Integrand size = 21, antiderivative size = 184 \[ \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {6 a \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{b^7 d}-\frac {\left (5 a^4-9 a^2 b^2+3 b^4\right ) \sin (c+d x)}{b^6 d}+\frac {a \left (2 a^2-3 b^2\right ) \sin ^2(c+d x)}{b^5 d}-\frac {\left (a^2-b^2\right ) \sin ^3(c+d x)}{b^4 d}+\frac {a \sin ^4(c+d x)}{2 b^3 d}-\frac {\sin ^5(c+d x)}{5 b^2 d}+\frac {\left (a^2-b^2\right )^3}{b^7 d (a+b \sin (c+d x))} \]
6*a*(a^2-b^2)^2*ln(a+b*sin(d*x+c))/b^7/d-(5*a^4-9*a^2*b^2+3*b^4)*sin(d*x+c )/b^6/d+a*(2*a^2-3*b^2)*sin(d*x+c)^2/b^5/d-(a^2-b^2)*sin(d*x+c)^3/b^4/d+1/ 2*a*sin(d*x+c)^4/b^3/d-1/5*sin(d*x+c)^5/b^2/d+(a^2-b^2)^3/b^7/d/(a+b*sin(d *x+c))
Time = 0.34 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.28 \[ \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {2 b^6 \cos ^6(c+d x)+4 \left (a^2-b^2\right )^2 \left (4 a^2-4 b^2+15 a^2 \log (a+b \sin (c+d x))\right )+4 a b \left (-11 a^4+18 a^2 b^2-4 b^4+15 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))\right ) \sin (c+d x)-2 b^2 \left (15 a^4-29 a^2 b^2+8 b^4\right ) \sin ^2(c+d x)+2 a b^3 \left (5 a^2-7 b^2\right ) \sin ^3(c+d x)-4 a^2 b^4 \sin ^4(c+d x)+b^4 \cos ^4(c+d x) \left (-a^2+4 b^2+3 a b \sin (c+d x)\right )}{10 b^7 d (a+b \sin (c+d x))} \]
(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*Lo g[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]))/(10*b^7 *d*(a + b*Sin[c + d*x]))
Time = 0.35 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.88, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3147, 476, 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.
| \(\displaystyle \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^7}{(a+b \sin (c+d x))^2}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {\int \frac {\left (b^2-b^2 \sin ^2(c+d x)\right )^3}{(a+b \sin (c+d x))^2}d(b \sin (c+d x))}{b^7 d}\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \frac {\int \left (-5 \left (\frac {3 \left (b^2-3 a^2\right ) b^2}{5 a^4}+1\right ) a^4+2 b^3 \sin ^3(c+d x) a+2 b \left (2 a^2-3 b^2\right ) \sin (c+d x) a+\frac {6 \left (a^2-b^2\right )^2 a}{a+b \sin (c+d x)}-b^4 \sin ^4(c+d x)-3 b^2 \left (a^2-b^2\right ) \sin ^2(c+d x)-\frac {\left (a^2-b^2\right )^3}{(a+b \sin (c+d x))^2}\right )d(b \sin (c+d x))}{b^7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a b^2 \left (2 a^2-3 b^2\right ) \sin ^2(c+d x)+\frac {\left (a^2-b^2\right )^3}{a+b \sin (c+d x)}+6 a \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))-b^3 \left (a^2-b^2\right ) \sin ^3(c+d x)-b \left (5 a^4-9 a^2 b^2+3 b^4\right ) \sin (c+d x)+\frac {1}{2} a b^4 \sin ^4(c+d x)-\frac {1}{5} b^5 \sin ^5(c+d x)}{b^7 d}\) |
(6*a*(a^2 - b^2)^2*Log[a + b*Sin[c + d*x]] - b*(5*a^4 - 9*a^2*b^2 + 3*b^4) *Sin[c + d*x] + a*b^2*(2*a^2 - 3*b^2)*Sin[c + d*x]^2 - b^3*(a^2 - b^2)*Sin [c + d*x]^3 + (a*b^4*Sin[c + d*x]^4)/2 - (b^5*Sin[c + d*x]^5)/5 + (a^2 - b ^2)^3/(a + b*Sin[c + d*x]))/(b^7*d)
3.5.37.3.1 Defintions of rubi rules used
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ ExpandIntegrand[(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[p, 0]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Time = 1.79 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.11
| method | result | size |
| derivativedivides | \(-\frac {\frac {\frac {\left (\sin ^{5}\left (d x +c \right )\right ) b^{4}}{5}-\frac {a \,b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{2}+a^{2} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )-b^{4} \left (\sin ^{3}\left (d x +c \right )\right )-2 a^{3} b \left (\sin ^{2}\left (d x +c \right )\right )+3 a \,b^{3} \left (\sin ^{2}\left (d x +c \right )\right )+5 a^{4} \sin \left (d x +c \right )-9 a^{2} b^{2} \sin \left (d x +c \right )+3 b^{4} \sin \left (d x +c \right )}{b^{6}}-\frac {a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}{b^{7} \left (a +b \sin \left (d x +c \right )\right )}-\frac {6 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{7}}}{d}\) | \(205\) |
| default | \(-\frac {\frac {\frac {\left (\sin ^{5}\left (d x +c \right )\right ) b^{4}}{5}-\frac {a \,b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{2}+a^{2} b^{2} \left (\sin ^{3}\left (d x +c \right )\right )-b^{4} \left (\sin ^{3}\left (d x +c \right )\right )-2 a^{3} b \left (\sin ^{2}\left (d x +c \right )\right )+3 a \,b^{3} \left (\sin ^{2}\left (d x +c \right )\right )+5 a^{4} \sin \left (d x +c \right )-9 a^{2} b^{2} \sin \left (d x +c \right )+3 b^{4} \sin \left (d x +c \right )}{b^{6}}-\frac {a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}{b^{7} \left (a +b \sin \left (d x +c \right )\right )}-\frac {6 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{b^{7}}}{d}\) | \(205\) |
| parallelrisch | \(\frac {960 a^{2} \left (a -b \right )^{2} \left (a +b \right )^{2} \left (a +b \sin \left (d x +c \right )\right ) \ln \left (2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )-960 a^{2} \left (a -b \right )^{2} \left (a +b \right )^{2} \left (a +b \sin \left (d x +c \right )\right ) \ln \left (\sec ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (240 b^{2} a^{5}-440 b^{4} a^{3}+175 b^{6} a \right ) \cos \left (2 d x +2 c \right )+\left (-10 b^{4} a^{3}+14 b^{6} a \right ) \cos \left (4 d x +4 c \right )+\left (-40 a^{4} b^{3}+65 a^{2} b^{5}\right ) \sin \left (3 d x +3 c \right )+b^{6} a \cos \left (6 d x +6 c \right )+3 a^{2} b^{5} \sin \left (5 d x +5 c \right )+\left (-960 a^{6} b +2040 a^{4} b^{3}-1170 a^{2} b^{5}+160 b^{7}\right ) \sin \left (d x +c \right )-240 b^{2} a^{5}+450 b^{4} a^{3}-190 b^{6} a}{160 b^{7} d a \left (a +b \sin \left (d x +c \right )\right )}\) | \(291\) |
| risch | \(-\frac {a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{2 b^{5} d}+\frac {5 a \,{\mathrm e}^{2 i \left (d x +c \right )}}{8 b^{3} d}+\frac {19 i {\mathrm e}^{i \left (d x +c \right )}}{16 b^{2} d}-\frac {19 i {\mathrm e}^{-i \left (d x +c \right )}}{16 b^{2} d}+\frac {6 a^{5} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}-1\right )}{b^{7} d}-\frac {12 a^{3} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}-1\right )}{b^{5} d}+\frac {6 a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}-1\right )}{b^{3} d}+\frac {\sin \left (3 d x +3 c \right ) a^{2}}{4 b^{4} d}-\frac {6 i a^{5} x}{b^{7}}+\frac {12 i a^{3} x}{b^{5}}-\frac {6 i a x}{b^{3}}+\frac {2 \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) {\mathrm e}^{i \left (d x +c \right )}}{b^{7} d \left (-i b \,{\mathrm e}^{2 i \left (d x +c \right )}+2 a \,{\mathrm e}^{i \left (d x +c \right )}+i b \right )}+\frac {5 i {\mathrm e}^{i \left (d x +c \right )} a^{4}}{2 b^{6} d}-\frac {33 i {\mathrm e}^{i \left (d x +c \right )} a^{2}}{8 b^{4} d}-\frac {5 i {\mathrm e}^{-i \left (d x +c \right )} a^{4}}{2 b^{6} d}+\frac {33 i {\mathrm e}^{-i \left (d x +c \right )} a^{2}}{8 b^{4} d}-\frac {\sin \left (5 d x +5 c \right )}{80 b^{2} d}-\frac {3 \sin \left (3 d x +3 c \right )}{16 b^{2} d}-\frac {12 i a^{5} c}{b^{7} d}+\frac {24 i a^{3} c}{b^{5} d}-\frac {12 i a c}{b^{3} d}+\frac {a \cos \left (4 d x +4 c \right )}{16 b^{3} d}-\frac {a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{2 b^{5} d}+\frac {5 a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{8 b^{3} d}\) | \(524\) |
| norman | \(\frac {-\frac {2 \left (600 a^{4}-1080 a^{2} b^{2}+424 b^{4}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 b^{5} d}-\frac {4 \left (225 a^{4}-410 a^{2} b^{2}+161 b^{4}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 b^{5} d}-\frac {4 \left (225 a^{4}-410 a^{2} b^{2}+161 b^{4}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 b^{5} d}-\frac {4 \left (18 a^{4}-34 a^{2} b^{2}+14 b^{4}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5} d}-\frac {4 \left (18 a^{4}-34 a^{2} b^{2}+14 b^{4}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5} d}-\frac {4 \left (3 a^{4}-6 a^{2} b^{2}+3 b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5} d}-\frac {4 \left (3 a^{4}-6 a^{2} b^{2}+3 b^{4}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{5} d}-\frac {2 \left (1050 a^{6}-2300 a^{4} b^{2}+1378 a^{2} b^{4}-175 b^{6}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a \,b^{6} d}-\frac {2 \left (1050 a^{6}-2300 a^{4} b^{2}+1378 a^{2} b^{4}-175 b^{6}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a \,b^{6} d}-\frac {2 \left (630 a^{6}-1360 a^{4} b^{2}+806 a^{2} b^{4}-105 b^{6}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a \,b^{6} d}-\frac {2 \left (630 a^{6}-1360 a^{4} b^{2}+806 a^{2} b^{4}-105 b^{6}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 a \,b^{6} d}-\frac {2 \left (42 a^{6}-88 a^{4} b^{2}+50 a^{2} b^{4}-7 b^{6}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,b^{6} d}-\frac {2 \left (42 a^{6}-88 a^{4} b^{2}+50 a^{2} b^{4}-7 b^{6}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \,b^{6} d}-\frac {2 \left (6 a^{6}-12 a^{4} b^{2}+6 a^{2} b^{4}-b^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{b^{6} d a}-\frac {2 \left (6 a^{6}-12 a^{4} b^{2}+6 a^{2} b^{4}-b^{6}\right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{6} d a}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}-\frac {6 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{7} d}+\frac {6 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \ln \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{b^{7} d}\) | \(794\) |
-1/d*(1/b^6*(1/5*sin(d*x+c)^5*b^4-1/2*a*b^3*sin(d*x+c)^4+a^2*b^2*sin(d*x+c )^3-b^4*sin(d*x+c)^3-2*a^3*b*sin(d*x+c)^2+3*a*b^3*sin(d*x+c)^2+5*a^4*sin(d *x+c)-9*a^2*b^2*sin(d*x+c)+3*b^4*sin(d*x+c))-1/b^7*(a^6-3*a^4*b^2+3*a^2*b^ 4-b^6)/(a+b*sin(d*x+c))-6*a/b^7*(a^4-2*a^2*b^2+b^4)*ln(a+b*sin(d*x+c)))
Time = 0.31 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.32 \[ \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {16 \, b^{6} \cos \left (d x + c\right )^{6} + 80 \, a^{6} - 560 \, a^{4} b^{2} + 785 \, a^{2} b^{4} - 256 \, b^{6} - 8 \, {\left (5 \, a^{2} b^{4} - 4 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 16 \, {\left (15 \, a^{4} b^{2} - 25 \, a^{2} b^{4} + 8 \, b^{6}\right )} \cos \left (d x + c\right )^{2} + 480 \, {\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4} + {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left (24 \, a b^{5} \cos \left (d x + c\right )^{4} - 400 \, a^{5} b + 720 \, a^{3} b^{3} - 271 \, a b^{5} - 16 \, {\left (5 \, a^{3} b^{3} - 7 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{80 \, {\left (b^{8} d \sin \left (d x + c\right ) + a b^{7} d\right )}} \]
1/80*(16*b^6*cos(d*x + c)^6 + 80*a^6 - 560*a^4*b^2 + 785*a^2*b^4 - 256*b^6 - 8*(5*a^2*b^4 - 4*b^6)*cos(d*x + c)^4 + 16*(15*a^4*b^2 - 25*a^2*b^4 + 8* b^6)*cos(d*x + c)^2 + 480*(a^6 - 2*a^4*b^2 + a^2*b^4 + (a^5*b - 2*a^3*b^3 + a*b^5)*sin(d*x + c))*log(b*sin(d*x + c) + a) + (24*a*b^5*cos(d*x + c)^4 - 400*a^5*b + 720*a^3*b^3 - 271*a*b^5 - 16*(5*a^3*b^3 - 7*a*b^5)*cos(d*x + c)^2)*sin(d*x + c))/(b^8*d*sin(d*x + c) + a*b^7*d)
Timed out. \[ \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\text {Timed out} \]
Time = 0.19 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.03 \[ \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {10 \, {\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )}}{b^{8} \sin \left (d x + c\right ) + a b^{7}} - \frac {2 \, b^{4} \sin \left (d x + c\right )^{5} - 5 \, a b^{3} \sin \left (d x + c\right )^{4} + 10 \, {\left (a^{2} b^{2} - b^{4}\right )} \sin \left (d x + c\right )^{3} - 10 \, {\left (2 \, a^{3} b - 3 \, a b^{3}\right )} \sin \left (d x + c\right )^{2} + 10 \, {\left (5 \, a^{4} - 9 \, a^{2} b^{2} + 3 \, b^{4}\right )} \sin \left (d x + c\right )}{b^{6}} + \frac {60 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{b^{7}}}{10 \, d} \]
1/10*(10*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)/(b^8*sin(d*x + c) + a*b^7) - (2*b^4*sin(d*x + c)^5 - 5*a*b^3*sin(d*x + c)^4 + 10*(a^2*b^2 - b^4)*sin(d* x + c)^3 - 10*(2*a^3*b - 3*a*b^3)*sin(d*x + c)^2 + 10*(5*a^4 - 9*a^2*b^2 + 3*b^4)*sin(d*x + c))/b^6 + 60*(a^5 - 2*a^3*b^2 + a*b^4)*log(b*sin(d*x + c ) + a)/b^7)/d
Time = 0.33 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.36 \[ \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {\frac {60 \, {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{b^{7}} - \frac {10 \, {\left (6 \, a^{5} b \sin \left (d x + c\right ) - 12 \, a^{3} b^{3} \sin \left (d x + c\right ) + 6 \, a b^{5} \sin \left (d x + c\right ) + 5 \, a^{6} - 9 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} b^{7}} - \frac {2 \, b^{8} \sin \left (d x + c\right )^{5} - 5 \, a b^{7} \sin \left (d x + c\right )^{4} + 10 \, a^{2} b^{6} \sin \left (d x + c\right )^{3} - 10 \, b^{8} \sin \left (d x + c\right )^{3} - 20 \, a^{3} b^{5} \sin \left (d x + c\right )^{2} + 30 \, a b^{7} \sin \left (d x + c\right )^{2} + 50 \, a^{4} b^{4} \sin \left (d x + c\right ) - 90 \, a^{2} b^{6} \sin \left (d x + c\right ) + 30 \, b^{8} \sin \left (d x + c\right )}{b^{10}}}{10 \, d} \]
1/10*(60*(a^5 - 2*a^3*b^2 + a*b^4)*log(abs(b*sin(d*x + c) + a))/b^7 - 10*( 6*a^5*b*sin(d*x + c) - 12*a^3*b^3*sin(d*x + c) + 6*a*b^5*sin(d*x + c) + 5* a^6 - 9*a^4*b^2 + 3*a^2*b^4 + b^6)/((b*sin(d*x + c) + a)*b^7) - (2*b^8*sin (d*x + c)^5 - 5*a*b^7*sin(d*x + c)^4 + 10*a^2*b^6*sin(d*x + c)^3 - 10*b^8* sin(d*x + c)^3 - 20*a^3*b^5*sin(d*x + c)^2 + 30*a*b^7*sin(d*x + c)^2 + 50* a^4*b^4*sin(d*x + c) - 90*a^2*b^6*sin(d*x + c) + 30*b^8*sin(d*x + c))/b^10 )/d
Time = 0.14 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.41 \[ \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^2} \, dx=\frac {{\sin \left (c+d\,x\right )}^3\,\left (\frac {1}{b^2}-\frac {a^2}{b^4}\right )}{d}-\frac {{\sin \left (c+d\,x\right )}^5}{5\,b^2\,d}-\frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {a^3}{b^5}+\frac {a\,\left (\frac {3}{b^2}-\frac {3\,a^2}{b^4}\right )}{b}\right )}{d}-\frac {\sin \left (c+d\,x\right )\,\left (\frac {3}{b^2}+\frac {a^2\,\left (\frac {3}{b^2}-\frac {3\,a^2}{b^4}\right )}{b^2}-\frac {2\,a\,\left (\frac {2\,a^3}{b^5}+\frac {2\,a\,\left (\frac {3}{b^2}-\frac {3\,a^2}{b^4}\right )}{b}\right )}{b}\right )}{d}+\frac {a\,{\sin \left (c+d\,x\right )}^4}{2\,b^3\,d}+\frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (6\,a^5-12\,a^3\,b^2+6\,a\,b^4\right )}{b^7\,d}+\frac {a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}{b\,d\,\left (\sin \left (c+d\,x\right )\,b^7+a\,b^6\right )} \]
(sin(c + d*x)^3*(1/b^2 - a^2/b^4))/d - sin(c + d*x)^5/(5*b^2*d) - (sin(c + d*x)^2*(a^3/b^5 + (a*(3/b^2 - (3*a^2)/b^4))/b))/d - (sin(c + d*x)*(3/b^2 + (a^2*(3/b^2 - (3*a^2)/b^4))/b^2 - (2*a*((2*a^3)/b^5 + (2*a*(3/b^2 - (3*a ^2)/b^4))/b))/b))/d + (a*sin(c + d*x)^4)/(2*b^3*d) + (log(a + b*sin(c + d* x))*(6*a*b^4 + 6*a^5 - 12*a^3*b^2))/(b^7*d) + (a^6 - b^6 + 3*a^2*b^4 - 3*a ^4*b^2)/(b*d*(a*b^6 + b^7*sin(c + d*x)))