Integrand size = 29, antiderivative size = 129 \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {11 x}{16 a^2}+\frac {2 \cos (c+d x)}{a^2 d}-\frac {4 \cos ^3(c+d x)}{3 a^2 d}+\frac {2 \cos ^5(c+d x)}{5 a^2 d}-\frac {11 \cos (c+d x) \sin (c+d x)}{16 a^2 d}-\frac {11 \cos (c+d x) \sin ^3(c+d x)}{24 a^2 d}-\frac {\cos (c+d x) \sin ^5(c+d x)}{6 a^2 d} \]
11/16*x/a^2+2*cos(d*x+c)/a^2/d-4/3*cos(d*x+c)^3/a^2/d+2/5*cos(d*x+c)^5/a^2 /d-11/16*cos(d*x+c)*sin(d*x+c)/a^2/d-11/24*cos(d*x+c)*sin(d*x+c)^3/a^2/d-1 /6*cos(d*x+c)*sin(d*x+c)^5/a^2/d
Time = 0.76 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.59 \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {660 c+660 d x+1200 \cos (c+d x)-200 \cos (3 (c+d x))+24 \cos (5 (c+d x))-465 \sin (2 (c+d x))+75 \sin (4 (c+d x))-5 \sin (6 (c+d x))}{960 a^2 d} \]
(660*c + 660*d*x + 1200*Cos[c + d*x] - 200*Cos[3*(c + d*x)] + 24*Cos[5*(c + d*x)] - 465*Sin[2*(c + d*x)] + 75*Sin[4*(c + d*x)] - 5*Sin[6*(c + d*x)]) /(960*a^2*d)
Time = 0.45 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3348, 3042, 3236, 2009}
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.
| \(\displaystyle \int \frac {\sin ^4(c+d x) \cos ^4(c+d x)}{(a \sin (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^4 \cos (c+d x)^4}{(a \sin (c+d x)+a)^2}dx\) |
| \(\Big \downarrow \) 3348 |
| \(\displaystyle \frac {\int \sin ^4(c+d x) (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin (c+d x)^4 (a-a \sin (c+d x))^2dx}{a^4}\) |
| \(\Big \downarrow \) 3236 |
| \(\displaystyle \frac {\int \left (a^2 \sin ^6(c+d x)-2 a^2 \sin ^5(c+d x)+a^2 \sin ^4(c+d x)\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {2 a^2 \cos ^5(c+d x)}{5 d}-\frac {4 a^2 \cos ^3(c+d x)}{3 d}+\frac {2 a^2 \cos (c+d x)}{d}-\frac {a^2 \sin ^5(c+d x) \cos (c+d x)}{6 d}-\frac {11 a^2 \sin ^3(c+d x) \cos (c+d x)}{24 d}-\frac {11 a^2 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {11 a^2 x}{16}}{a^4}\) |
((11*a^2*x)/16 + (2*a^2*Cos[c + d*x])/d - (4*a^2*Cos[c + d*x]^3)/(3*d) + ( 2*a^2*Cos[c + d*x]^5)/(5*d) - (11*a^2*Cos[c + d*x]*Sin[c + d*x])/(16*d) - (11*a^2*Cos[c + d*x]*Sin[c + d*x]^3)/(24*d) - (a^2*Cos[c + d*x]*Sin[c + d* x]^5)/(6*d))/a^4
3.5.21.3.1 Defintions of rubi rules used
Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*( x_)])^(m_.), x_Symbol] :> Int[ExpandTrig[(a + b*sin[e + f*x])^m*(d*sin[e + f*x])^n, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IGt Q[m, 0] && RationalQ[n]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[a^(2*m) Int[(d* Sin[e + f*x])^n/(a - b*Sin[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p] && EqQ[2*m + p, 0]
Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60
| method | result | size |
| parallelrisch | \(\frac {660 d x -200 \cos \left (3 d x +3 c \right )+1200 \cos \left (d x +c \right )-5 \sin \left (6 d x +6 c \right )+24 \cos \left (5 d x +5 c \right )+75 \sin \left (4 d x +4 c \right )-465 \sin \left (2 d x +2 c \right )+1024}{960 d \,a^{2}}\) | \(78\) |
| risch | \(\frac {11 x}{16 a^{2}}+\frac {5 \cos \left (d x +c \right )}{4 a^{2} d}-\frac {\sin \left (6 d x +6 c \right )}{192 d \,a^{2}}+\frac {\cos \left (5 d x +5 c \right )}{40 d \,a^{2}}+\frac {5 \sin \left (4 d x +4 c \right )}{64 d \,a^{2}}-\frac {5 \cos \left (3 d x +3 c \right )}{24 d \,a^{2}}-\frac {31 \sin \left (2 d x +2 c \right )}{64 d \,a^{2}}\) | \(107\) |
| derivativedivides | \(\frac {\frac {32 \left (\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {187 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}+\frac {47 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {47 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {187 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256}+\frac {1}{15}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d \,a^{2}}\) | \(153\) |
| default | \(\frac {\frac {32 \left (\frac {11 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256}+\frac {187 \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}+\frac {47 \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {2 \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3}-\frac {47 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {187 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768}+\frac {2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5}-\frac {11 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{256}+\frac {1}{15}\right )}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}+\frac {11 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{d \,a^{2}}\) | \(153\) |
1/960*(660*d*x-200*cos(3*d*x+3*c)+1200*cos(d*x+c)-5*sin(6*d*x+6*c)+24*cos( 5*d*x+5*c)+75*sin(4*d*x+4*c)-465*sin(2*d*x+2*c)+1024)/d/a^2
Time = 0.26 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60 \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {96 \, \cos \left (d x + c\right )^{5} - 320 \, \cos \left (d x + c\right )^{3} + 165 \, d x - 5 \, {\left (8 \, \cos \left (d x + c\right )^{5} - 38 \, \cos \left (d x + c\right )^{3} + 63 \, \cos \left (d x + c\right )\right )} \sin \left (d x + c\right ) + 480 \, \cos \left (d x + c\right )}{240 \, a^{2} d} \]
1/240*(96*cos(d*x + c)^5 - 320*cos(d*x + c)^3 + 165*d*x - 5*(8*cos(d*x + c )^5 - 38*cos(d*x + c)^3 + 63*cos(d*x + c))*sin(d*x + c) + 480*cos(d*x + c) )/(a^2*d)
Leaf count of result is larger than twice the leaf count of optimal. 2271 vs. \(2 (122) = 244\).
Time = 53.17 (sec) , antiderivative size = 2271, normalized size of antiderivative = 17.60 \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\text {Too large to display} \]
Piecewise((165*d*x*tan(c/2 + d*x/2)**12/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800 *a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2* d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 990*d*x*tan(c/2 + d*x/2)**10/(240*a* *2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d *tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c /2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 2475*d*x* tan(c/2 + d*x/2)**8/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d* x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)** 2 + 240*a**2*d) + 3300*d*x*tan(c/2 + d*x/2)**6/(240*a**2*d*tan(c/2 + d*x/2 )**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 144 0*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 2475*d*x*tan(c/2 + d*x/2)**4/ (240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*tan(c/2 + d*x/2)**10 + 3600 *a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/2 + d*x/2)**6 + 3600*a**2* d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 + d*x/2)**2 + 240*a**2*d) + 99 0*d*x*tan(c/2 + d*x/2)**2/(240*a**2*d*tan(c/2 + d*x/2)**12 + 1440*a**2*d*t an(c/2 + d*x/2)**10 + 3600*a**2*d*tan(c/2 + d*x/2)**8 + 4800*a**2*d*tan(c/ 2 + d*x/2)**6 + 3600*a**2*d*tan(c/2 + d*x/2)**4 + 1440*a**2*d*tan(c/2 +...
Leaf count of result is larger than twice the leaf count of optimal. 353 vs. \(2 (117) = 234\).
Time = 0.30 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.74 \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=-\frac {\frac {\frac {165 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1536 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {935 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {3840 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {1410 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {2560 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {1410 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {935 \, \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {165 \, \sin \left (d x + c\right )^{11}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{11}} - 256}{a^{2} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {20 \, a^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {15 \, a^{2} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}} + \frac {a^{2} \sin \left (d x + c\right )^{12}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{12}}} - \frac {165 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{2}}}{120 \, d} \]
-1/120*((165*sin(d*x + c)/(cos(d*x + c) + 1) - 1536*sin(d*x + c)^2/(cos(d* x + c) + 1)^2 + 935*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 3840*sin(d*x + c )^4/(cos(d*x + c) + 1)^4 + 1410*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 2560 *sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 1410*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 - 935*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 - 165*sin(d*x + c)^11/(cos (d*x + c) + 1)^11 - 256)/(a^2 + 6*a^2*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 15*a^2*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 20*a^2*sin(d*x + c)^6/(cos( d*x + c) + 1)^6 + 15*a^2*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 6*a^2*sin(d *x + c)^10/(cos(d*x + c) + 1)^10 + a^2*sin(d*x + c)^12/(cos(d*x + c) + 1)^ 12) - 165*arctan(sin(d*x + c)/(cos(d*x + c) + 1))/a^2)/d
Time = 0.45 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.19 \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {\frac {165 \, {\left (d x + c\right )}}{a^{2}} + \frac {2 \, {\left (165 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 935 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 1410 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2560 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1410 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3840 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 935 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1536 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 165 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 256\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6} a^{2}}}{240 \, d} \]
1/240*(165*(d*x + c)/a^2 + 2*(165*tan(1/2*d*x + 1/2*c)^11 + 935*tan(1/2*d* x + 1/2*c)^9 + 1410*tan(1/2*d*x + 1/2*c)^7 + 2560*tan(1/2*d*x + 1/2*c)^6 - 1410*tan(1/2*d*x + 1/2*c)^5 + 3840*tan(1/2*d*x + 1/2*c)^4 - 935*tan(1/2*d *x + 1/2*c)^3 + 1536*tan(1/2*d*x + 1/2*c)^2 - 165*tan(1/2*d*x + 1/2*c) + 2 56)/((tan(1/2*d*x + 1/2*c)^2 + 1)^6*a^2))/d
Time = 12.73 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.13 \[ \int \frac {\cos ^4(c+d x) \sin ^4(c+d x)}{(a+a \sin (c+d x))^2} \, dx=\frac {11\,x}{16\,a^2}+\frac {\frac {11\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {187\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}+\frac {47\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}-\frac {47\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {187\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}-\frac {11\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}+\frac {32}{15}}{a^2\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]
(11*x)/(16*a^2) + ((64*tan(c/2 + (d*x)/2)^2)/5 - (11*tan(c/2 + (d*x)/2))/8 - (187*tan(c/2 + (d*x)/2)^3)/24 + 32*tan(c/2 + (d*x)/2)^4 - (47*tan(c/2 + (d*x)/2)^5)/4 + (64*tan(c/2 + (d*x)/2)^6)/3 + (47*tan(c/2 + (d*x)/2)^7)/4 + (187*tan(c/2 + (d*x)/2)^9)/24 + (11*tan(c/2 + (d*x)/2)^11)/8 + 32/15)/( a^2*d*(tan(c/2 + (d*x)/2)^2 + 1)^6)