Integrand size = 29, antiderivative size = 212 \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^8 d}+\frac {a^2 \left (a^2-b^2\right )^2 \sin (c+d x)}{b^7 d}-\frac {a \left (a^2-b^2\right )^2 \sin ^2(c+d x)}{2 b^6 d}+\frac {\left (a^2-b^2\right )^2 \sin ^3(c+d x)}{3 b^5 d}-\frac {a \left (a^2-2 b^2\right ) \sin ^4(c+d x)}{4 b^4 d}+\frac {\left (a^2-2 b^2\right ) \sin ^5(c+d x)}{5 b^3 d}-\frac {a \sin ^6(c+d x)}{6 b^2 d}+\frac {\sin ^7(c+d x)}{7 b d} \] Output:
-a^3*(a^2-b^2)^2*ln(a+b*sin(d*x+c))/b^8/d+a^2*(a^2-b^2)^2*sin(d*x+c)/b^7/d -1/2*a*(a^2-b^2)^2*sin(d*x+c)^2/b^6/d+1/3*(a^2-b^2)^2*sin(d*x+c)^3/b^5/d-1 /4*a*(a^2-2*b^2)*sin(d*x+c)^4/b^4/d+1/5*(a^2-2*b^2)*sin(d*x+c)^5/b^3/d-1/6 *a*sin(d*x+c)^6/b^2/d+1/7*sin(d*x+c)^7/b/d
Time = 1.29 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.85 \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {-420 a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))+420 b \left (a^3-a b^2\right )^2 \sin (c+d x)-210 a b^2 \left (a^2-b^2\right )^2 \sin ^2(c+d x)+140 b^3 \left (a^2-b^2\right )^2 \sin ^3(c+d x)-105 a b^4 \left (a^2-2 b^2\right ) \sin ^4(c+d x)+84 b^5 \left (a^2-2 b^2\right ) \sin ^5(c+d x)-70 a b^6 \sin ^6(c+d x)+60 b^7 \sin ^7(c+d x)}{420 b^8 d} \] Input:
Integrate[(Cos[c + d*x]^5*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]
Output:
(-420*a^3*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]] + 420*b*(a^3 - a*b^2)^2*Si n[c + d*x] - 210*a*b^2*(a^2 - b^2)^2*Sin[c + d*x]^2 + 140*b^3*(a^2 - b^2)^ 2*Sin[c + d*x]^3 - 105*a*b^4*(a^2 - 2*b^2)*Sin[c + d*x]^4 + 84*b^5*(a^2 - 2*b^2)*Sin[c + d*x]^5 - 70*a*b^6*Sin[c + d*x]^6 + 60*b^7*Sin[c + d*x]^7)/( 420*b^8*d)
Time = 0.43 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3316, 27, 522, 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.
| \(\displaystyle \int \frac {\sin ^3(c+d x) \cos ^5(c+d x)}{a+b \sin (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin (c+d x)^3 \cos (c+d x)^5}{a+b \sin (c+d x)}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {\int \frac {\sin ^3(c+d x) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{a+b \sin (c+d x)}d(b \sin (c+d x))}{b^5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {b^3 \sin ^3(c+d x) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{a+b \sin (c+d x)}d(b \sin (c+d x))}{b^8 d}\) |
| \(\Big \downarrow \) 522 |
| \(\displaystyle \frac {\int \left (b^6 \sin ^6(c+d x)-a b^5 \sin ^5(c+d x)+b^4 \left (a^2-2 b^2\right ) \sin ^4(c+d x)-a b^3 \left (a^2-2 b^2\right ) \sin ^3(c+d x)+b^2 \left (a^2-b^2\right )^2 \sin ^2(c+d x)-a b \left (a^2-b^2\right )^2 \sin (c+d x)+\left (a^3-a b^2\right )^2-\frac {a^3 \left (a^2-b^2\right )^2}{a+b \sin (c+d x)}\right )d(b \sin (c+d x))}{b^8 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {1}{2} a b^2 \left (a^2-b^2\right )^2 \sin ^2(c+d x)+a^2 b \left (a^2-b^2\right )^2 \sin (c+d x)+\frac {1}{5} b^5 \left (a^2-2 b^2\right ) \sin ^5(c+d x)-\frac {1}{4} a b^4 \left (a^2-2 b^2\right ) \sin ^4(c+d x)+\frac {1}{3} b^3 \left (a^2-b^2\right )^2 \sin ^3(c+d x)-a^3 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))-\frac {1}{6} a b^6 \sin ^6(c+d x)+\frac {1}{7} b^7 \sin ^7(c+d x)}{b^8 d}\) |
Input:
Int[(Cos[c + d*x]^5*Sin[c + d*x]^3)/(a + b*Sin[c + d*x]),x]
Output:
(-(a^3*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]]) + a^2*b*(a^2 - b^2)^2*Sin[c + d*x] - (a*b^2*(a^2 - b^2)^2*Sin[c + d*x]^2)/2 + (b^3*(a^2 - b^2)^2*Sin[c + d*x]^3)/3 - (a*b^4*(a^2 - 2*b^2)*Sin[c + d*x]^4)/4 + (b^5*(a^2 - 2*b^2) *Sin[c + d*x]^5)/5 - (a*b^6*Sin[c + d*x]^6)/6 + (b^7*Sin[c + d*x]^7)/7)/(b ^8*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x, b* Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1) /2] && NeQ[a^2 - b^2, 0]
Time = 1.50 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.21
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {\sin \left (d x +c \right )^{7} b^{6}}{7}-\frac {a \,b^{5} \sin \left (d x +c \right )^{6}}{6}+\frac {a^{2} b^{4} \sin \left (d x +c \right )^{5}}{5}-\frac {2 b^{6} \sin \left (d x +c \right )^{5}}{5}-\frac {a^{3} b^{3} \sin \left (d x +c \right )^{4}}{4}+\frac {a \,b^{5} \sin \left (d x +c \right )^{4}}{2}+\frac {a^{4} b^{2} \sin \left (d x +c \right )^{3}}{3}-\frac {2 a^{2} b^{4} \sin \left (d x +c \right )^{3}}{3}+\frac {b^{6} \sin \left (d x +c \right )^{3}}{3}-\frac {a^{5} b \sin \left (d x +c \right )^{2}}{2}+a^{3} b^{3} \sin \left (d x +c \right )^{2}-\frac {a \,b^{5} \sin \left (d x +c \right )^{2}}{2}+a^{6} \sin \left (d x +c \right )-2 a^{4} b^{2} \sin \left (d x +c \right )+a^{2} b^{4} \sin \left (d x +c \right )}{b^{7}}-\frac {a^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{8}}}{d}\) | \(256\) |
| default | \(\frac {\frac {\frac {\sin \left (d x +c \right )^{7} b^{6}}{7}-\frac {a \,b^{5} \sin \left (d x +c \right )^{6}}{6}+\frac {a^{2} b^{4} \sin \left (d x +c \right )^{5}}{5}-\frac {2 b^{6} \sin \left (d x +c \right )^{5}}{5}-\frac {a^{3} b^{3} \sin \left (d x +c \right )^{4}}{4}+\frac {a \,b^{5} \sin \left (d x +c \right )^{4}}{2}+\frac {a^{4} b^{2} \sin \left (d x +c \right )^{3}}{3}-\frac {2 a^{2} b^{4} \sin \left (d x +c \right )^{3}}{3}+\frac {b^{6} \sin \left (d x +c \right )^{3}}{3}-\frac {a^{5} b \sin \left (d x +c \right )^{2}}{2}+a^{3} b^{3} \sin \left (d x +c \right )^{2}-\frac {a \,b^{5} \sin \left (d x +c \right )^{2}}{2}+a^{6} \sin \left (d x +c \right )-2 a^{4} b^{2} \sin \left (d x +c \right )+a^{2} b^{4} \sin \left (d x +c \right )}{b^{7}}-\frac {a^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{8}}}{d}\) | \(256\) |
| parallelrisch | \(\frac {-6720 a^{3} \left (a -b \right )^{2} \left (a +b \right )^{2} \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+6720 a^{3} \left (a -b \right )^{2} \left (a +b \right )^{2} \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\left (1680 a^{5} b^{2}-2520 a^{3} b^{4}+525 a \,b^{6}\right ) \cos \left (2 d x +2 c \right )+\left (-560 a^{4} b^{3}+700 a^{2} b^{5}-35 b^{7}\right ) \sin \left (3 d x +3 c \right )+\left (-210 a^{3} b^{4}+210 a \,b^{6}\right ) \cos \left (4 d x +4 c \right )+\left (84 a^{2} b^{5}-63 b^{7}\right ) \sin \left (5 d x +5 c \right )+35 a \,b^{6} \cos \left (6 d x +6 c \right )+6720 b \left (-\frac {b^{6} \sin \left (7 d x +7 c \right )}{448}+\left (a^{6}-\frac {7}{4} a^{4} b^{2}+\frac {5}{8} a^{2} b^{4}+\frac {5}{64} b^{6}\right ) \sin \left (d x +c \right )-\frac {a^{5} b}{4}+\frac {13 a^{3} b^{3}}{32}-\frac {11 a \,b^{5}}{96}\right )}{6720 b^{8} d}\) | \(282\) |
| risch | \(\frac {2 i a^{3} c}{b^{4} d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{6}}{2 b^{7} d}-\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} a^{4}}{8 b^{5} d}+\frac {5 i {\mathrm e}^{-i \left (d x +c \right )} a^{2}}{16 b^{3} d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{6}}{2 b^{7} d}+\frac {7 i {\mathrm e}^{i \left (d x +c \right )} a^{4}}{8 b^{5} d}+\frac {2 i a^{7} c}{b^{8} d}-\frac {5 i {\mathrm e}^{i \left (d x +c \right )}}{128 b d}+\frac {a \cos \left (6 d x +6 c \right )}{192 b^{2} d}-\frac {4 i a^{5} c}{b^{6} d}-\frac {5 i {\mathrm e}^{i \left (d x +c \right )} a^{2}}{16 b^{3} d}+\frac {\sin \left (5 d x +5 c \right ) a^{2}}{80 b^{3} d}-\frac {a^{3} \cos \left (4 d x +4 c \right )}{32 b^{4} d}+\frac {a \cos \left (4 d x +4 c \right )}{32 b^{2} d}-\frac {\sin \left (3 d x +3 c \right ) a^{4}}{12 b^{5} d}+\frac {5 \sin \left (3 d x +3 c \right ) a^{2}}{48 b^{3} d}+\frac {i a^{3} x}{b^{4}}+\frac {i a^{7} x}{b^{8}}-\frac {a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{b^{4} d}+\frac {5 a \,{\mathrm e}^{2 i \left (d x +c \right )}}{128 b^{2} d}+\frac {5 a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{128 b^{2} d}-\frac {2 i a^{5} x}{b^{6}}-\frac {a^{7} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{b^{8} d}+\frac {2 a^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}\right )}{b^{6} d}+\frac {a^{5} {\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{6} d}-\frac {3 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{16 b^{4} d}+\frac {a^{5} {\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{6} d}-\frac {3 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{16 b^{4} d}+\frac {5 i {\mathrm e}^{-i \left (d x +c \right )}}{128 b d}-\frac {3 \sin \left (5 d x +5 c \right )}{320 b d}-\frac {\sin \left (3 d x +3 c \right )}{192 b d}-\frac {\sin \left (7 d x +7 c \right )}{448 b d}\) | \(633\) |
| norman | \(\frac {\frac {2 \left (315 a^{6}-530 a^{4} b^{2}+163 a^{2} b^{4}+4 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 b^{7} d}+\frac {2 \left (315 a^{6}-530 a^{4} b^{2}+163 a^{2} b^{4}+4 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{15 b^{7} d}+\frac {2 \left (3675 a^{6}-5950 a^{4} b^{2}+1883 a^{2} b^{4}+344 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{105 b^{7} d}+\frac {2 \left (3675 a^{6}-5950 a^{4} b^{2}+1883 a^{2} b^{4}+344 b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{105 b^{7} d}-\frac {2 \left (60 a^{5}-84 a^{3} b^{2}+20 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3 d \,b^{6}}-\frac {2 \left (45 a^{5}-66 a^{3} b^{2}+13 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d \,b^{6}}-\frac {2 \left (45 a^{5}-66 a^{3} b^{2}+13 a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{3 d \,b^{6}}-\frac {4 \left (3 a^{5}-5 a^{3} b^{2}+a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d \,b^{6}}-\frac {4 \left (3 a^{5}-5 a^{3} b^{2}+a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d \,b^{6}}-\frac {2 \left (a^{5}-2 a^{3} b^{2}+a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{b^{6} d}-\frac {2 \left (a^{5}-2 a^{3} b^{2}+a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{b^{6} d}+\frac {2 a^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{7} d}+\frac {2 a^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{b^{7} d}+\frac {2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left (21 a^{2}+4 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 b^{7} d}+\frac {2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left (21 a^{2}+4 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{3 b^{7} d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}+\frac {a^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{b^{8} d}-\frac {a^{3} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{b^{8} d}\) | \(723\) |
Input:
int(cos(d*x+c)^5*sin(d*x+c)^3/(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(1/b^7*(1/7*sin(d*x+c)^7*b^6-1/6*a*b^5*sin(d*x+c)^6+1/5*a^2*b^4*sin(d* x+c)^5-2/5*b^6*sin(d*x+c)^5-1/4*a^3*b^3*sin(d*x+c)^4+1/2*a*b^5*sin(d*x+c)^ 4+1/3*a^4*b^2*sin(d*x+c)^3-2/3*a^2*b^4*sin(d*x+c)^3+1/3*b^6*sin(d*x+c)^3-1 /2*a^5*b*sin(d*x+c)^2+a^3*b^3*sin(d*x+c)^2-1/2*a*b^5*sin(d*x+c)^2+a^6*sin( d*x+c)-2*a^4*b^2*sin(d*x+c)+a^2*b^4*sin(d*x+c))-a^3*(a^4-2*a^2*b^2+b^4)/b^ 8*ln(a+b*sin(d*x+c)))
Time = 0.13 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.94 \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {70 \, a b^{6} \cos \left (d x + c\right )^{6} - 105 \, a^{3} b^{4} \cos \left (d x + c\right )^{4} + 210 \, {\left (a^{5} b^{2} - a^{3} b^{4}\right )} \cos \left (d x + c\right )^{2} - 420 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 4 \, {\left (15 \, b^{7} \cos \left (d x + c\right )^{6} - 105 \, a^{6} b + 175 \, a^{4} b^{3} - 56 \, a^{2} b^{5} - 8 \, b^{7} - 3 \, {\left (7 \, a^{2} b^{5} + b^{7}\right )} \cos \left (d x + c\right )^{4} + {\left (35 \, a^{4} b^{3} - 28 \, a^{2} b^{5} - 4 \, b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{420 \, b^{8} d} \] Input:
integrate(cos(d*x+c)^5*sin(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/420*(70*a*b^6*cos(d*x + c)^6 - 105*a^3*b^4*cos(d*x + c)^4 + 210*(a^5*b^2 - a^3*b^4)*cos(d*x + c)^2 - 420*(a^7 - 2*a^5*b^2 + a^3*b^4)*log(b*sin(d*x + c) + a) - 4*(15*b^7*cos(d*x + c)^6 - 105*a^6*b + 175*a^4*b^3 - 56*a^2*b ^5 - 8*b^7 - 3*(7*a^2*b^5 + b^7)*cos(d*x + c)^4 + (35*a^4*b^3 - 28*a^2*b^5 - 4*b^7)*cos(d*x + c)^2)*sin(d*x + c))/(b^8*d)
Timed out. \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**5*sin(d*x+c)**3/(a+b*sin(d*x+c)),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.97 \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {\frac {60 \, b^{6} \sin \left (d x + c\right )^{7} - 70 \, a b^{5} \sin \left (d x + c\right )^{6} + 84 \, {\left (a^{2} b^{4} - 2 \, b^{6}\right )} \sin \left (d x + c\right )^{5} - 105 \, {\left (a^{3} b^{3} - 2 \, a b^{5}\right )} \sin \left (d x + c\right )^{4} + 140 \, {\left (a^{4} b^{2} - 2 \, a^{2} b^{4} + b^{6}\right )} \sin \left (d x + c\right )^{3} - 210 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )^{2} + 420 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \sin \left (d x + c\right )}{b^{7}} - \frac {420 \, {\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{8}}}{420 \, d} \] Input:
integrate(cos(d*x+c)^5*sin(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="maxima" )
Output:
1/420*((60*b^6*sin(d*x + c)^7 - 70*a*b^5*sin(d*x + c)^6 + 84*(a^2*b^4 - 2* b^6)*sin(d*x + c)^5 - 105*(a^3*b^3 - 2*a*b^5)*sin(d*x + c)^4 + 140*(a^4*b^ 2 - 2*a^2*b^4 + b^6)*sin(d*x + c)^3 - 210*(a^5*b - 2*a^3*b^3 + a*b^5)*sin( d*x + c)^2 + 420*(a^6 - 2*a^4*b^2 + a^2*b^4)*sin(d*x + c))/b^7 - 420*(a^7 - 2*a^5*b^2 + a^3*b^4)*log(b*sin(d*x + c) + a)/b^8)/d
Time = 0.24 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.45 \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {{\left (a^{7} - 2 \, a^{5} b^{2} + a^{3} b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{8} d} + \frac {60 \, b^{6} d^{6} \sin \left (d x + c\right )^{7} - 70 \, a b^{5} d^{6} \sin \left (d x + c\right )^{6} + 84 \, a^{2} b^{4} d^{6} \sin \left (d x + c\right )^{5} - 168 \, b^{6} d^{6} \sin \left (d x + c\right )^{5} - 105 \, a^{3} b^{3} d^{6} \sin \left (d x + c\right )^{4} + 210 \, a b^{5} d^{6} \sin \left (d x + c\right )^{4} + 140 \, a^{4} b^{2} d^{6} \sin \left (d x + c\right )^{3} - 280 \, a^{2} b^{4} d^{6} \sin \left (d x + c\right )^{3} + 140 \, b^{6} d^{6} \sin \left (d x + c\right )^{3} - 210 \, a^{5} b d^{6} \sin \left (d x + c\right )^{2} + 420 \, a^{3} b^{3} d^{6} \sin \left (d x + c\right )^{2} - 210 \, a b^{5} d^{6} \sin \left (d x + c\right )^{2} + 420 \, a^{6} d^{6} \sin \left (d x + c\right ) - 840 \, a^{4} b^{2} d^{6} \sin \left (d x + c\right ) + 420 \, a^{2} b^{4} d^{6} \sin \left (d x + c\right )}{420 \, b^{7} d^{7}} \] Input:
integrate(cos(d*x+c)^5*sin(d*x+c)^3/(a+b*sin(d*x+c)),x, algorithm="giac")
Output:
-(a^7 - 2*a^5*b^2 + a^3*b^4)*log(abs(b*sin(d*x + c) + a))/(b^8*d) + 1/420* (60*b^6*d^6*sin(d*x + c)^7 - 70*a*b^5*d^6*sin(d*x + c)^6 + 84*a^2*b^4*d^6* sin(d*x + c)^5 - 168*b^6*d^6*sin(d*x + c)^5 - 105*a^3*b^3*d^6*sin(d*x + c) ^4 + 210*a*b^5*d^6*sin(d*x + c)^4 + 140*a^4*b^2*d^6*sin(d*x + c)^3 - 280*a ^2*b^4*d^6*sin(d*x + c)^3 + 140*b^6*d^6*sin(d*x + c)^3 - 210*a^5*b*d^6*sin (d*x + c)^2 + 420*a^3*b^3*d^6*sin(d*x + c)^2 - 210*a*b^5*d^6*sin(d*x + c)^ 2 + 420*a^6*d^6*sin(d*x + c) - 840*a^4*b^2*d^6*sin(d*x + c) + 420*a^2*b^4* d^6*sin(d*x + c))/(b^7*d^7)
Time = 0.08 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=-\frac {{\sin \left (c+d\,x\right )}^5\,\left (\frac {2}{5\,b}-\frac {a^2}{5\,b^3}\right )-\frac {{\sin \left (c+d\,x\right )}^7}{7\,b}-{\sin \left (c+d\,x\right )}^3\,\left (\frac {1}{3\,b}-\frac {a^2\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{3\,b^2}\right )+\frac {a\,{\sin \left (c+d\,x\right )}^6}{6\,b^2}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (a^7-2\,a^5\,b^2+a^3\,b^4\right )}{b^8}-\frac {a\,{\sin \left (c+d\,x\right )}^4\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{4\,b}+\frac {a\,{\sin \left (c+d\,x\right )}^2\,\left (\frac {1}{b}-\frac {a^2\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{b^2}\right )}{2\,b}-\frac {a^2\,\sin \left (c+d\,x\right )\,\left (\frac {1}{b}-\frac {a^2\,\left (\frac {2}{b}-\frac {a^2}{b^3}\right )}{b^2}\right )}{b^2}}{d} \] Input:
int((cos(c + d*x)^5*sin(c + d*x)^3)/(a + b*sin(c + d*x)),x)
Output:
-(sin(c + d*x)^5*(2/(5*b) - a^2/(5*b^3)) - sin(c + d*x)^7/(7*b) - sin(c + d*x)^3*(1/(3*b) - (a^2*(2/b - a^2/b^3))/(3*b^2)) + (a*sin(c + d*x)^6)/(6*b ^2) + (log(a + b*sin(c + d*x))*(a^7 + a^3*b^4 - 2*a^5*b^2))/b^8 - (a*sin(c + d*x)^4*(2/b - a^2/b^3))/(4*b) + (a*sin(c + d*x)^2*(1/b - (a^2*(2/b - a^ 2/b^3))/b^2))/(2*b) - (a^2*sin(c + d*x)*(1/b - (a^2*(2/b - a^2/b^3))/b^2)) /b^2)/d
Time = 0.20 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.96 \[ \int \frac {\cos ^5(c+d x) \sin ^3(c+d x)}{a+b \sin (c+d x)} \, dx=\frac {420 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) a^{7}-840 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) a^{5} b^{2}+420 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) a^{3} b^{4}-420 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) a^{7}+840 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) a^{5} b^{2}-420 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) a^{3} b^{4}+60 \sin \left (d x +c \right )^{7} b^{7}-70 \sin \left (d x +c \right )^{6} a \,b^{6}+84 \sin \left (d x +c \right )^{5} a^{2} b^{5}-168 \sin \left (d x +c \right )^{5} b^{7}-105 \sin \left (d x +c \right )^{4} a^{3} b^{4}+210 \sin \left (d x +c \right )^{4} a \,b^{6}+140 \sin \left (d x +c \right )^{3} a^{4} b^{3}-280 \sin \left (d x +c \right )^{3} a^{2} b^{5}+140 \sin \left (d x +c \right )^{3} b^{7}-210 \sin \left (d x +c \right )^{2} a^{5} b^{2}+420 \sin \left (d x +c \right )^{2} a^{3} b^{4}-210 \sin \left (d x +c \right )^{2} a \,b^{6}+420 \sin \left (d x +c \right ) a^{6} b -840 \sin \left (d x +c \right ) a^{4} b^{3}+420 \sin \left (d x +c \right ) a^{2} b^{5}+120 a^{5} b^{2}-240 a^{3} b^{4}+120 a \,b^{6}}{420 b^{8} d} \] Input:
int(cos(d*x+c)^5*sin(d*x+c)^3/(a+b*sin(d*x+c)),x)
Output:
(420*log(tan((c + d*x)/2)**2 + 1)*a**7 - 840*log(tan((c + d*x)/2)**2 + 1)* a**5*b**2 + 420*log(tan((c + d*x)/2)**2 + 1)*a**3*b**4 - 420*log(tan((c + d*x)/2)**2*a + 2*tan((c + d*x)/2)*b + a)*a**7 + 840*log(tan((c + d*x)/2)** 2*a + 2*tan((c + d*x)/2)*b + a)*a**5*b**2 - 420*log(tan((c + d*x)/2)**2*a + 2*tan((c + d*x)/2)*b + a)*a**3*b**4 + 60*sin(c + d*x)**7*b**7 - 70*sin(c + d*x)**6*a*b**6 + 84*sin(c + d*x)**5*a**2*b**5 - 168*sin(c + d*x)**5*b** 7 - 105*sin(c + d*x)**4*a**3*b**4 + 210*sin(c + d*x)**4*a*b**6 + 140*sin(c + d*x)**3*a**4*b**3 - 280*sin(c + d*x)**3*a**2*b**5 + 140*sin(c + d*x)**3 *b**7 - 210*sin(c + d*x)**2*a**5*b**2 + 420*sin(c + d*x)**2*a**3*b**4 - 21 0*sin(c + d*x)**2*a*b**6 + 420*sin(c + d*x)*a**6*b - 840*sin(c + d*x)*a**4 *b**3 + 420*sin(c + d*x)*a**2*b**5 + 120*a**5*b**2 - 240*a**3*b**4 + 120*a *b**6)/(420*b**8*d)