Integrand size = 12, antiderivative size = 108 \[ \int \frac {1}{(5-3 \sin (c+d x))^4} \, dx=\frac {385 x}{32768}-\frac {385 \arctan \left (\frac {\cos (c+d x)}{3-\sin (c+d x)}\right )}{16384 d}-\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac {25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac {311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))} \] Output:
385/32768*x-385/16384*arctan(cos(d*x+c)/(3-sin(d*x+c)))/d-1/16*cos(d*x+c)/ d/(5-3*sin(d*x+c))^3-25/512*cos(d*x+c)/d/(5-3*sin(d*x+c))^2-311/8192*cos(d *x+c)/d/(5-3*sin(d*x+c))
Time = 0.67 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(5-3 \sin (c+d x))^4} \, dx=\frac {-1925 \arctan \left (\frac {2 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}\right )+\frac {-239470+219735 \cos (c+d x)+83970 \cos (2 (c+d x))-13995 \cos (3 (c+d x))+305091 \sin (c+d x)-105300 \sin (2 (c+d x))-8397 \sin (3 (c+d x))}{2 (-5+3 \sin (c+d x))^3}}{81920 d} \] Input:
Integrate[(5 - 3*Sin[c + d*x])^(-4),x]
Output:
(-1925*ArcTan[(2*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2]))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])] + (-239470 + 219735*Cos[c + d*x] + 83970*Cos[2*(c + d *x)] - 13995*Cos[3*(c + d*x)] + 305091*Sin[c + d*x] - 105300*Sin[2*(c + d* x)] - 8397*Sin[3*(c + d*x)])/(2*(-5 + 3*Sin[c + d*x])^3))/(81920*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.
| \(\displaystyle \int \frac {1}{(5-3 \sin (c+d x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(5-3 \sin (c+d x))^4}dx\) |
| \(\Big \downarrow \) 3143 |
| \(\displaystyle -\frac {1}{48} \int -\frac {3 (2 \sin (c+d x)+5)}{(5-3 \sin (c+d x))^3}dx-\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \int \frac {2 \sin (c+d x)+5}{(5-3 \sin (c+d x))^3}dx-\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{16} \int \frac {2 \sin (c+d x)+5}{(5-3 \sin (c+d x))^3}dx-\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{16} \left (-\frac {1}{32} \int -\frac {25 \sin (c+d x)+62}{(5-3 \sin (c+d x))^2}dx-\frac {25 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}\right )-\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \int \frac {25 \sin (c+d x)+62}{(5-3 \sin (c+d x))^2}dx-\frac {25 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}\right )-\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \int \frac {25 \sin (c+d x)+62}{(5-3 \sin (c+d x))^2}dx-\frac {25 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}\right )-\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (-\frac {1}{16} \int -\frac {385}{5-3 \sin (c+d x)}dx-\frac {311 \cos (c+d x)}{16 d (5-3 \sin (c+d x))}\right )-\frac {25 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}\right )-\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {385}{16} \int \frac {1}{5-3 \sin (c+d x)}dx-\frac {311 \cos (c+d x)}{16 d (5-3 \sin (c+d x))}\right )-\frac {25 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}\right )-\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {385}{16} \int \frac {1}{5-3 \sin (c+d x)}dx-\frac {311 \cos (c+d x)}{16 d (5-3 \sin (c+d x))}\right )-\frac {25 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}\right )-\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}\) |
| \(\Big \downarrow \) 3136 |
| \(\displaystyle \frac {1}{16} \left (\frac {1}{32} \left (\frac {385}{16} \left (\frac {x}{4}-\frac {\arctan \left (\frac {\cos (c+d x)}{3-\sin (c+d x)}\right )}{2 d}\right )-\frac {311 \cos (c+d x)}{16 d (5-3 \sin (c+d x))}\right )-\frac {25 \cos (c+d x)}{32 d (5-3 \sin (c+d x))^2}\right )-\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}\) |
Input:
Int[(5 - 3*Sin[c + d*x])^(-4),x]
Output:
(((385*(x/4 - ArcTan[Cos[c + d*x]/(3 - Sin[c + d*x])]/(2*d)))/16 - (311*Co s[c + d*x])/(16*d*(5 - 3*Sin[c + d*x])))/32 - (25*Cos[c + d*x])/(32*d*(5 - 3*Sin[c + d*x])^2))/16 - Cos[c + d*x]/(16*d*(5 - 3*Sin[c + d*x])^3)
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 = 0.54 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.08
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {39933 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20480}-\frac {672723 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{102400}+\frac {2870073 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{256000}-\frac {604899 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{51200}+\frac {145233 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{20480}-\frac {10287}{4096}}{\left (5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )^{3}}+\frac {385 \arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {3}{4}\right )}{16384}}{d}\) | \(117\) |
| default | \(\frac {\frac {\frac {39933 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20480}-\frac {672723 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{102400}+\frac {2870073 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{256000}-\frac {604899 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{51200}+\frac {145233 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{20480}-\frac {10287}{4096}}{\left (5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )^{3}}+\frac {385 \arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {3}{4}\right )}{16384}}{d}\) | \(117\) |
| risch | \(-\frac {-239470 \,{\mathrm e}^{3 i \left (d x +c \right )}-86625 i {\mathrm e}^{4 i \left (d x +c \right )}+218466 i {\mathrm e}^{2 i \left (d x +c \right )}+10395 \,{\mathrm e}^{5 i \left (d x +c \right )}+73575 \,{\mathrm e}^{i \left (d x +c \right )}-8397 i}{12288 \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-3-10 i {\mathrm e}^{i \left (d x +c \right )}\right )^{3} d}+\frac {385 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}-3 i\right )}{32768 d}-\frac {385 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i}{3}\right )}{32768 d}\) | \(132\) |
| parallelrisch | \(\frac {31683960+48125 i \left (27 \sin \left (3 d x +3 c \right )-981 \sin \left (d x +c \right )-270 \cos \left (2 d x +2 c \right )+770\right ) \ln \left (5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3-4 i\right )+48125 i \left (-27 \sin \left (3 d x +3 c \right )+981 \sin \left (d x +c \right )+270 \cos \left (2 d x +2 c \right )-770\right ) \ln \left (5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3+4 i\right )+21973500 \cos \left (d x +c \right )-11109960 \cos \left (2 d x +2 c \right )-1399500 \cos \left (3 d x +3 c \right )-40366188 \sin \left (d x +c \right )-10530000 \sin \left (2 d x +2 c \right )+1110996 \sin \left (3 d x +3 c \right )}{4096000 d \left (-27 \sin \left (3 d x +3 c \right )+981 \sin \left (d x +c \right )+270 \cos \left (2 d x +2 c \right )-770\right )}\) | \(204\) |
Input:
int(1/(5-3*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/d*(250*(39933/5120000*tan(1/2*d*x+1/2*c)^5-672723/25600000*tan(1/2*d*x+1 /2*c)^4+2870073/64000000*tan(1/2*d*x+1/2*c)^3-604899/12800000*tan(1/2*d*x+ 1/2*c)^2+145233/5120000*tan(1/2*d*x+1/2*c)-10287/1024000)/(5*tan(1/2*d*x+1 /2*c)^2-6*tan(1/2*d*x+1/2*c)+5)^3+385/16384*arctan(5/4*tan(1/2*d*x+1/2*c)- 3/4))
Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(5-3 \sin (c+d x))^4} \, dx=-\frac {11196 \, \cos \left (d x + c\right )^{3} - 385 \, {\left (135 \, \cos \left (d x + c\right )^{2} - 9 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 28\right )} \sin \left (d x + c\right ) - 260\right )} \arctan \left (\frac {5 \, \sin \left (d x + c\right ) - 3}{4 \, \cos \left (d x + c\right )}\right ) + 42120 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 52344 \, \cos \left (d x + c\right )}{32768 \, {\left (135 \, d \cos \left (d x + c\right )^{2} - 9 \, {\left (3 \, d \cos \left (d x + c\right )^{2} - 28 \, d\right )} \sin \left (d x + c\right ) - 260 \, d\right )}} \] Input:
integrate(1/(5-3*sin(d*x+c))^4,x, algorithm="fricas")
Output:
-1/32768*(11196*cos(d*x + c)^3 - 385*(135*cos(d*x + c)^2 - 9*(3*cos(d*x + c)^2 - 28)*sin(d*x + c) - 260)*arctan(1/4*(5*sin(d*x + c) - 3)/cos(d*x + c )) + 42120*cos(d*x + c)*sin(d*x + c) - 52344*cos(d*x + c))/(135*d*cos(d*x + c)^2 - 9*(3*d*cos(d*x + c)^2 - 28*d)*sin(d*x + c) - 260*d)
Result contains complex when optimal does not.
Time = 4.66 (sec) , antiderivative size = 1690, normalized size of antiderivative = 15.65 \[ \int \frac {1}{(5-3 \sin (c+d x))^4} \, dx=\text {Too large to display} \] Input:
integrate(1/(5-3*sin(d*x+c))**4,x)
Output:
Piecewise((x/(5 - 3*sin(2*atan(3/5 - 4*I/5)))**4, Eq(c, -d*x + 2*atan(3/5 - 4*I/5))), (x/(5 - 3*sin(2*atan(3/5 + 4*I/5)))**4, Eq(c, -d*x + 2*atan(3/ 5 + 4*I/5))), (x/(5 - 3*sin(c))**4, Eq(d, 0)), (6015625*(atan(5*tan(c/2 + d*x/2)/4 - 3/4) + pi*floor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)**6/( 256000000*d*tan(c/2 + d*x/2)**6 - 921600000*d*tan(c/2 + d*x/2)**5 + 187392 0000*d*tan(c/2 + d*x/2)**4 - 2285568000*d*tan(c/2 + d*x/2)**3 + 1873920000 *d*tan(c/2 + d*x/2)**2 - 921600000*d*tan(c/2 + d*x/2) + 256000000*d) - 216 56250*(atan(5*tan(c/2 + d*x/2)/4 - 3/4) + pi*floor((c/2 + d*x/2 - pi/2)/pi ))*tan(c/2 + d*x/2)**5/(256000000*d*tan(c/2 + d*x/2)**6 - 921600000*d*tan( c/2 + d*x/2)**5 + 1873920000*d*tan(c/2 + d*x/2)**4 - 2285568000*d*tan(c/2 + d*x/2)**3 + 1873920000*d*tan(c/2 + d*x/2)**2 - 921600000*d*tan(c/2 + d*x /2) + 256000000*d) + 44034375*(atan(5*tan(c/2 + d*x/2)/4 - 3/4) + pi*floor ((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)**4/(256000000*d*tan(c/2 + d*x/ 2)**6 - 921600000*d*tan(c/2 + d*x/2)**5 + 1873920000*d*tan(c/2 + d*x/2)**4 - 2285568000*d*tan(c/2 + d*x/2)**3 + 1873920000*d*tan(c/2 + d*x/2)**2 - 9 21600000*d*tan(c/2 + d*x/2) + 256000000*d) - 53707500*(atan(5*tan(c/2 + d* x/2)/4 - 3/4) + pi*floor((c/2 + d*x/2 - pi/2)/pi))*tan(c/2 + d*x/2)**3/(25 6000000*d*tan(c/2 + d*x/2)**6 - 921600000*d*tan(c/2 + d*x/2)**5 + 18739200 00*d*tan(c/2 + d*x/2)**4 - 2285568000*d*tan(c/2 + d*x/2)**3 + 1873920000*d *tan(c/2 + d*x/2)**2 - 921600000*d*tan(c/2 + d*x/2) + 256000000*d) + 44...
Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (97) = 194\).
Time = 0.11 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.34 \[ \int \frac {1}{(5-3 \sin (c+d x))^4} \, dx=-\frac {\frac {36 \, {\left (\frac {403425 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {672110 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {637794 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {373735 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {110925 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 142875\right )}}{\frac {450 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {915 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1116 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {915 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {450 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {125 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 125} - 48125 \, \arctan \left (\frac {5 \, \sin \left (d x + c\right )}{4 \, {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {3}{4}\right )}{2048000 \, d} \] Input:
integrate(1/(5-3*sin(d*x+c))^4,x, algorithm="maxima")
Output:
-1/2048000*(36*(403425*sin(d*x + c)/(cos(d*x + c) + 1) - 672110*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 637794*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 3 73735*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 110925*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 142875)/(450*sin(d*x + c)/(cos(d*x + c) + 1) - 915*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1116*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 9 15*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 450*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 125*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 125) - 48125*arctan(5/4 *sin(d*x + c)/(cos(d*x + c) + 1) - 3/4))/d
Time = 0.12 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(5-3 \sin (c+d x))^4} \, dx=\frac {48125 \, d x + 48125 \, c + \frac {72 \, {\left (110925 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 373735 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 637794 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 672110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 403425 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 142875\right )}}{{\left (5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5\right )}^{3}} + 96250 \, \arctan \left (\frac {3 \, \cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 3}{\cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) - 9}\right )}{4096000 \, d} \] Input:
integrate(1/(5-3*sin(d*x+c))^4,x, algorithm="giac")
Output:
1/4096000*(48125*d*x + 48125*c + 72*(110925*tan(1/2*d*x + 1/2*c)^5 - 37373 5*tan(1/2*d*x + 1/2*c)^4 + 637794*tan(1/2*d*x + 1/2*c)^3 - 672110*tan(1/2* d*x + 1/2*c)^2 + 403425*tan(1/2*d*x + 1/2*c) - 142875)/(5*tan(1/2*d*x + 1/ 2*c)^2 - 6*tan(1/2*d*x + 1/2*c) + 5)^3 + 96250*arctan((3*cos(d*x + c) - si n(d*x + c) + 3)/(cos(d*x + c) + 3*sin(d*x + c) - 9)))/d
Time = 26.42 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.73 \[ \int \frac {1}{(5-3 \sin (c+d x))^4} \, dx=\frac {385\,\mathrm {atan}\left (\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}-\frac {3}{4}\right )}{16384\,d}-\frac {385\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{16384\,d}+\frac {\frac {39933\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2560000}-\frac {672723\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{12800000}+\frac {2870073\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32000000}-\frac {604899\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6400000}+\frac {145233\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2560000}-\frac {10287}{512000}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+\frac {183\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{25}-\frac {1116\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{125}+\frac {183\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{25}-\frac {18\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}+1\right )} \] Input:
int(1/(3*sin(c + d*x) - 5)^4,x)
Output:
(385*atan((5*tan(c/2 + (d*x)/2))/4 - 3/4))/(16384*d) - (385*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(16384*d) + ((145233*tan(c/2 + (d*x)/2))/2560000 - (604899*tan(c/2 + (d*x)/2)^2)/6400000 + (2870073*tan(c/2 + (d*x)/2)^3)/32 000000 - (672723*tan(c/2 + (d*x)/2)^4)/12800000 + (39933*tan(c/2 + (d*x)/2 )^5)/2560000 - 10287/512000)/(d*((183*tan(c/2 + (d*x)/2)^2)/25 - (18*tan(c /2 + (d*x)/2))/5 - (1116*tan(c/2 + (d*x)/2)^3)/125 + (183*tan(c/2 + (d*x)/ 2)^4)/25 - (18*tan(c/2 + (d*x)/2)^5)/5 + tan(c/2 + (d*x)/2)^6 + 1))
Time = 0.16 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.77 \[ \int \frac {1}{(5-3 \sin (c+d x))^4} \, dx=\frac {51975 \mathit {atan} \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {3}{4}\right ) \sin \left (d x +c \right )^{3}-259875 \mathit {atan} \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {3}{4}\right ) \sin \left (d x +c \right )^{2}+433125 \mathit {atan} \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {3}{4}\right ) \sin \left (d x +c \right )-240625 \mathit {atan} \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {3}{4}\right )+27990 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-105300 \cos \left (d x +c \right ) \sin \left (d x +c \right )+102870 \cos \left (d x +c \right )-12636 \sin \left (d x +c \right )^{3}+63180 \sin \left (d x +c \right )^{2}-105300 \sin \left (d x +c \right )+58500}{81920 d \left (27 \sin \left (d x +c \right )^{3}-135 \sin \left (d x +c \right )^{2}+225 \sin \left (d x +c \right )-125\right )} \] Input:
int(1/(5-3*sin(d*x+c))^4,x)
Output:
(51975*atan((5*tan((c + d*x)/2) - 3)/4)*sin(c + d*x)**3 - 259875*atan((5*t an((c + d*x)/2) - 3)/4)*sin(c + d*x)**2 + 433125*atan((5*tan((c + d*x)/2) - 3)/4)*sin(c + d*x) - 240625*atan((5*tan((c + d*x)/2) - 3)/4) + 27990*cos (c + d*x)*sin(c + d*x)**2 - 105300*cos(c + d*x)*sin(c + d*x) + 102870*cos( c + d*x) - 12636*sin(c + d*x)**3 + 63180*sin(c + d*x)**2 - 105300*sin(c + d*x) + 58500)/(81920*d*(27*sin(c + d*x)**3 - 135*sin(c + d*x)**2 + 225*sin (c + d*x) - 125))