Integrand size = 12, antiderivative size = 108 \[ \int \frac {1}{(5-3 \cos (c+d x))^4} \, dx=\frac {385 x}{32768}+\frac {385 \arctan \left (\frac {\sin (c+d x)}{3-\cos (c+d x)}\right )}{16384 d}+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}+\frac {25 \sin (c+d x)}{512 d (5-3 \cos (c+d x))^2}+\frac {311 \sin (c+d x)}{8192 d (5-3 \cos (c+d x))} \]
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 = 0.35 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {1}{(5-3 \cos (c+d x))^4} \, dx=\frac {770 \arctan \left (2 \tan \left (\frac {1}{2} (c+d x)\right )\right )-\frac {9 (4883 \sin (c+d x)-2340 \sin (2 (c+d x))+311 \sin (3 (c+d x)))}{(-5+3 \cos (c+d x))^3}}{32768 d} \]
(770*ArcTan[2*Tan[(c + d*x)/2]] - (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)
Time = 0.51 (sec) , antiderivative size = 123, 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.
| \(\displaystyle \int \frac {1}{(5-3 \cos (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (5-3 \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3143 |
| \(\displaystyle \frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}-\frac {1}{48} \int -\frac {3 (2 \cos (c+d x)+5)}{(5-3 \cos (c+d x))^3}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \int \frac {2 \cos (c+d x)+5}{(5-3 \cos (c+d x))^3}dx+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{16} \int \frac {2 \sin \left (c+d x+\frac {\pi }{2}\right )+5}{\left (5-3 \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3}dx+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{16} \left (\frac {25 \sin (c+d x)}{32 d (5-3 \cos (c+d x))^2}-\frac {1}{32} \int -\frac {25 \cos (c+d x)+62}{(5-3 \cos (c+d x))^2}dx\right )+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \int \frac {25 \cos (c+d x)+62}{(5-3 \cos (c+d x))^2}dx+\frac {25 \sin (c+d x)}{32 d (5-3 \cos (c+d x))^2}\right )+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \int \frac {25 \sin \left (c+d x+\frac {\pi }{2}\right )+62}{\left (5-3 \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx+\frac {25 \sin (c+d x)}{32 d (5-3 \cos (c+d x))^2}\right )+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {311 \sin (c+d x)}{16 d (5-3 \cos (c+d x))}-\frac {1}{16} \int -\frac {385}{5-3 \cos (c+d x)}dx\right )+\frac {25 \sin (c+d x)}{32 d (5-3 \cos (c+d x))^2}\right )+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {385}{16} \int \frac {1}{5-3 \cos (c+d x)}dx+\frac {311 \sin (c+d x)}{16 d (5-3 \cos (c+d x))}\right )+\frac {25 \sin (c+d x)}{32 d (5-3 \cos (c+d x))^2}\right )+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {385}{16} \int \frac {1}{5-3 \sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {311 \sin (c+d x)}{16 d (5-3 \cos (c+d x))}\right )+\frac {25 \sin (c+d x)}{32 d (5-3 \cos (c+d x))^2}\right )+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}\) |
| \(\Big \downarrow \) 3136 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {385}{16} \left (\frac {\arctan \left (\frac {\sin (c+d x)}{3-\cos (c+d x)}\right )}{2 d}+\frac {x}{4}\right )+\frac {311 \sin (c+d x)}{16 d (5-3 \cos (c+d x))}\right )+\frac {25 \sin (c+d x)}{32 d (5-3 \cos (c+d x))^2}\right )+\frac {\sin (c+d x)}{16 d (5-3 \cos (c+d x))^3}\) |
Sin[c + d*x]/(16*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
3.1.25.3.1 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 = 0.84 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.71
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {369 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}+\frac {117 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {639 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8192}}{{\left (4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right )}^{3}}+\frac {385 \arctan \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16384}}{d}\) | \(77\) |
| default | \(\frac {\frac {\frac {369 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{512}+\frac {117 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {639 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8192}}{{\left (4 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right )}^{3}}+\frac {385 \arctan \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16384}}{d}\) | \(77\) |
| risch | \(\frac {i \left (10395 \,{\mathrm e}^{5 i \left (d x +c \right )}-86625 \,{\mathrm e}^{4 i \left (d x +c \right )}+239470 \,{\mathrm e}^{3 i \left (d x +c \right )}-218466 \,{\mathrm e}^{2 i \left (d x +c \right )}+73575 \,{\mathrm e}^{i \left (d x +c \right )}-8397\right )}{12288 d \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-10 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )^{3}}-\frac {385 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {1}{3}\right )}{32768 d}+\frac {385 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}-3\right )}{32768 d}\) | \(127\) |
| parallelrisch | \(\frac {385 i \left (770-27 \cos \left (3 d x +3 c \right )-981 \cos \left (d x +c \right )+270 \cos \left (2 d x +2 c \right )\right ) \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )+385 i \left (27 \cos \left (3 d x +3 c \right )+981 \cos \left (d x +c \right )-270 \cos \left (2 d x +2 c \right )-770\right ) \ln \left (2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )-175788 \sin \left (d x +c \right )+84240 \sin \left (2 d x +2 c \right )-11196 \sin \left (3 d x +3 c \right )}{32768 d \left (27 \cos \left (3 d x +3 c \right )+981 \cos \left (d x +c \right )-270 \cos \left (2 d x +2 c \right )-770\right )}\) | \(171\) |
1/d*(8*(369/4096*tan(1/2*d*x+1/2*c)^5+117/2048*tan(1/2*d*x+1/2*c)^3+639/65 536*tan(1/2*d*x+1/2*c))/(4*tan(1/2*d*x+1/2*c)^2+1)^3+385/16384*arctan(2*ta n(1/2*d*x+1/2*c)))
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.12 \[ \int \frac {1}{(5-3 \cos (c+d x))^4} \, dx=-\frac {385 \, {\left (27 \, \cos \left (d x + c\right )^{3} - 135 \, \cos \left (d x + c\right )^{2} + 225 \, \cos \left (d x + c\right ) - 125\right )} \arctan \left (\frac {5 \, \cos \left (d x + c\right ) - 3}{4 \, \sin \left (d x + c\right )}\right ) + 36 \, {\left (311 \, \cos \left (d x + c\right )^{2} - 1170 \, \cos \left (d x + c\right ) + 1143\right )} \sin \left (d x + c\right )}{32768 \, {\left (27 \, d \cos \left (d x + c\right )^{3} - 135 \, d \cos \left (d x + c\right )^{2} + 225 \, d \cos \left (d x + c\right ) - 125 \, d\right )}} \]
-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 = 2.85 (sec) , antiderivative size = 597, normalized size of antiderivative = 5.53 \[ \int \frac {1}{(5-3 \cos (c+d x))^4} \, dx=\begin {cases} \frac {x}{\left (5 - 3 \cosh {\left (2 \operatorname {atanh}{\left (\frac {1}{2} \right )} \right )}\right )^{4}} & \text {for}\: c = - d x - 2 i \operatorname {atanh}{\left (\frac {1}{2} \right )} \vee c = - d x + 2 i \operatorname {atanh}{\left (\frac {1}{2} \right )} \\\frac {x}{\left (5 - 3 \cos {\left (c \right )}\right )^{4}} & \text {for}\: d = 0 \\\frac {24640 \left (\operatorname {atan}{\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \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 )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {18480 \left (\operatorname {atan}{\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \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 )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {4620 \left (\operatorname {atan}{\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \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 )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {385 \left (\operatorname {atan}{\left (2 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {11808 \tan ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {7488 \tan ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {1278 \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1048576 d \tan ^{6}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 786432 d \tan ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 196608 d \tan ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} & \text {otherwise} \end {cases} \]
Piecewise((x/(5 - 3*cosh(2*atanh(1/2)))**4, Eq(c, -d*x - 2*I*atanh(1/2)) | Eq(c, -d*x + 2*I*atanh(1/2))), (x/(5 - 3*cos(c))**4, Eq(d, 0)), (24640*(a tan(2*tan(c/2 + d*x/2)) + pi*floor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x /2)**6/(1048576*d*tan(c/2 + d*x/2)**6 + 786432*d*tan(c/2 + d*x/2)**4 + 196 608*d*tan(c/2 + d*x/2)**2 + 16384*d) + 18480*(atan(2*tan(c/2 + d*x/2)) + p i*floor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)**4/(1048576*d*tan(c/2 + d*x/2)**6 + 786432*d*tan(c/2 + d*x/2)**4 + 196608*d*tan(c/2 + d*x/2)**2 + 16384*d) + 4620*(atan(2*tan(c/2 + d*x/2)) + pi*floor((c/2 + d*x/2 - pi/2) /pi))*tan(c/2 + d*x/2)**2/(1048576*d*tan(c/2 + d*x/2)**6 + 786432*d*tan(c/ 2 + d*x/2)**4 + 196608*d*tan(c/2 + d*x/2)**2 + 16384*d) + 385*(atan(2*tan( c/2 + d*x/2)) + pi*floor((c/2 + d*x/2 - pi/2)/pi))/(1048576*d*tan(c/2 + d* x/2)**6 + 786432*d*tan(c/2 + d*x/2)**4 + 196608*d*tan(c/2 + d*x/2)**2 + 16 384*d) + 11808*tan(c/2 + d*x/2)**5/(1048576*d*tan(c/2 + d*x/2)**6 + 786432 *d*tan(c/2 + d*x/2)**4 + 196608*d*tan(c/2 + d*x/2)**2 + 16384*d) + 7488*ta n(c/2 + d*x/2)**3/(1048576*d*tan(c/2 + d*x/2)**6 + 786432*d*tan(c/2 + d*x/ 2)**4 + 196608*d*tan(c/2 + d*x/2)**2 + 16384*d) + 1278*tan(c/2 + d*x/2)/(1 048576*d*tan(c/2 + d*x/2)**6 + 786432*d*tan(c/2 + d*x/2)**4 + 196608*d*tan (c/2 + d*x/2)**2 + 16384*d), True))
Time = 0.35 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.41 \[ \int \frac {1}{(5-3 \cos (c+d x))^4} \, dx=\frac {\frac {18 \, {\left (\frac {71 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {416 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {656 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {12 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {48 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {64 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 1} + 385 \, \arctan \left (\frac {2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{16384 \, d} \]
1/16384*(18*(71*sin(d*x + c)/(cos(d*x + c) + 1) + 416*sin(d*x + c)^3/(cos( d*x + c) + 1)^3 + 656*sin(d*x + c)^5/(cos(d*x + c) + 1)^5)/(12*sin(d*x + c )^2/(cos(d*x + c) + 1)^2 + 48*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 64*sin (d*x + c)^6/(cos(d*x + c) + 1)^6 + 1) + 385*arctan(2*sin(d*x + c)/(cos(d*x + c) + 1)))/d
Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(5-3 \cos (c+d x))^4} \, dx=\frac {385 \, d x + 385 \, c + \frac {36 \, {\left (656 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 416 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 71 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (4 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{3}} - 770 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) - 3}\right )}{32768 \, d} \]
1/32768*(385*d*x + 385*c + 36*(656*tan(1/2*d*x + 1/2*c)^5 + 416*tan(1/2*d* x + 1/2*c)^3 + 71*tan(1/2*d*x + 1/2*c))/(4*tan(1/2*d*x + 1/2*c)^2 + 1)^3 - 770*arctan(sin(d*x + c)/(cos(d*x + c) - 3)))/d
Time = 15.63 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(5-3 \cos (c+d x))^4} \, dx=\frac {385\,\mathrm {atan}\left (2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{16384\,d}-\frac {385\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{16384\,d}+\frac {\frac {369\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{512}+\frac {117\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{256}+\frac {639\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8192}}{d\,{\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3} \]