Integrand size = 31, antiderivative size = 206 \[ \int \frac {\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=-\frac {\left (a^2-b^2\right ) \left (4 a A b-5 a^2 B+b^2 B\right ) \log (a+b \sin (c+d x))}{b^6 d}+\frac {\left (3 a^2 A b-2 A b^3-4 a^3 B+4 a b^2 B\right ) \sin (c+d x)}{b^5 d}-\frac {\left (2 a A b-3 a^2 B+2 b^2 B\right ) \sin ^2(c+d x)}{2 b^4 d}+\frac {(A b-2 a B) \sin ^3(c+d x)}{3 b^3 d}+\frac {B \sin ^4(c+d x)}{4 b^2 d}-\frac {\left (a^2-b^2\right )^2 (A b-a B)}{b^6 d (a+b \sin (c+d x))} \] Output:
-(a^2-b^2)*(4*A*a*b-5*B*a^2+B*b^2)*ln(a+b*sin(d*x+c))/b^6/d+(3*A*a^2*b-2*A *b^3-4*B*a^3+4*B*a*b^2)*sin(d*x+c)/b^5/d-1/2*(2*A*a*b-3*B*a^2+2*B*b^2)*sin (d*x+c)^2/b^4/d+1/3*(A*b-2*B*a)*sin(d*x+c)^3/b^3/d+1/4*B*sin(d*x+c)^4/b^2/ d-(a^2-b^2)^2*(A*b-B*a)/b^6/d/(a+b*sin(d*x+c))
Time = 2.16 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.14 \[ \int \frac {\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {B \left (3 b^3 \cos ^4(c+d x)+\frac {12 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b}-12 a \left (a^2-2 b^2\right ) \sin (c+d x)+6 b \left (a^2-b^2\right ) \sin ^2(c+d x)-4 a b^2 \sin ^3(c+d x)\right )+4 \left (A-\frac {a B}{b}\right ) \left (\left (8 a^2 b-4 b^3\right ) \sin (c+d x)-2 a b^2 \sin ^2(c+d x)+\frac {b^4 \cos ^4(c+d x)-4 \left (a^2-b^2\right ) \left (a^2-b^2+3 a^2 \log (a+b \sin (c+d x))+3 a b \log (a+b \sin (c+d x)) \sin (c+d x)\right )}{a+b \sin (c+d x)}\right )}{12 b^5 d} \] Input:
Integrate[(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]
Output:
(B*(3*b^3*Cos[c + d*x]^4 + (12*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]])/b - 12*a*(a^2 - 2*b^2)*Sin[c + d*x] + 6*b*(a^2 - b^2)*Sin[c + d*x]^2 - 4*a*b^2 *Sin[c + d*x]^3) + 4*(A - (a*B)/b)*((8*a^2*b - 4*b^3)*Sin[c + d*x] - 2*a*b ^2*Sin[c + d*x]^2 + (b^4*Cos[c + d*x]^4 - 4*(a^2 - b^2)*(a^2 - b^2 + 3*a^2 *Log[a + b*Sin[c + d*x]] + 3*a*b*Log[a + b*Sin[c + d*x]]*Sin[c + d*x]))/(a + b*Sin[c + d*x])))/(12*b^5*d)
Time = 0.49 (sec) , antiderivative size = 188, 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.
| \(\displaystyle \int \frac {\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^5 (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3316 |
| \(\displaystyle \frac {\int \frac {(A b+B \sin (c+d x) b) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{b (a+b \sin (c+d x))^2}d(b \sin (c+d x))}{b^5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {(A b+B \sin (c+d x) b) \left (b^2-b^2 \sin ^2(c+d x)\right )^2}{(a+b \sin (c+d x))^2}d(b \sin (c+d x))}{b^6 d}\) |
| \(\Big \downarrow \) 652 |
| \(\displaystyle \frac {\int \left (B (a+b \sin (c+d x))^3+(A b-5 a B) (a+b \sin (c+d x))^2+2 \left (5 B a^2-2 A b a-b^2 B\right ) (a+b \sin (c+d x))-2 \left (5 B a^3-3 A b a^2-3 b^2 B a+A b^3\right )+\frac {\left (a^2-b^2\right ) \left (5 B a^2-4 A b a-b^2 B\right )}{a+b \sin (c+d x)}-\frac {\left (a^2-b^2\right )^2 (a B-A b)}{(a+b \sin (c+d x))^2}\right )d(b \sin (c+d x))}{b^6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\left (-5 a^2 B+2 a A b+b^2 B\right ) (a+b \sin (c+d x))^2-\frac {\left (a^2-b^2\right )^2 (A b-a B)}{a+b \sin (c+d x)}-\left (a^2-b^2\right ) \left (-5 a^2 B+4 a A b+b^2 B\right ) \log (a+b \sin (c+d x))+2 b \left (-5 a^3 B+3 a^2 A b+3 a b^2 B-A b^3\right ) \sin (c+d x)+\frac {1}{3} (A b-5 a B) (a+b \sin (c+d x))^3+\frac {1}{4} B (a+b \sin (c+d x))^4}{b^6 d}\) |
Input:
Int[(Cos[c + d*x]^5*(A + B*Sin[c + d*x]))/(a + b*Sin[c + d*x])^2,x]
Output:
(-((a^2 - b^2)*(4*a*A*b - 5*a^2*B + b^2*B)*Log[a + b*Sin[c + d*x]]) + 2*b* (3*a^2*A*b - A*b^3 - 5*a^3*B + 3*a*b^2*B)*Sin[c + d*x] - ((a^2 - b^2)^2*(A *b - a*B))/(a + b*Sin[c + d*x]) - (2*a*A*b - 5*a^2*B + b^2*B)*(a + b*Sin[c + d*x])^2 + ((A*b - 5*a*B)*(a + b*Sin[c + d*x])^3)/3 + (B*(a + b*Sin[c + d*x])^4)/4)/(b^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[((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 = 3.67 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.25
| method | result | size |
| derivativedivides | \(\frac {\frac {\frac {B \,b^{3} \sin \left (d x +c \right )^{4}}{4}+\frac {A \,b^{3} \sin \left (d x +c \right )^{3}}{3}-\frac {2 B a \,b^{2} \sin \left (d x +c \right )^{3}}{3}-A a \,b^{2} \sin \left (d x +c \right )^{2}+\frac {3 B \,a^{2} b \sin \left (d x +c \right )^{2}}{2}-B \,b^{3} \sin \left (d x +c \right )^{2}+3 \sin \left (d x +c \right ) A \,a^{2} b -2 \sin \left (d x +c \right ) A \,b^{3}-4 \sin \left (d x +c \right ) a^{3} B +4 \sin \left (d x +c \right ) B a \,b^{2}}{b^{5}}+\frac {\left (-4 A \,a^{3} b +4 A a \,b^{3}+5 B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{6}}-\frac {A \,a^{4} b -2 A \,a^{2} b^{3}+A \,b^{5}-B \,a^{5}+2 B \,a^{3} b^{2}-B a \,b^{4}}{b^{6} \left (a +b \sin \left (d x +c \right )\right )}}{d}\) | \(258\) |
| default | \(\frac {\frac {\frac {B \,b^{3} \sin \left (d x +c \right )^{4}}{4}+\frac {A \,b^{3} \sin \left (d x +c \right )^{3}}{3}-\frac {2 B a \,b^{2} \sin \left (d x +c \right )^{3}}{3}-A a \,b^{2} \sin \left (d x +c \right )^{2}+\frac {3 B \,a^{2} b \sin \left (d x +c \right )^{2}}{2}-B \,b^{3} \sin \left (d x +c \right )^{2}+3 \sin \left (d x +c \right ) A \,a^{2} b -2 \sin \left (d x +c \right ) A \,b^{3}-4 \sin \left (d x +c \right ) a^{3} B +4 \sin \left (d x +c \right ) B a \,b^{2}}{b^{5}}+\frac {\left (-4 A \,a^{3} b +4 A a \,b^{3}+5 B \,a^{4}-6 B \,a^{2} b^{2}+B \,b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{6}}-\frac {A \,a^{4} b -2 A \,a^{2} b^{3}+A \,b^{5}-B \,a^{5}+2 B \,a^{3} b^{2}-B a \,b^{4}}{b^{6} \left (a +b \sin \left (d x +c \right )\right )}}{d}\) | \(258\) |
| parallelrisch | \(\frac {-768 \left (A a b -\frac {5}{4} a^{2} B +\frac {1}{4} B \,b^{2}\right ) \left (a -b \right ) \left (a +b \right ) \left (a +b \sin \left (d x +c \right )\right ) \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+768 \left (A a b -\frac {5}{4} a^{2} B +\frac {1}{4} B \,b^{2}\right ) \left (a -b \right ) \left (a +b \right ) \left (a +b \sin \left (d x +c \right )\right ) \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+\left (-192 A \,a^{2} b^{3}+160 A \,b^{5}+240 B \,a^{3} b^{2}-248 B a \,b^{4}\right ) \cos \left (2 d x +2 c \right )+\left (32 A a \,b^{4}-40 B \,a^{2} b^{3}+33 B \,b^{5}\right ) \sin \left (3 d x +3 c \right )+\left (8 A \,b^{5}-10 B a \,b^{4}\right ) \cos \left (4 d x +4 c \right )+3 \sin \left (5 d x +5 c \right ) B \,b^{5}+\left (-96 A a \,b^{4}+120 B \,a^{2} b^{3}-114 B \,b^{5}\right ) \sin \left (d x +c \right )-768 A \,a^{4} b +960 A \,a^{2} b^{3}-360 A \,b^{5}+960 B \,a^{5}-1392 B \,a^{3} b^{2}+450 B a \,b^{4}}{192 b^{6} d \left (a +b \sin \left (d x +c \right )\right )}\) | \(323\) |
| risch | \(\frac {\sin \left (3 d x +3 c \right ) B a}{6 b^{3} d}+\frac {8 i A \,a^{3} c}{b^{5} d}-\frac {8 i A a c}{b^{3} d}-\frac {10 i B \,a^{4} c}{b^{6} d}+\frac {\cos \left (4 d x +4 c \right ) B}{32 b^{2} d}-\frac {\sin \left (3 d x +3 c \right ) A}{12 b^{2} d}+\frac {12 i B \,a^{2} c}{b^{4} d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} A \,a^{2}}{2 b^{4} d}+\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a^{3} B}{b^{5} d}-\frac {7 i {\mathrm e}^{i \left (d x +c \right )} B a}{4 b^{3} d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} A \,a^{2}}{2 b^{4} d}-\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a^{3} B}{b^{5} d}+\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} B a}{4 b^{3} d}+\frac {4 i x A \,a^{3}}{b^{5}}-\frac {4 i x A a}{b^{3}}-\frac {5 i x B \,a^{4}}{b^{6}}+\frac {6 i x B \,a^{2}}{b^{4}}-\frac {7 i {\mathrm e}^{-i \left (d x +c \right )} A}{8 b^{2} d}+\frac {{\mathrm e}^{-2 i \left (d x +c \right )} A a}{4 b^{3} d}-\frac {3 \,{\mathrm e}^{-2 i \left (d x +c \right )} B \,a^{2}}{8 b^{4} d}-\frac {4 \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 ) A \,a^{3}}{b^{5} d}+\frac {4 \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 ) A a}{b^{3} d}+\frac {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 \,a^{4}}{b^{6} d}-\frac {6 \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 \,a^{2}}{b^{4} d}+\frac {7 i {\mathrm e}^{i \left (d x +c \right )} A}{8 b^{2} d}+\frac {{\mathrm e}^{2 i \left (d x +c \right )} A a}{4 b^{3} d}-\frac {3 \,{\mathrm e}^{2 i \left (d x +c \right )} B \,a^{2}}{8 b^{4} d}+\frac {2 \left (-A \,a^{4} b +2 A \,a^{2} b^{3}-A \,b^{5}+B \,a^{5}-2 B \,a^{3} b^{2}+B a \,b^{4}\right ) {\mathrm e}^{i \left (d x +c \right )}}{b^{6} d \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )}-\frac {2 i B c}{b^{2} d}+\frac {3 \,{\mathrm e}^{-2 i \left (d x +c \right )} B}{16 b^{2} d}+\frac {\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}{b^{2} d}+\frac {3 \,{\mathrm e}^{2 i \left (d x +c \right )} B}{16 b^{2} d}-\frac {i B x}{b^{2}}\) | \(740\) |
| norman | \(\frac {\frac {\left (80 A \,a^{2} b -64 A \,b^{3}-100 a^{3} B +100 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{b^{4} d}+\frac {\left (80 A \,a^{2} b -64 A \,b^{3}-100 a^{3} B +100 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{b^{4} d}+\frac {\left (8 A \,a^{2} b -8 A \,b^{3}-10 a^{3} B +12 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{b^{4} d}+\frac {\left (8 A \,a^{2} b -8 A \,b^{3}-10 a^{3} B +12 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{b^{4} d}+\frac {\left (120 A \,a^{2} b -104 A \,b^{3}-150 a^{3} B +160 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 b^{4} d}+\frac {\left (120 A \,a^{2} b -104 A \,b^{3}-150 a^{3} B +160 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{3 b^{4} d}+\frac {8 \left (20 A \,a^{4} b -24 A \,a^{2} b^{3}+5 A \,b^{5}-25 B \,a^{5}+35 B \,a^{3} b^{2}-9 B a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{b^{5} d a}+\frac {4 \left (36 A \,a^{4} b -40 A \,a^{2} b^{3}+9 A \,b^{5}-45 B \,a^{5}+59 B \,a^{3} b^{2}-15 B a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 b^{5} d a}+\frac {4 \left (36 A \,a^{4} b -40 A \,a^{2} b^{3}+9 A \,b^{5}-45 B \,a^{5}+59 B \,a^{3} b^{2}-15 B a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{3 b^{5} d a}+\frac {2 \left (180 A \,a^{4} b -212 A \,a^{2} b^{3}+45 A \,b^{5}-225 B \,a^{5}+310 B \,a^{3} b^{2}-81 B a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 b^{5} d a}+\frac {2 \left (180 A \,a^{4} b -212 A \,a^{2} b^{3}+45 A \,b^{5}-225 B \,a^{5}+310 B \,a^{3} b^{2}-81 B a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 b^{5} d a}+\frac {2 \left (4 A \,a^{4} b -4 A \,a^{2} b^{3}+A \,b^{5}-5 B \,a^{5}+6 B \,a^{3} b^{2}-B a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{5} a d}+\frac {2 \left (4 A \,a^{4} b -4 A \,a^{2} b^{3}+A \,b^{5}-5 B \,a^{5}+6 B \,a^{3} b^{2}-B a \,b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{b^{5} a d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6} \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 )}+\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-5 B \,a^{4}+6 B \,a^{2} b^{2}-B \,b^{4}\right ) \ln \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{b^{6} d}-\frac {\left (4 A \,a^{3} b -4 A a \,b^{3}-5 B \,a^{4}+6 B \,a^{2} b^{2}-B \,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^{6} d}\) | \(914\) |
Input:
int(cos(d*x+c)^5*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x,method=_RETURNVERBO SE)
Output:
1/d*(1/b^5*(1/4*B*b^3*sin(d*x+c)^4+1/3*A*b^3*sin(d*x+c)^3-2/3*B*a*b^2*sin( d*x+c)^3-A*a*b^2*sin(d*x+c)^2+3/2*B*a^2*b*sin(d*x+c)^2-B*b^3*sin(d*x+c)^2+ 3*sin(d*x+c)*A*a^2*b-2*sin(d*x+c)*A*b^3-4*sin(d*x+c)*a^3*B+4*sin(d*x+c)*B* a*b^2)+(-4*A*a^3*b+4*A*a*b^3+5*B*a^4-6*B*a^2*b^2+B*b^4)/b^6*ln(a+b*sin(d*x +c))-1/b^6*(A*a^4*b-2*A*a^2*b^3+A*b^5-B*a^5+2*B*a^3*b^2-B*a*b^4)/(a+b*sin( d*x+c)))
Time = 0.12 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.56 \[ \int \frac {\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {96 \, B a^{5} - 96 \, A a^{4} b - 504 \, B a^{3} b^{2} + 432 \, A a^{2} b^{3} + 383 \, B a b^{4} - 256 \, A b^{5} - 8 \, {\left (5 \, B a b^{4} - 4 \, A b^{5}\right )} \cos \left (d x + c\right )^{4} + 16 \, {\left (15 \, B a^{3} b^{2} - 12 \, A a^{2} b^{3} - 13 \, B a b^{4} + 8 \, A b^{5}\right )} \cos \left (d x + c\right )^{2} + 96 \, {\left (5 \, B a^{5} - 4 \, A a^{4} b - 6 \, B a^{3} b^{2} + 4 \, A a^{2} b^{3} + B a b^{4} + {\left (5 \, B a^{4} b - 4 \, A a^{3} b^{2} - 6 \, B a^{2} b^{3} + 4 \, A a b^{4} + B b^{5}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left (24 \, B b^{5} \cos \left (d x + c\right )^{4} - 384 \, B a^{4} b + 288 \, A a^{3} b^{2} + 392 \, B a^{2} b^{3} - 208 \, A a b^{4} - 33 \, B b^{5} - 16 \, {\left (5 \, B a^{2} b^{3} - 4 \, A a b^{4} - 3 \, B b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{96 \, {\left (b^{7} d \sin \left (d x + c\right ) + a b^{6} d\right )}} \] Input:
integrate(cos(d*x+c)^5*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="f ricas")
Output:
1/96*(96*B*a^5 - 96*A*a^4*b - 504*B*a^3*b^2 + 432*A*a^2*b^3 + 383*B*a*b^4 - 256*A*b^5 - 8*(5*B*a*b^4 - 4*A*b^5)*cos(d*x + c)^4 + 16*(15*B*a^3*b^2 - 12*A*a^2*b^3 - 13*B*a*b^4 + 8*A*b^5)*cos(d*x + c)^2 + 96*(5*B*a^5 - 4*A*a^ 4*b - 6*B*a^3*b^2 + 4*A*a^2*b^3 + B*a*b^4 + (5*B*a^4*b - 4*A*a^3*b^2 - 6*B *a^2*b^3 + 4*A*a*b^4 + B*b^5)*sin(d*x + c))*log(b*sin(d*x + c) + a) + (24* B*b^5*cos(d*x + c)^4 - 384*B*a^4*b + 288*A*a^3*b^2 + 392*B*a^2*b^3 - 208*A *a*b^4 - 33*B*b^5 - 16*(5*B*a^2*b^3 - 4*A*a*b^4 - 3*B*b^5)*cos(d*x + c)^2) *sin(d*x + c))/(b^7*d*sin(d*x + c) + a*b^6*d)
Timed out. \[ \int \frac {\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**5*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))**2,x)
Output:
Timed out
Time = 0.04 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.11 \[ \int \frac {\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {12 \, {\left (B a^{5} - A a^{4} b - 2 \, B a^{3} b^{2} + 2 \, A a^{2} b^{3} + B a b^{4} - A b^{5}\right )}}{b^{7} \sin \left (d x + c\right ) + a b^{6}} + \frac {3 \, B b^{3} \sin \left (d x + c\right )^{4} - 4 \, {\left (2 \, B a b^{2} - A b^{3}\right )} \sin \left (d x + c\right )^{3} + 6 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2} - 2 \, B b^{3}\right )} \sin \left (d x + c\right )^{2} - 12 \, {\left (4 \, B a^{3} - 3 \, A a^{2} b - 4 \, B a b^{2} + 2 \, A b^{3}\right )} \sin \left (d x + c\right )}{b^{5}} + \frac {12 \, {\left (5 \, B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{6}}}{12 \, d} \] Input:
integrate(cos(d*x+c)^5*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="m axima")
Output:
1/12*(12*(B*a^5 - A*a^4*b - 2*B*a^3*b^2 + 2*A*a^2*b^3 + B*a*b^4 - A*b^5)/( b^7*sin(d*x + c) + a*b^6) + (3*B*b^3*sin(d*x + c)^4 - 4*(2*B*a*b^2 - A*b^3 )*sin(d*x + c)^3 + 6*(3*B*a^2*b - 2*A*a*b^2 - 2*B*b^3)*sin(d*x + c)^2 - 12 *(4*B*a^3 - 3*A*a^2*b - 4*B*a*b^2 + 2*A*b^3)*sin(d*x + c))/b^5 + 12*(5*B*a ^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*b^4)*log(b*sin(d*x + c) + a)/ b^6)/d
Time = 0.36 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.46 \[ \int \frac {\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {{\left (5 \, B a^{4} - 4 \, A a^{3} b - 6 \, B a^{2} b^{2} + 4 \, A a b^{3} + B b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{6} d} + \frac {B a^{5} - A a^{4} b - 2 \, B a^{3} b^{2} + 2 \, A a^{2} b^{3} + B a b^{4} - A b^{5}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{6} d} + \frac {3 \, B b^{6} d^{3} \sin \left (d x + c\right )^{4} - 8 \, B a b^{5} d^{3} \sin \left (d x + c\right )^{3} + 4 \, A b^{6} d^{3} \sin \left (d x + c\right )^{3} + 18 \, B a^{2} b^{4} d^{3} \sin \left (d x + c\right )^{2} - 12 \, A a b^{5} d^{3} \sin \left (d x + c\right )^{2} - 12 \, B b^{6} d^{3} \sin \left (d x + c\right )^{2} - 48 \, B a^{3} b^{3} d^{3} \sin \left (d x + c\right ) + 36 \, A a^{2} b^{4} d^{3} \sin \left (d x + c\right ) + 48 \, B a b^{5} d^{3} \sin \left (d x + c\right ) - 24 \, A b^{6} d^{3} \sin \left (d x + c\right )}{12 \, b^{8} d^{4}} \] Input:
integrate(cos(d*x+c)^5*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x, algorithm="g iac")
Output:
(5*B*a^4 - 4*A*a^3*b - 6*B*a^2*b^2 + 4*A*a*b^3 + B*b^4)*log(abs(b*sin(d*x + c) + a))/(b^6*d) + (B*a^5 - A*a^4*b - 2*B*a^3*b^2 + 2*A*a^2*b^3 + B*a*b^ 4 - A*b^5)/((b*sin(d*x + c) + a)*b^6*d) + 1/12*(3*B*b^6*d^3*sin(d*x + c)^4 - 8*B*a*b^5*d^3*sin(d*x + c)^3 + 4*A*b^6*d^3*sin(d*x + c)^3 + 18*B*a^2*b^ 4*d^3*sin(d*x + c)^2 - 12*A*a*b^5*d^3*sin(d*x + c)^2 - 12*B*b^6*d^3*sin(d* x + c)^2 - 48*B*a^3*b^3*d^3*sin(d*x + c) + 36*A*a^2*b^4*d^3*sin(d*x + c) + 48*B*a*b^5*d^3*sin(d*x + c) - 24*A*b^6*d^3*sin(d*x + c))/(b^8*d^4)
Time = 0.13 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.41 \[ \int \frac {\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {A}{3\,b^2}-\frac {2\,B\,a}{3\,b^3}\right )}{d}-\frac {\sin \left (c+d\,x\right )\,\left (\frac {2\,A}{b^2}+\frac {a^2\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b^2}-\frac {2\,a\,\left (\frac {2\,B}{b^2}+\frac {2\,a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{b^4}\right )}{b}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {B}{b^2}+\frac {a\,\left (\frac {A}{b^2}-\frac {2\,B\,a}{b^3}\right )}{b}+\frac {B\,a^2}{2\,b^4}\right )}{d}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (5\,B\,a^4-4\,A\,a^3\,b-6\,B\,a^2\,b^2+4\,A\,a\,b^3+B\,b^4\right )}{b^6\,d}-\frac {-B\,a^5+A\,a^4\,b+2\,B\,a^3\,b^2-2\,A\,a^2\,b^3-B\,a\,b^4+A\,b^5}{b\,d\,\left (\sin \left (c+d\,x\right )\,b^6+a\,b^5\right )}+\frac {B\,{\sin \left (c+d\,x\right )}^4}{4\,b^2\,d} \] Input:
int((cos(c + d*x)^5*(A + B*sin(c + d*x)))/(a + b*sin(c + d*x))^2,x)
Output:
(sin(c + d*x)^3*(A/(3*b^2) - (2*B*a)/(3*b^3)))/d - (sin(c + d*x)*((2*A)/b^ 2 + (a^2*(A/b^2 - (2*B*a)/b^3))/b^2 - (2*a*((2*B)/b^2 + (2*a*(A/b^2 - (2*B *a)/b^3))/b + (B*a^2)/b^4))/b))/d - (sin(c + d*x)^2*(B/b^2 + (a*(A/b^2 - ( 2*B*a)/b^3))/b + (B*a^2)/(2*b^4)))/d + (log(a + b*sin(c + d*x))*(5*B*a^4 + B*b^4 - 6*B*a^2*b^2 + 4*A*a*b^3 - 4*A*a^3*b))/(b^6*d) - (A*b^5 - B*a^5 - 2*A*a^2*b^3 + 2*B*a^3*b^2 + A*a^4*b - B*a*b^4)/(b*d*(a*b^5 + b^6*sin(c + d *x))) + (B*sin(c + d*x)^4)/(4*b^2*d)
Time = 0.18 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.28 \[ \int \frac {\cos ^5(c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx=\frac {-12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) a^{4}+24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) a^{2} b^{2}-12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+1\right ) b^{4}+12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) a^{4}-24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) a^{2} b^{2}+12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) b +a \right ) b^{4}+3 \sin \left (d x +c \right )^{4} b^{4}-4 \sin \left (d x +c \right )^{3} a \,b^{3}+6 \sin \left (d x +c \right )^{2} a^{2} b^{2}-12 \sin \left (d x +c \right )^{2} b^{4}-12 \sin \left (d x +c \right ) a^{3} b +24 \sin \left (d x +c \right ) a \,b^{3}-6 a^{2} b^{2}+12 b^{4}}{12 b^{5} d} \] Input:
int(cos(d*x+c)^5*(A+B*sin(d*x+c))/(a+b*sin(d*x+c))^2,x)
Output:
( - 12*log(tan((c + d*x)/2)**2 + 1)*a**4 + 24*log(tan((c + d*x)/2)**2 + 1) *a**2*b**2 - 12*log(tan((c + d*x)/2)**2 + 1)*b**4 + 12*log(tan((c + d*x)/2 )**2*a + 2*tan((c + d*x)/2)*b + a)*a**4 - 24*log(tan((c + d*x)/2)**2*a + 2 *tan((c + d*x)/2)*b + a)*a**2*b**2 + 12*log(tan((c + d*x)/2)**2*a + 2*tan( (c + d*x)/2)*b + a)*b**4 + 3*sin(c + d*x)**4*b**4 - 4*sin(c + d*x)**3*a*b* *3 + 6*sin(c + d*x)**2*a**2*b**2 - 12*sin(c + d*x)**2*b**4 - 12*sin(c + d* x)*a**3*b + 24*sin(c + d*x)*a*b**3 - 6*a**2*b**2 + 12*b**4)/(12*b**5*d)