac {117\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{256}+\frac {369\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+4\right )}^3} \]