Integrand size = 27, antiderivative size = 111 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {10 \log (1+\sin (c+d x))}{a^3 d}+\frac {6 \sin (c+d x)}{a^3 d}-\frac {3 \sin ^2(c+d x)}{2 a^3 d}+\frac {\sin ^3(c+d x)}{3 a^3 d}+\frac {1}{2 a d (a+a \sin (c+d x))^2}-\frac {5}{d \left (a^3+a^3 \sin (c+d x)\right )} \] Output:
-10*ln(1+sin(d*x+c))/a^3/d+6*sin(d*x+c)/a^3/d-3/2*sin(d*x+c)^2/a^3/d+1/3*s in(d*x+c)^3/a^3/d+1/2/a/d/(a+a*sin(d*x+c))^2-5/d/(a^3+a^3*sin(d*x+c))
Time = 0.50 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.95 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-417-960 \log (1+\sin (c+d x))-6 (-21+320 \log (1+\sin (c+d x))) \sin (c+d x)+(1023-960 \log (1+\sin (c+d x))) \sin ^2(c+d x)+320 \sin ^3(c+d x)-80 \sin ^4(c+d x)+32 \sin ^5(c+d x)}{96 a^3 d (1+\sin (c+d x))^2} \] Input:
Integrate[(Cos[c + d*x]*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^3,x]
Output:
(-417 - 960*Log[1 + Sin[c + d*x]] - 6*(-21 + 320*Log[1 + Sin[c + d*x]])*Si n[c + d*x] + (1023 - 960*Log[1 + Sin[c + d*x]])*Sin[c + d*x]^2 + 320*Sin[c + d*x]^3 - 80*Sin[c + d*x]^4 + 32*Sin[c + d*x]^5)/(96*a^3*d*(1 + Sin[c + d*x])^2)
Time = 0.32 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {3042, 3312, 27, 49, 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.
| \(\displaystyle \int \frac {\sin ^5(c+d x) \cos (c+d x)}{(a \sin (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^5 \cos (c+d x)}{(a \sin (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3312 |
| \(\displaystyle \frac {\int \frac {\sin ^5(c+d x)}{(\sin (c+d x) a+a)^3}d(a \sin (c+d x))}{a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a^5 \sin ^5(c+d x)}{(\sin (c+d x) a+a)^3}d(a \sin (c+d x))}{a^6 d}\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {\int \left (-\frac {a^5}{(\sin (c+d x) a+a)^3}+\frac {5 a^4}{(\sin (c+d x) a+a)^2}-\frac {10 a^3}{\sin (c+d x) a+a}+\sin ^2(c+d x) a^2-3 \sin (c+d x) a^2+6 a^2\right )d(a \sin (c+d x))}{a^6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {a^5}{2 (a \sin (c+d x)+a)^2}-\frac {5 a^4}{a \sin (c+d x)+a}+\frac {1}{3} a^3 \sin ^3(c+d x)-\frac {3}{2} a^3 \sin ^2(c+d x)+6 a^3 \sin (c+d x)-10 a^3 \log (a \sin (c+d x)+a)}{a^6 d}\) |
Input:
Int[(Cos[c + d*x]*Sin[c + d*x]^5)/(a + a*Sin[c + d*x])^3,x]
Output:
(-10*a^3*Log[a + a*Sin[c + d*x]] + 6*a^3*Sin[c + d*x] - (3*a^3*Sin[c + d*x ]^2)/2 + (a^3*Sin[c + d*x]^3)/3 + a^5/(2*(a + a*Sin[c + d*x])^2) - (5*a^4) /(a + a*Sin[c + d*x]))/(a^6*d)
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_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0] && IGtQ[m + n + 2, 0]
Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*(( c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b*f) Su bst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 1.13 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.65
| method | result | size |
| derivativedivides | \(\frac {\frac {\sin \left (d x +c \right )^{3}}{3}-\frac {3 \sin \left (d x +c \right )^{2}}{2}+6 \sin \left (d x +c \right )+\frac {1}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {5}{1+\sin \left (d x +c \right )}-10 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{3}}\) | \(72\) |
| default | \(\frac {\frac {\sin \left (d x +c \right )^{3}}{3}-\frac {3 \sin \left (d x +c \right )^{2}}{2}+6 \sin \left (d x +c \right )+\frac {1}{2 \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {5}{1+\sin \left (d x +c \right )}-10 \ln \left (1+\sin \left (d x +c \right )\right )}{d \,a^{3}}\) | \(72\) |
| parallelrisch | \(\frac {\left (240 \cos \left (2 d x +2 c \right )-960 \sin \left (d x +c \right )-720\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\left (-480 \cos \left (2 d x +2 c \right )+1920 \sin \left (d x +c \right )+1440\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+340 \cos \left (2 d x +2 c \right )+5 \cos \left (4 d x +4 c \right )-610 \sin \left (d x +c \right )+45 \sin \left (3 d x +3 c \right )-\sin \left (5 d x +5 c \right )-345}{24 d \,a^{3} \left (-3+\cos \left (2 d x +2 c \right )-4 \sin \left (d x +c \right )\right )}\) | \(152\) |
| risch | \(\frac {10 i x}{a^{3}}+\frac {i {\mathrm e}^{3 i \left (d x +c \right )}}{24 d \,a^{3}}+\frac {3 \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {25 i {\mathrm e}^{i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {25 i {\mathrm e}^{-i \left (d x +c \right )}}{8 d \,a^{3}}+\frac {3 \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 d \,a^{3}}-\frac {i {\mathrm e}^{-3 i \left (d x +c \right )}}{24 d \,a^{3}}+\frac {20 i c}{d \,a^{3}}-\frac {2 i \left (-5 \,{\mathrm e}^{i \left (d x +c \right )}+9 i {\mathrm e}^{2 i \left (d x +c \right )}+5 \,{\mathrm e}^{3 i \left (d x +c \right )}\right )}{d \,a^{3} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{4}}-\frac {20 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d \,a^{3}}\) | \(204\) |
| norman | \(\frac {\frac {20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a}+\frac {20 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{d a}+\frac {80 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d a}+\frac {80 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{d a}+\frac {680 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d a}+\frac {680 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{3 d a}+\frac {1540 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 d a}+\frac {1540 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{3 d a}+\frac {5536 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d a}+\frac {5536 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{3 d a}+\frac {4232 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d a}+\frac {4232 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 d a}+\frac {2732 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d a}+\frac {2732 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{3 d a}+\frac {6356 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d a}+\frac {6356 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6} a^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {20 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{3}}+\frac {10 \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{d \,a^{3}}\) | \(379\) |
Input:
int(cos(d*x+c)*sin(d*x+c)^5/(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d/a^3*(1/3*sin(d*x+c)^3-3/2*sin(d*x+c)^2+6*sin(d*x+c)+1/2/(1+sin(d*x+c)) ^2-5/(1+sin(d*x+c))-10*ln(1+sin(d*x+c)))
Time = 0.08 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.05 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {10 \, \cos \left (d x + c\right )^{4} + 115 \, \cos \left (d x + c\right )^{2} - 120 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) - 2\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, \cos \left (d x + c\right )^{4} - 24 \, \cos \left (d x + c\right )^{2} + 37\right )} \sin \left (d x + c\right ) - 80}{12 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="fricas" )
Output:
1/12*(10*cos(d*x + c)^4 + 115*cos(d*x + c)^2 - 120*(cos(d*x + c)^2 - 2*sin (d*x + c) - 2)*log(sin(d*x + c) + 1) - 2*(2*cos(d*x + c)^4 - 24*cos(d*x + c)^2 + 37)*sin(d*x + c) - 80)/(a^3*d*cos(d*x + c)^2 - 2*a^3*d*sin(d*x + c) - 2*a^3*d)
Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (97) = 194\).
Time = 1.48 (sec) , antiderivative size = 394, normalized size of antiderivative = 3.55 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\begin {cases} - \frac {60 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin ^{2}{\left (c + d x \right )}}{6 a^{3} d \sin ^{2}{\left (c + d x \right )} + 12 a^{3} d \sin {\left (c + d x \right )} + 6 a^{3} d} - \frac {120 \log {\left (\sin {\left (c + d x \right )} + 1 \right )} \sin {\left (c + d x \right )}}{6 a^{3} d \sin ^{2}{\left (c + d x \right )} + 12 a^{3} d \sin {\left (c + d x \right )} + 6 a^{3} d} - \frac {60 \log {\left (\sin {\left (c + d x \right )} + 1 \right )}}{6 a^{3} d \sin ^{2}{\left (c + d x \right )} + 12 a^{3} d \sin {\left (c + d x \right )} + 6 a^{3} d} + \frac {2 \sin ^{5}{\left (c + d x \right )}}{6 a^{3} d \sin ^{2}{\left (c + d x \right )} + 12 a^{3} d \sin {\left (c + d x \right )} + 6 a^{3} d} - \frac {5 \sin ^{4}{\left (c + d x \right )}}{6 a^{3} d \sin ^{2}{\left (c + d x \right )} + 12 a^{3} d \sin {\left (c + d x \right )} + 6 a^{3} d} + \frac {20 \sin ^{3}{\left (c + d x \right )}}{6 a^{3} d \sin ^{2}{\left (c + d x \right )} + 12 a^{3} d \sin {\left (c + d x \right )} + 6 a^{3} d} - \frac {120 \sin {\left (c + d x \right )}}{6 a^{3} d \sin ^{2}{\left (c + d x \right )} + 12 a^{3} d \sin {\left (c + d x \right )} + 6 a^{3} d} - \frac {90}{6 a^{3} d \sin ^{2}{\left (c + d x \right )} + 12 a^{3} d \sin {\left (c + d x \right )} + 6 a^{3} d} & \text {for}\: d \neq 0 \\\frac {x \sin ^{5}{\left (c \right )} \cos {\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{3}} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)**5/(a+a*sin(d*x+c))**3,x)
Output:
Piecewise((-60*log(sin(c + d*x) + 1)*sin(c + d*x)**2/(6*a**3*d*sin(c + d*x )**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d) - 120*log(sin(c + d*x) + 1)*sin( c + d*x)/(6*a**3*d*sin(c + d*x)**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d) - 60*log(sin(c + d*x) + 1)/(6*a**3*d*sin(c + d*x)**2 + 12*a**3*d*sin(c + d*x ) + 6*a**3*d) + 2*sin(c + d*x)**5/(6*a**3*d*sin(c + d*x)**2 + 12*a**3*d*si n(c + d*x) + 6*a**3*d) - 5*sin(c + d*x)**4/(6*a**3*d*sin(c + d*x)**2 + 12* a**3*d*sin(c + d*x) + 6*a**3*d) + 20*sin(c + d*x)**3/(6*a**3*d*sin(c + d*x )**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d) - 120*sin(c + d*x)/(6*a**3*d*sin (c + d*x)**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d) - 90/(6*a**3*d*sin(c + d *x)**2 + 12*a**3*d*sin(c + d*x) + 6*a**3*d), Ne(d, 0)), (x*sin(c)**5*cos(c )/(a*sin(c) + a)**3, True))
Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.86 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {\frac {3 \, {\left (10 \, \sin \left (d x + c\right ) + 9\right )}}{a^{3} \sin \left (d x + c\right )^{2} + 2 \, a^{3} \sin \left (d x + c\right ) + a^{3}} - \frac {2 \, \sin \left (d x + c\right )^{3} - 9 \, \sin \left (d x + c\right )^{2} + 36 \, \sin \left (d x + c\right )}{a^{3}} + \frac {60 \, \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3}}}{6 \, d} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="maxima" )
Output:
-1/6*(3*(10*sin(d*x + c) + 9)/(a^3*sin(d*x + c)^2 + 2*a^3*sin(d*x + c) + a ^3) - (2*sin(d*x + c)^3 - 9*sin(d*x + c)^2 + 36*sin(d*x + c))/a^3 + 60*log (sin(d*x + c) + 1)/a^3)/d
Time = 0.20 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.92 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=-\frac {10 \, \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} d} - \frac {10 \, \sin \left (d x + c\right ) + 9}{2 \, a^{3} d {\left (\sin \left (d x + c\right ) + 1\right )}^{2}} + \frac {2 \, a^{6} d^{2} \sin \left (d x + c\right )^{3} - 9 \, a^{6} d^{2} \sin \left (d x + c\right )^{2} + 36 \, a^{6} d^{2} \sin \left (d x + c\right )}{6 \, a^{9} d^{3}} \] Input:
integrate(cos(d*x+c)*sin(d*x+c)^5/(a+a*sin(d*x+c))^3,x, algorithm="giac")
Output:
-10*log(abs(sin(d*x + c) + 1))/(a^3*d) - 1/2*(10*sin(d*x + c) + 9)/(a^3*d* (sin(d*x + c) + 1)^2) + 1/6*(2*a^6*d^2*sin(d*x + c)^3 - 9*a^6*d^2*sin(d*x + c)^2 + 36*a^6*d^2*sin(d*x + c))/(a^9*d^3)
Time = 0.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.97 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {6\,\sin \left (c+d\,x\right )}{a^3\,d}-\frac {5\,\sin \left (c+d\,x\right )+\frac {9}{2}}{d\,\left (a^3\,{\sin \left (c+d\,x\right )}^2+2\,a^3\,\sin \left (c+d\,x\right )+a^3\right )}-\frac {10\,\ln \left (\sin \left (c+d\,x\right )+1\right )}{a^3\,d}-\frac {3\,{\sin \left (c+d\,x\right )}^2}{2\,a^3\,d}+\frac {{\sin \left (c+d\,x\right )}^3}{3\,a^3\,d} \] Input:
int((cos(c + d*x)*sin(c + d*x)^5)/(a + a*sin(c + d*x))^3,x)
Output:
(6*sin(c + d*x))/(a^3*d) - (5*sin(c + d*x) + 9/2)/(d*(2*a^3*sin(c + d*x) + a^3 + a^3*sin(c + d*x)^2)) - (10*log(sin(c + d*x) + 1))/(a^3*d) - (3*sin( c + d*x)^2)/(2*a^3*d) + sin(c + d*x)^3/(3*a^3*d)
Time = 0.16 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.36 \[ \int \frac {\cos (c+d x) \sin ^5(c+d x)}{(a+a \sin (c+d x))^3} \, dx=\frac {-10 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}-20 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )-10 \cos \left (d x +c \right )^{2}-60 \,\mathrm {log}\left (\sin \left (d x +c \right )+1\right ) \sin \left (d x +c \right )^{2}-120 \,\mathrm {log}\left (\sin \left (d x +c \right )+1\right ) \sin \left (d x +c \right )-60 \,\mathrm {log}\left (\sin \left (d x +c \right )+1\right )+2 \sin \left (d x +c \right )^{5}-15 \sin \left (d x +c \right )^{4}+50 \sin \left (d x +c \right )^{2}-30}{6 a^{3} d \left (\sin \left (d x +c \right )^{2}+2 \sin \left (d x +c \right )+1\right )} \] Input:
int(cos(d*x+c)*sin(d*x+c)^5/(a+a*sin(d*x+c))^3,x)
Output:
( - 10*cos(c + d*x)**2*sin(c + d*x)**2 - 20*cos(c + d*x)**2*sin(c + d*x) - 10*cos(c + d*x)**2 - 60*log(sin(c + d*x) + 1)*sin(c + d*x)**2 - 120*log(s in(c + d*x) + 1)*sin(c + d*x) - 60*log(sin(c + d*x) + 1) + 2*sin(c + d*x)* *5 - 15*sin(c + d*x)**4 + 50*sin(c + d*x)**2 - 30)/(6*a**3*d*(sin(c + d*x) **2 + 2*sin(c + d*x) + 1))