Integrand size = 31, antiderivative size = 324 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {\left (a^2-b^2\right )^2 \left (6 a A b-7 a^2 B+b^2 B\right ) \log (a+b \sin (c+d x))}{b^8 d}-\frac {\left (5 a^4 A b-9 a^2 A b^3+3 A b^5-6 a^5 B+12 a^3 b^2 B-6 a b^4 B\right ) \sin (c+d x)}{b^7 d}+\frac {\left (4 a^3 A b-6 a A b^3-5 a^4 B+9 a^2 b^2 B-3 b^4 B\right ) \sin ^2(c+d x)}{2 b^6 d}-\frac {\left (3 a^2 A b-3 A b^3-4 a^3 B+6 a b^2 B\right ) \sin ^3(c+d x)}{3 b^5 d}+\frac {\left (2 a A b-3 a^2 B+3 b^2 B\right ) \sin ^4(c+d x)}{4 b^4 d}-\frac {(A b-2 a B) \sin ^5(c+d x)}{5 b^3 d}-\frac {B \sin ^6(c+d x)}{6 b^2 d}+\frac {\left (a^2-b^2\right )^3 (A b-a B)}{b^8 d (a+b \sin (c+d x))} \]
(a^2-b^2)^2*(6*A*a*b-7*B*a^2+B*b^2)*ln(a+b*sin(d*x+c))/b^8/d-(5*A*a^4*b-9* A*a^2*b^3+3*A*b^5-6*B*a^5+12*B*a^3*b^2-6*B*a*b^4)*sin(d*x+c)/b^7/d+1/2*(4* A*a^3*b-6*A*a*b^3-5*B*a^4+9*B*a^2*b^2-3*B*b^4)*sin(d*x+c)^2/b^6/d-1/3*(3*A *a^2*b-3*A*b^3-4*B*a^3+6*B*a*b^2)*sin(d*x+c)^3/b^5/d+1/4*(2*A*a*b-3*B*a^2+ 3*B*b^2)*sin(d*x+c)^4/b^4/d-1/5*(A*b-2*B*a)*sin(d*x+c)^5/b^3/d-1/6*B*sin(d *x+c)^6/b^2/d+(a^2-b^2)^3*(A*b-B*a)/b^8/d/(a+b*sin(d*x+c))
Time = 1.07 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.22 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {B \left (15 b^4 \left (-a^2+b^2\right ) \cos ^4(c+d x)+10 b^6 \cos ^6(c+d x)-60 \left (a^2-b^2\right )^3 \log (a+b \sin (c+d x))+60 a b \left (a^4-3 a^2 b^2+3 b^4\right ) \sin (c+d x)-30 b^2 \left (a^2-b^2\right )^2 \sin ^2(c+d x)+20 a b^3 \left (a^2-3 b^2\right ) \sin ^3(c+d x)+12 a b^5 \sin ^5(c+d x)\right )+\frac {6 (A b-a B) \left (2 b^6 \cos ^6(c+d x)+4 \left (a^2-b^2\right )^2 \left (4 a^2-4 b^2+15 a^2 \log (a+b \sin (c+d x))\right )+4 a b \left (-11 a^4+18 a^2 b^2-4 b^4+15 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))\right ) \sin (c+d x)-2 b^2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin ^2(c+d x)+2 a b^3 \left (5 a^2-7 b^2\right ) \sin ^3(c+d x)-4 a^2 b^4 \sin ^4(c+d x)+b^4 \cos ^4(c+d x) \left (-a^2+4 b^2+3 a b \sin (c+d x)\right )\right )}{a+b \sin (c+d x)}}{60 b^8 d} \]
(B*(15*b^4*(-a^2 + b^2)*Cos[c + d*x]^4 + 10*b^6*Cos[c + d*x]^6 - 60*(a^2 - b^2)^3*Log[a + b*Sin[c + d*x]] + 60*a*b*(a^4 - 3*a^2*b^2 + 3*b^4)*Sin[c + d*x] - 30*b^2*(a^2 - b^2)^2*Sin[c + d*x]^2 + 20*a*b^3*(a^2 - 3*b^2)*Sin[c + d*x]^3 + 12*a*b^5*Sin[c + d*x]^5) + (6*(A*b - a*B)*(2*b^6*Cos[c + d*x]^ 6 + 4*(a^2 - b^2)^2*(4*a^2 - 4*b^2 + 15*a^2*Log[a + b*Sin[c + d*x]]) + 4*a *b*(-11*a^4 + 18*a^2*b^2 - 4*b^4 + 15*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x] ])*Sin[c + d*x] - 2*b^2*(15*a^4 - 29*a^2*b^2 + 8*b^4)*Sin[c + d*x]^2 + 2*a *b^3*(5*a^2 - 7*b^2)*Sin[c + d*x]^3 - 4*a^2*b^4*Sin[c + d*x]^4 + b^4*Cos[c + d*x]^4*(-a^2 + 4*b^2 + 3*a*b*Sin[c + d*x])))/(a + b*Sin[c + d*x]))/(60* b^8*d)
Time = 0.63 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3042, 3316, 27, 652, 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.
| \(\displaystyle \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^7 (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {\int \frac {(A b+B \sin (c+d x) b) \left (b^2-b^2 \sin ^2(c+d x)\right )^3}{b (a+b \sin (c+d x))^2}d(b \sin (c+d x))}{b^7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(A b+B \sin (c+d x) b) \left (b^2-b^2 \sin ^2(c+d x)\right )^3}{(a+b \sin (c+d x))^2}d(b \sin (c+d x))}{b^8 d}\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \frac {\int \left (-B (a+b \sin (c+d x))^5+(7 a B-A b) (a+b \sin (c+d x))^4-3 \left (7 B a^2-2 A b a-b^2 B\right ) (a+b \sin (c+d x))^3+\left (35 B a^3-15 A b a^2-15 b^2 B a+3 A b^3\right ) (a+b \sin (c+d x))^2+\left (-35 B a^4+20 A b a^3+30 b^2 B a^2-12 A b^3 a-3 b^4 B\right ) (a+b \sin (c+d x))+3 \left (a^2-b^2\right ) \left (7 B a^3-5 A b a^2-3 b^2 B a+A b^3\right )-\frac {\left (a^2-b^2\right )^2 \left (7 B a^2-6 A b a-b^2 B\right )}{a+b \sin (c+d x)}+\frac {\left (a^2-b^2\right )^3 (a B-A b)}{(a+b \sin (c+d x))^2}\right )d(b \sin (c+d x))}{b^8 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {3}{4} \left (-7 a^2 B+2 a A b+b^2 B\right ) (a+b \sin (c+d x))^4+\frac {\left (a^2-b^2\right )^3 (A b-a B)}{a+b \sin (c+d x)}+\left (a^2-b^2\right )^2 \left (-7 a^2 B+6 a A b+b^2 B\right ) \log (a+b \sin (c+d x))-\frac {1}{3} \left (-35 a^3 B+15 a^2 A b+15 a b^2 B-3 A b^3\right ) (a+b \sin (c+d x))^3-3 b \left (a^2-b^2\right ) \left (-7 a^3 B+5 a^2 A b+3 a b^2 B-A b^3\right ) \sin (c+d x)+\frac {1}{2} \left (-35 a^4 B+20 a^3 A b+30 a^2 b^2 B-12 a A b^3-3 b^4 B\right ) (a+b \sin (c+d x))^2-\frac {1}{5} (A b-7 a B) (a+b \sin (c+d x))^5-\frac {1}{6} B (a+b \sin (c+d x))^6}{b^8 d}\) |
((a^2 - b^2)^2*(6*a*A*b - 7*a^2*B + b^2*B)*Log[a + b*Sin[c + d*x]] - 3*b*( a^2 - b^2)*(5*a^2*A*b - A*b^3 - 7*a^3*B + 3*a*b^2*B)*Sin[c + d*x] + ((a^2 - b^2)^3*(A*b - a*B))/(a + b*Sin[c + d*x]) + ((20*a^3*A*b - 12*a*A*b^3 - 3 5*a^4*B + 30*a^2*b^2*B - 3*b^4*B)*(a + b*Sin[c + d*x])^2)/2 - ((15*a^2*A*b - 3*A*b^3 - 35*a^3*B + 15*a*b^2*B)*(a + b*Sin[c + d*x])^3)/3 + (3*(2*a*A* b - 7*a^2*B + b^2*B)*(a + b*Sin[c + d*x])^4)/4 - ((A*b - 7*a*B)*(a + b*Sin [c + d*x])^5)/5 - (B*(a + b*Sin[c + d*x])^6)/6)/(b^8*d)
3.16.52.3.1 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[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ )^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c *x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, 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 = 1.85 (sec) , antiderivative size = 468, normalized size of antiderivative = 1.44
| method | result | size |
| derivativedivides | \(-\frac {\frac {\frac {B \left (\sin ^{6}\left (d x +c \right )\right ) b^{5}}{6}+\frac {A \,b^{5} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {2 B a \,b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {A a \,b^{4} \left (\sin ^{4}\left (d x +c \right )\right )}{2}+\frac {3 B \,a^{2} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {3 B \,b^{5} \left (\sin ^{4}\left (d x +c \right )\right )}{4}+A \,a^{2} b^{3} \left (\sin ^{3}\left (d x +c \right )\right )-A \,b^{5} \left (\sin ^{3}\left (d x +c \right )\right )-\frac {4 B \,a^{3} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3}+2 B a \,b^{4} \left (\sin ^{3}\left (d x +c \right )\right )-2 A \,a^{3} b^{2} \left (\sin ^{2}\left (d x +c \right )\right )+3 A a \,b^{4} \left (\sin ^{2}\left (d x +c \right )\right )+\frac {5 B \,a^{4} b \left (\sin ^{2}\left (d x +c \right )\right )}{2}-\frac {9 B \,a^{2} b^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{2}+\frac {3 B \,b^{5} \left (\sin ^{2}\left (d x +c \right )\right )}{2}+5 \sin \left (d x +c \right ) A \,a^{4} b -9 \sin \left (d x +c \right ) A \,a^{2} b^{3}+3 \sin \left (d x +c \right ) A \,b^{5}-6 \sin \left (d x +c \right ) B \,a^{5}+12 \sin \left (d x +c \right ) B \,a^{3} b^{2}-6 \sin \left (d x +c \right ) B a \,b^{4}}{b^{7}}+\frac {\left (-6 A \,a^{5} b +12 A \,a^{3} b^{3}-6 A a \,b^{5}+7 B \,a^{6}-15 B \,a^{4} b^{2}+9 B \,a^{2} b^{4}-B \,b^{6}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{8}}-\frac {A \,a^{6} b -3 A \,a^{4} b^{3}+3 A \,a^{2} b^{5}-A \,b^{7}-B \,a^{7}+3 B \,a^{5} b^{2}-3 B \,a^{3} b^{4}+B a \,b^{6}}{b^{8} \left (a +b \sin \left (d x +c \right )\right )}}{d}\) | \(468\) |
| default | \(-\frac {\frac {\frac {B \left (\sin ^{6}\left (d x +c \right )\right ) b^{5}}{6}+\frac {A \,b^{5} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {2 B a \,b^{4} \left (\sin ^{5}\left (d x +c \right )\right )}{5}-\frac {A a \,b^{4} \left (\sin ^{4}\left (d x +c \right )\right )}{2}+\frac {3 B \,a^{2} b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{4}-\frac {3 B \,b^{5} \left (\sin ^{4}\left (d x +c \right )\right )}{4}+A \,a^{2} b^{3} \left (\sin ^{3}\left (d x +c \right )\right )-A \,b^{5} \left (\sin ^{3}\left (d x +c \right )\right )-\frac {4 B \,a^{3} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{3}+2 B a \,b^{4} \left (\sin ^{3}\left (d x +c \right )\right )-2 A \,a^{3} b^{2} \left (\sin ^{2}\left (d x +c \right )\right )+3 A a \,b^{4} \left (\sin ^{2}\left (d x +c \right )\right )+\frac {5 B \,a^{4} b \left (\sin ^{2}\left (d x +c \right )\right )}{2}-\frac {9 B \,a^{2} b^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{2}+\frac {3 B \,b^{5} \left (\sin ^{2}\left (d x +c \right )\right )}{2}+5 \sin \left (d x +c \right ) A \,a^{4} b -9 \sin \left (d x +c \right ) A \,a^{2} b^{3}+3 \sin \left (d x +c \right ) A \,b^{5}-6 \sin \left (d x +c \right ) B \,a^{5}+12 \sin \left (d x +c \right ) B \,a^{3} b^{2}-6 \sin \left (d x +c \right ) B a \,b^{4}}{b^{7}}+\frac {\left (-6 A \,a^{5} b +12 A \,a^{3} b^{3}-6 A a \,b^{5}+7 B \,a^{6}-15 B \,a^{4} b^{2}+9 B \,a^{2} b^{4}-B \,b^{6}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{8}}-\frac {A \,a^{6} b -3 A \,a^{4} b^{3}+3 A \,a^{2} b^{5}-A \,b^{7}-B \,a^{7}+3 B \,a^{5} b^{2}-3 B \,a^{3} b^{4}+B a \,b^{6}}{b^{8} \left (a +b \sin \left (d x +c \right )\right )}}{d}\) | \(468\) |
| parallelrisch | \(\frac {11520 \left (a +b \sin \left (d x +c \right )\right ) \left (A a b -\frac {7}{6} B \,a^{2}+\frac {1}{6} B \,b^{2}\right ) a \left (a +b \right )^{2} \left (a -b \right )^{2} \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-11520 \left (a +b \sin \left (d x +c \right )\right ) \left (A a b -\frac {7}{6} B \,a^{2}+\frac {1}{6} B \,b^{2}\right ) a \left (a +b \right )^{2} \left (a -b \right )^{2} \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+12 b \left (\left (-280 B \,a^{6} b +\frac {1660}{3} B \,a^{4} b^{3}-\frac {555}{2} B \,a^{2} b^{5}+240 A \,a^{5} b^{2}-440 A \,a^{3} b^{4}+175 a A \,b^{6}\right ) \cos \left (2 d x +2 c \right )-40 \left (A \,a^{3} b -\frac {13}{8} A a \,b^{3}-\frac {7}{6} B \,a^{4}+\frac {33}{16} B \,a^{2} b^{2}-\frac {25}{32} B \,b^{4}\right ) b^{2} a \sin \left (3 d x +3 c \right )+\left (\frac {35}{3} B \,a^{4} b^{3}-18 B \,a^{2} b^{5}-10 A \,a^{3} b^{4}+14 a A \,b^{6}\right ) \cos \left (4 d x +4 c \right )+\left (-\frac {7}{2} B \,a^{3} b^{4}+\frac {55}{12} B a \,b^{6}+3 A \,a^{2} b^{5}\right ) \sin \left (5 d x +5 c \right )+a \,b^{5} \left (A b -\frac {7 B a}{6}\right ) \cos \left (6 d x +6 c \right )+\frac {5 B a \,b^{6} \sin \left (7 d x +7 c \right )}{12}+\left (-960 A \,a^{6} b +2040 A \,a^{4} b^{3}-1170 A \,a^{2} b^{5}-2540 B \,a^{5} b^{2}+1705 B \,a^{3} b^{4}-\frac {3355}{12} B a \,b^{6}+160 A \,b^{7}+1120 B \,a^{7}\right ) \sin \left (d x +c \right )-240 A \,a^{5} b^{2}+450 A \,a^{3} b^{4}-190 a A \,b^{6}+280 B \,a^{6} b -565 B \,a^{4} b^{3}+\frac {890 B \,a^{2} b^{5}}{3}\right )}{1920 a \,b^{8} d \left (a +b \sin \left (d x +c \right )\right )}\) | \(502\) |
| risch | \(\text {Expression too large to display}\) | \(1197\) |
-1/d*(1/b^7*(1/6*B*sin(d*x+c)^6*b^5+1/5*A*b^5*sin(d*x+c)^5-2/5*B*a*b^4*sin (d*x+c)^5-1/2*A*a*b^4*sin(d*x+c)^4+3/4*B*a^2*b^3*sin(d*x+c)^4-3/4*B*b^5*si n(d*x+c)^4+A*a^2*b^3*sin(d*x+c)^3-A*b^5*sin(d*x+c)^3-4/3*B*a^3*b^2*sin(d*x +c)^3+2*B*a*b^4*sin(d*x+c)^3-2*A*a^3*b^2*sin(d*x+c)^2+3*A*a*b^4*sin(d*x+c) ^2+5/2*B*a^4*b*sin(d*x+c)^2-9/2*B*a^2*b^3*sin(d*x+c)^2+3/2*B*b^5*sin(d*x+c )^2+5*sin(d*x+c)*A*a^4*b-9*sin(d*x+c)*A*a^2*b^3+3*sin(d*x+c)*A*b^5-6*sin(d *x+c)*B*a^5+12*sin(d*x+c)*B*a^3*b^2-6*sin(d*x+c)*B*a*b^4)+(-6*A*a^5*b+12*A *a^3*b^3-6*A*a*b^5+7*B*a^6-15*B*a^4*b^2+9*B*a^2*b^4-B*b^6)/b^8*ln(a+b*sin( d*x+c))-1/b^8*(A*a^6*b-3*A*a^4*b^3+3*A*a^2*b^5-A*b^7-B*a^7+3*B*a^5*b^2-3*B *a^3*b^4+B*a*b^6)/(a+b*sin(d*x+c)))
Time = 0.35 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.57 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=-\frac {480 \, B a^{7} - 480 \, A a^{6} b - 3720 \, B a^{5} b^{2} + 3360 \, A a^{4} b^{3} + 5705 \, B a^{3} b^{4} - 4710 \, A a^{2} b^{5} - 2402 \, B a b^{6} + 1536 \, A b^{7} + 16 \, {\left (7 \, B a b^{6} - 6 \, A b^{7}\right )} \cos \left (d x + c\right )^{6} - 8 \, {\left (35 \, B a^{3} b^{4} - 30 \, A a^{2} b^{5} - 33 \, B a b^{6} + 24 \, A b^{7}\right )} \cos \left (d x + c\right )^{4} + 16 \, {\left (105 \, B a^{5} b^{2} - 90 \, A a^{4} b^{3} - 190 \, B a^{3} b^{4} + 150 \, A a^{2} b^{5} + 81 \, B a b^{6} - 48 \, A b^{7}\right )} \cos \left (d x + c\right )^{2} + 480 \, {\left (7 \, B a^{7} - 6 \, A a^{6} b - 15 \, B a^{5} b^{2} + 12 \, A a^{4} b^{3} + 9 \, B a^{3} b^{4} - 6 \, A a^{2} b^{5} - B a b^{6} + {\left (7 \, B a^{6} b - 6 \, A a^{5} b^{2} - 15 \, B a^{4} b^{3} + 12 \, A a^{3} b^{4} + 9 \, B a^{2} b^{5} - 6 \, A a b^{6} - B b^{7}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - {\left (80 \, B b^{7} \cos \left (d x + c\right )^{6} + 2880 \, B a^{6} b - 2400 \, A a^{5} b^{2} - 5720 \, B a^{4} b^{3} + 4320 \, A a^{3} b^{4} + 2967 \, B a^{2} b^{5} - 1626 \, A a b^{6} - 190 \, B b^{7} - 24 \, {\left (7 \, B a^{2} b^{5} - 6 \, A a b^{6} - 5 \, B b^{7}\right )} \cos \left (d x + c\right )^{4} + 16 \, {\left (35 \, B a^{4} b^{3} - 30 \, A a^{3} b^{4} - 54 \, B a^{2} b^{5} + 42 \, A a b^{6} + 15 \, B b^{7}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{480 \, {\left (b^{9} d \sin \left (d x + c\right ) + a b^{8} d\right )}} \]
-1/480*(480*B*a^7 - 480*A*a^6*b - 3720*B*a^5*b^2 + 3360*A*a^4*b^3 + 5705*B *a^3*b^4 - 4710*A*a^2*b^5 - 2402*B*a*b^6 + 1536*A*b^7 + 16*(7*B*a*b^6 - 6* A*b^7)*cos(d*x + c)^6 - 8*(35*B*a^3*b^4 - 30*A*a^2*b^5 - 33*B*a*b^6 + 24*A *b^7)*cos(d*x + c)^4 + 16*(105*B*a^5*b^2 - 90*A*a^4*b^3 - 190*B*a^3*b^4 + 150*A*a^2*b^5 + 81*B*a*b^6 - 48*A*b^7)*cos(d*x + c)^2 + 480*(7*B*a^7 - 6*A *a^6*b - 15*B*a^5*b^2 + 12*A*a^4*b^3 + 9*B*a^3*b^4 - 6*A*a^2*b^5 - B*a*b^6 + (7*B*a^6*b - 6*A*a^5*b^2 - 15*B*a^4*b^3 + 12*A*a^3*b^4 + 9*B*a^2*b^5 - 6*A*a*b^6 - B*b^7)*sin(d*x + c))*log(b*sin(d*x + c) + a) - (80*B*b^7*cos(d *x + c)^6 + 2880*B*a^6*b - 2400*A*a^5*b^2 - 5720*B*a^4*b^3 + 4320*A*a^3*b^ 4 + 2967*B*a^2*b^5 - 1626*A*a*b^6 - 190*B*b^7 - 24*(7*B*a^2*b^5 - 6*A*a*b^ 6 - 5*B*b^7)*cos(d*x + c)^4 + 16*(35*B*a^4*b^3 - 30*A*a^3*b^4 - 54*B*a^2*b ^5 + 42*A*a*b^6 + 15*B*b^7)*cos(d*x + c)^2)*sin(d*x + c))/(b^9*d*sin(d*x + c) + a*b^8*d)
Timed out. \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.16 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {60 \, {\left (B a^{7} - A a^{6} b - 3 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} + 3 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - B a b^{6} + A b^{7}\right )}}{b^{9} \sin \left (d x + c\right ) + a b^{8}} + \frac {10 \, B b^{5} \sin \left (d x + c\right )^{6} - 12 \, {\left (2 \, B a b^{4} - A b^{5}\right )} \sin \left (d x + c\right )^{5} + 15 \, {\left (3 \, B a^{2} b^{3} - 2 \, A a b^{4} - 3 \, B b^{5}\right )} \sin \left (d x + c\right )^{4} - 20 \, {\left (4 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - 6 \, B a b^{4} + 3 \, A b^{5}\right )} \sin \left (d x + c\right )^{3} + 30 \, {\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2} - 9 \, B a^{2} b^{3} + 6 \, A a b^{4} + 3 \, B b^{5}\right )} \sin \left (d x + c\right )^{2} - 60 \, {\left (6 \, B a^{5} - 5 \, A a^{4} b - 12 \, B a^{3} b^{2} + 9 \, A a^{2} b^{3} + 6 \, B a b^{4} - 3 \, A b^{5}\right )} \sin \left (d x + c\right )}{b^{7}} + \frac {60 \, {\left (7 \, B a^{6} - 6 \, A a^{5} b - 15 \, B a^{4} b^{2} + 12 \, A a^{3} b^{3} + 9 \, B a^{2} b^{4} - 6 \, A a b^{5} - B b^{6}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{8}}}{60 \, d} \]
-1/60*(60*(B*a^7 - A*a^6*b - 3*B*a^5*b^2 + 3*A*a^4*b^3 + 3*B*a^3*b^4 - 3*A *a^2*b^5 - B*a*b^6 + A*b^7)/(b^9*sin(d*x + c) + a*b^8) + (10*B*b^5*sin(d*x + c)^6 - 12*(2*B*a*b^4 - A*b^5)*sin(d*x + c)^5 + 15*(3*B*a^2*b^3 - 2*A*a* b^4 - 3*B*b^5)*sin(d*x + c)^4 - 20*(4*B*a^3*b^2 - 3*A*a^2*b^3 - 6*B*a*b^4 + 3*A*b^5)*sin(d*x + c)^3 + 30*(5*B*a^4*b - 4*A*a^3*b^2 - 9*B*a^2*b^3 + 6* A*a*b^4 + 3*B*b^5)*sin(d*x + c)^2 - 60*(6*B*a^5 - 5*A*a^4*b - 12*B*a^3*b^2 + 9*A*a^2*b^3 + 6*B*a*b^4 - 3*A*b^5)*sin(d*x + c))/b^7 + 60*(7*B*a^6 - 6* A*a^5*b - 15*B*a^4*b^2 + 12*A*a^3*b^3 + 9*B*a^2*b^4 - 6*A*a*b^5 - B*b^6)*l og(b*sin(d*x + c) + a)/b^8)/d
Time = 0.41 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.76 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=-\frac {\frac {60 \, {\left (7 \, B a^{6} - 6 \, A a^{5} b - 15 \, B a^{4} b^{2} + 12 \, A a^{3} b^{3} + 9 \, B a^{2} b^{4} - 6 \, A a b^{5} - B b^{6}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{8}} - \frac {60 \, {\left (7 \, B a^{6} b \sin \left (d x + c\right ) - 6 \, A a^{5} b^{2} \sin \left (d x + c\right ) - 15 \, B a^{4} b^{3} \sin \left (d x + c\right ) + 12 \, A a^{3} b^{4} \sin \left (d x + c\right ) + 9 \, B a^{2} b^{5} \sin \left (d x + c\right ) - 6 \, A a b^{6} \sin \left (d x + c\right ) - B b^{7} \sin \left (d x + c\right ) + 6 \, B a^{7} - 5 \, A a^{6} b - 12 \, B a^{5} b^{2} + 9 \, A a^{4} b^{3} + 6 \, B a^{3} b^{4} - 3 \, A a^{2} b^{5} - A b^{7}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{8}} + \frac {10 \, B b^{10} \sin \left (d x + c\right )^{6} - 24 \, B a b^{9} \sin \left (d x + c\right )^{5} + 12 \, A b^{10} \sin \left (d x + c\right )^{5} + 45 \, B a^{2} b^{8} \sin \left (d x + c\right )^{4} - 30 \, A a b^{9} \sin \left (d x + c\right )^{4} - 45 \, B b^{10} \sin \left (d x + c\right )^{4} - 80 \, B a^{3} b^{7} \sin \left (d x + c\right )^{3} + 60 \, A a^{2} b^{8} \sin \left (d x + c\right )^{3} + 120 \, B a b^{9} \sin \left (d x + c\right )^{3} - 60 \, A b^{10} \sin \left (d x + c\right )^{3} + 150 \, B a^{4} b^{6} \sin \left (d x + c\right )^{2} - 120 \, A a^{3} b^{7} \sin \left (d x + c\right )^{2} - 270 \, B a^{2} b^{8} \sin \left (d x + c\right )^{2} + 180 \, A a b^{9} \sin \left (d x + c\right )^{2} + 90 \, B b^{10} \sin \left (d x + c\right )^{2} - 360 \, B a^{5} b^{5} \sin \left (d x + c\right ) + 300 \, A a^{4} b^{6} \sin \left (d x + c\right ) + 720 \, B a^{3} b^{7} \sin \left (d x + c\right ) - 540 \, A a^{2} b^{8} \sin \left (d x + c\right ) - 360 \, B a b^{9} \sin \left (d x + c\right ) + 180 \, A b^{10} \sin \left (d x + c\right )}{b^{12}}}{60 \, d} \]
-1/60*(60*(7*B*a^6 - 6*A*a^5*b - 15*B*a^4*b^2 + 12*A*a^3*b^3 + 9*B*a^2*b^4 - 6*A*a*b^5 - B*b^6)*log(abs(b*sin(d*x + c) + a))/b^8 - 60*(7*B*a^6*b*sin (d*x + c) - 6*A*a^5*b^2*sin(d*x + c) - 15*B*a^4*b^3*sin(d*x + c) + 12*A*a^ 3*b^4*sin(d*x + c) + 9*B*a^2*b^5*sin(d*x + c) - 6*A*a*b^6*sin(d*x + c) - B *b^7*sin(d*x + c) + 6*B*a^7 - 5*A*a^6*b - 12*B*a^5*b^2 + 9*A*a^4*b^3 + 6*B *a^3*b^4 - 3*A*a^2*b^5 - A*b^7)/((b*sin(d*x + c) + a)*b^8) + (10*B*b^10*si n(d*x + c)^6 - 24*B*a*b^9*sin(d*x + c)^5 + 12*A*b^10*sin(d*x + c)^5 + 45*B *a^2*b^8*sin(d*x + c)^4 - 30*A*a*b^9*sin(d*x + c)^4 - 45*B*b^10*sin(d*x + c)^4 - 80*B*a^3*b^7*sin(d*x + c)^3 + 60*A*a^2*b^8*sin(d*x + c)^3 + 120*B*a *b^9*sin(d*x + c)^3 - 60*A*b^10*sin(d*x + c)^3 + 150*B*a^4*b^6*sin(d*x + c )^2 - 120*A*a^3*b^7*sin(d*x + c)^2 - 270*B*a^2*b^8*sin(d*x + c)^2 + 180*A* a*b^9*sin(d*x + c)^2 + 90*B*b^10*sin(d*x + c)^2 - 360*B*a^5*b^5*sin(d*x + c) + 300*A*a^4*b^6*sin(d*x + c) + 720*B*a^3*b^7*sin(d*x + c) - 540*A*a^2*b ^8*sin(d*x + c) - 360*B*a*b^9*sin(d*x + c) + 180*A*b^10*sin(d*x + c))/b^12 )/d
Time = 11.56 (sec) , antiderivative size = 682, normalized size of antiderivative = 2.10 \[ \int \frac {\cos ^7(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {A}{b^2}+\frac {a^2\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{3\,b^2}-\frac {2\,a\,\left (\frac {3\,B}{b^2}+\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{3\,b}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^5\,\left (\frac {A}{5\,b^2}-\frac {2\,B\,a}{5\,b^3}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {3\,B}{2\,b^2}+\frac {a\,\left (\frac {3\,A}{b^2}+\frac {a^2\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b^2}-\frac {2\,a\,\left (\frac {3\,B}{b^2}+\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{b}\right )}{b}+\frac {a^2\,\left (\frac {3\,B}{b^2}+\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{2\,b^2}\right )}{d}+\frac {{\sin \left (c+d\,x\right )}^4\,\left (\frac {3\,B}{4\,b^2}+\frac {a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{2\,b}+\frac {B\,a^2}{4\,b^4}\right )}{d}-\frac {\sin \left (c+d\,x\right )\,\left (\frac {3\,A}{b^2}-\frac {2\,a\,\left (\frac {3\,B}{b^2}+\frac {2\,a\,\left (\frac {3\,A}{b^2}+\frac {a^2\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b^2}-\frac {2\,a\,\left (\frac {3\,B}{b^2}+\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{b}\right )}{b}+\frac {a^2\,\left (\frac {3\,B}{b^2}+\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{b^2}\right )}{b}+\frac {a^2\,\left (\frac {3\,A}{b^2}+\frac {a^2\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b^2}-\frac {2\,a\,\left (\frac {3\,B}{b^2}+\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{b}\right )}{b^2}\right )}{d}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (-7\,B\,a^6+6\,A\,a^5\,b+15\,B\,a^4\,b^2-12\,A\,a^3\,b^3-9\,B\,a^2\,b^4+6\,A\,a\,b^5+B\,b^6\right )}{b^8\,d}-\frac {B\,a^7-A\,a^6\,b-3\,B\,a^5\,b^2+3\,A\,a^4\,b^3+3\,B\,a^3\,b^4-3\,A\,a^2\,b^5-B\,a\,b^6+A\,b^7}{b\,d\,\left (\sin \left (c+d\,x\right )\,b^8+a\,b^7\right )}-\frac {B\,{\sin \left (c+d\,x\right )}^6}{6\,b^2\,d} \]
(sin(c + d*x)^3*(A/b^2 + (a^2*(A/b^2 - (2*B*a)/b^3))/(3*b^2) - (2*a*((3*B) /b^2 + (2*a*(A/b^2 - (2*B*a)/b^3))/b + (B*a^2)/b^4))/(3*b)))/d - (sin(c + d*x)^5*(A/(5*b^2) - (2*B*a)/(5*b^3)))/d - (sin(c + d*x)^2*((3*B)/(2*b^2) + (a*((3*A)/b^2 + (a^2*(A/b^2 - (2*B*a)/b^3))/b^2 - (2*a*((3*B)/b^2 + (2*a* (A/b^2 - (2*B*a)/b^3))/b + (B*a^2)/b^4))/b))/b + (a^2*((3*B)/b^2 + (2*a*(A /b^2 - (2*B*a)/b^3))/b + (B*a^2)/b^4))/(2*b^2)))/d + (sin(c + d*x)^4*((3*B )/(4*b^2) + (a*(A/b^2 - (2*B*a)/b^3))/(2*b) + (B*a^2)/(4*b^4)))/d - (sin(c + d*x)*((3*A)/b^2 - (2*a*((3*B)/b^2 + (2*a*((3*A)/b^2 + (a^2*(A/b^2 - (2* B*a)/b^3))/b^2 - (2*a*((3*B)/b^2 + (2*a*(A/b^2 - (2*B*a)/b^3))/b + (B*a^2) /b^4))/b))/b + (a^2*((3*B)/b^2 + (2*a*(A/b^2 - (2*B*a)/b^3))/b + (B*a^2)/b ^4))/b^2))/b + (a^2*((3*A)/b^2 + (a^2*(A/b^2 - (2*B*a)/b^3))/b^2 - (2*a*(( 3*B)/b^2 + (2*a*(A/b^2 - (2*B*a)/b^3))/b + (B*a^2)/b^4))/b))/b^2))/d + (lo g(a + b*sin(c + d*x))*(B*b^6 - 7*B*a^6 - 12*A*a^3*b^3 - 9*B*a^2*b^4 + 15*B *a^4*b^2 + 6*A*a*b^5 + 6*A*a^5*b))/(b^8*d) - (A*b^7 + B*a^7 - 3*A*a^2*b^5 + 3*A*a^4*b^3 + 3*B*a^3*b^4 - 3*B*a^5*b^2 - A*a^6*b - B*a*b^6)/(b*d*(a*b^7 + b^8*sin(c + d*x))) - (B*sin(c + d*x)^6)/(6*b^2*d)