Integrand size = 21, antiderivative size = 207 \[ \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {\left (a^2-b^2\right )^3}{7 b^7 d (a+b \sin (c+d x))^7}-\frac {a \left (a^2-b^2\right )^2}{b^7 d (a+b \sin (c+d x))^6}+\frac {3 \left (5 a^4-6 a^2 b^2+b^4\right )}{5 b^7 d (a+b \sin (c+d x))^5}-\frac {a \left (5 a^2-3 b^2\right )}{b^7 d (a+b \sin (c+d x))^4}+\frac {5 a^2-b^2}{b^7 d (a+b \sin (c+d x))^3}-\frac {3 a}{b^7 d (a+b \sin (c+d x))^2}+\frac {1}{b^7 d (a+b \sin (c+d x))} \]
1/7*(a^2-b^2)^3/b^7/d/(a+b*sin(d*x+c))^7-a*(a^2-b^2)^2/b^7/d/(a+b*sin(d*x+ c))^6+3/5*(5*a^4-6*a^2*b^2+b^4)/b^7/d/(a+b*sin(d*x+c))^5-a*(5*a^2-3*b^2)/b ^7/d/(a+b*sin(d*x+c))^4+(5*a^2-b^2)/b^7/d/(a+b*sin(d*x+c))^3-3*a/b^7/d/(a+ b*sin(d*x+c))^2+1/b^7/d/(a+b*sin(d*x+c))
Time = 0.89 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {\frac {\left (a^2-b^2\right )^3}{7 (a+b \sin (c+d x))^7}-\frac {a \left (a^2-b^2\right )^2}{(a+b \sin (c+d x))^6}+\frac {3 \left (5 a^4-6 a^2 b^2+b^4\right )}{5 (a+b \sin (c+d x))^5}-\frac {a \left (5 a^2-3 b^2\right )}{(a+b \sin (c+d x))^4}+\frac {5 a^2-b^2}{(a+b \sin (c+d x))^3}-\frac {3 a}{(a+b \sin (c+d x))^2}+\frac {1}{a+b \sin (c+d x)}}{b^7 d} \]
((a^2 - b^2)^3/(7*(a + b*Sin[c + d*x])^7) - (a*(a^2 - b^2)^2)/(a + b*Sin[c + d*x])^6 + (3*(5*a^4 - 6*a^2*b^2 + b^4))/(5*(a + b*Sin[c + d*x])^5) - (a *(5*a^2 - 3*b^2))/(a + b*Sin[c + d*x])^4 + (5*a^2 - b^2)/(a + b*Sin[c + d* x])^3 - (3*a)/(a + b*Sin[c + d*x])^2 + (a + b*Sin[c + d*x])^(-1))/(b^7*d)
Time = 0.38 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.83, 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.
| \(\displaystyle \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^7}{(a+b \sin (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {\int \frac {\left (b^2-b^2 \sin ^2(c+d x)\right )^3}{(a+b \sin (c+d x))^8}d(b \sin (c+d x))}{b^7 d}\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \frac {\int \left (-\frac {\left (a^2-b^2\right )^3}{(a+b \sin (c+d x))^8}+\frac {6 a \left (a^2-b^2\right )^2}{(a+b \sin (c+d x))^7}-\frac {1}{(a+b \sin (c+d x))^2}+\frac {6 a}{(a+b \sin (c+d x))^3}-\frac {3 \left (5 a^2-b^2\right )}{(a+b \sin (c+d x))^4}+\frac {4 \left (5 a^3-3 a b^2\right )}{(a+b \sin (c+d x))^5}-\frac {3 \left (5 a^4-6 b^2 a^2+b^4\right )}{(a+b \sin (c+d x))^6}\right )d(b \sin (c+d x))}{b^7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\left (a^2-b^2\right )^3}{7 (a+b \sin (c+d x))^7}-\frac {a \left (a^2-b^2\right )^2}{(a+b \sin (c+d x))^6}+\frac {5 a^2-b^2}{(a+b \sin (c+d x))^3}-\frac {a \left (5 a^2-3 b^2\right )}{(a+b \sin (c+d x))^4}+\frac {3 \left (5 a^4-6 a^2 b^2+b^4\right )}{5 (a+b \sin (c+d x))^5}+\frac {1}{a+b \sin (c+d x)}-\frac {3 a}{(a+b \sin (c+d x))^2}}{b^7 d}\) |
((a^2 - b^2)^3/(7*(a + b*Sin[c + d*x])^7) - (a*(a^2 - b^2)^2)/(a + b*Sin[c + d*x])^6 + (3*(5*a^4 - 6*a^2*b^2 + b^4))/(5*(a + b*Sin[c + d*x])^5) - (a *(5*a^2 - 3*b^2))/(a + b*Sin[c + d*x])^4 + (5*a^2 - b^2)/(a + b*Sin[c + d* x])^3 - (3*a)/(a + b*Sin[c + d*x])^2 + (a + b*Sin[c + d*x])^(-1))/(b^7*d)
3.5.61.3.1 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 = 7.56 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00
| method | result | size |
| derivativedivides | \(-\frac {\frac {a \left (5 a^{2}-3 b^{2}\right )}{b^{7} \left (a +b \sin \left (d x +c \right )\right )^{4}}+\frac {a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}{b^{7} \left (a +b \sin \left (d x +c \right )\right )^{6}}-\frac {1}{b^{7} \left (a +b \sin \left (d x +c \right )\right )}-\frac {a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}{7 b^{7} \left (a +b \sin \left (d x +c \right )\right )^{7}}-\frac {15 a^{2}-3 b^{2}}{3 b^{7} \left (a +b \sin \left (d x +c \right )\right )^{3}}-\frac {15 a^{4}-18 a^{2} b^{2}+3 b^{4}}{5 b^{7} \left (a +b \sin \left (d x +c \right )\right )^{5}}+\frac {3 a}{b^{7} \left (a +b \sin \left (d x +c \right )\right )^{2}}}{d}\) | \(208\) |
| default | \(-\frac {\frac {a \left (5 a^{2}-3 b^{2}\right )}{b^{7} \left (a +b \sin \left (d x +c \right )\right )^{4}}+\frac {a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}{b^{7} \left (a +b \sin \left (d x +c \right )\right )^{6}}-\frac {1}{b^{7} \left (a +b \sin \left (d x +c \right )\right )}-\frac {a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}{7 b^{7} \left (a +b \sin \left (d x +c \right )\right )^{7}}-\frac {15 a^{2}-3 b^{2}}{3 b^{7} \left (a +b \sin \left (d x +c \right )\right )^{3}}-\frac {15 a^{4}-18 a^{2} b^{2}+3 b^{4}}{5 b^{7} \left (a +b \sin \left (d x +c \right )\right )^{5}}+\frac {3 a}{b^{7} \left (a +b \sin \left (d x +c \right )\right )^{2}}}{d}\) | \(208\) |
| parallelrisch | \(\frac {2 \left (\left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{6}+6 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5} b +\left (2 a^{6}+20 a^{4} b^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (22 a^{5} b +40 a^{3} b^{3}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {43}{5} a^{6}+48 a^{2} b^{4}+\frac {352}{5} a^{4} b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {212}{5} a^{5} b +\frac {568}{5} a^{3} b^{3}+32 a \,b^{5}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {212}{35} a^{6}+\frac {64}{7} b^{6}+\frac {3592}{35} a^{4} b^{2}+\frac {3296}{35} a^{2} b^{4}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {212}{5} a^{5} b +\frac {568}{5} a^{3} b^{3}+32 a \,b^{5}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {43}{5} a^{6}+48 a^{2} b^{4}+\frac {352}{5} a^{4} b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (22 a^{5} b +40 a^{3} b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (2 a^{6}+20 a^{4} b^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{5} b +a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a^{7} d {\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{7}}\) | \(377\) |
| risch | \(\frac {2 i {\mathrm e}^{i \left (d x +c \right )} \left (35 b^{6}-320 a^{6} {\mathrm e}^{6 i \left (d x +c \right )}+35 b^{6} {\mathrm e}^{12 i \left (d x +c \right )}-70 b^{6} {\mathrm e}^{10 i \left (d x +c \right )}+301 b^{6} {\mathrm e}^{8 i \left (d x +c \right )}-212 b^{6} {\mathrm e}^{6 i \left (d x +c \right )}+301 b^{6} {\mathrm e}^{4 i \left (d x +c \right )}-70 b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-3976 i b^{3} {\mathrm e}^{5 i \left (d x +c \right )} a^{3}+1120 i b \,{\mathrm e}^{7 i \left (d x +c \right )} a^{5}+1400 i b^{3} a^{3} {\mathrm e}^{3 i \left (d x +c \right )}-1120 i a^{5} {\mathrm e}^{5 i \left (d x +c \right )} b +210 i b^{5} {\mathrm e}^{11 i \left (d x +c \right )} a -770 i b^{5} {\mathrm e}^{9 i \left (d x +c \right )} a +1484 i b^{5} {\mathrm e}^{7 i \left (d x +c \right )} a -1400 i b^{3} {\mathrm e}^{9 i \left (d x +c \right )} a^{3}-1484 i b^{5} {\mathrm e}^{5 i \left (d x +c \right )} a +3976 i b^{3} {\mathrm e}^{7 i \left (d x +c \right )} a^{3}+770 i b^{5} {\mathrm e}^{3 i \left (d x +c \right )} a -210 i b^{5} a \,{\mathrm e}^{i \left (d x +c \right )}+2464 b^{4} {\mathrm e}^{8 i \left (d x +c \right )} a^{2}-3592 b^{4} {\mathrm e}^{6 i \left (d x +c \right )} a^{2}+1680 b^{2} {\mathrm e}^{8 i \left (d x +c \right )} a^{4}+2464 b^{4} {\mathrm e}^{4 i \left (d x +c \right )} a^{2}-3296 b^{2} {\mathrm e}^{6 i \left (d x +c \right )} a^{4}-700 b^{4} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+1680 b^{2} a^{4} {\mathrm e}^{4 i \left (d x +c \right )}-700 b^{4} {\mathrm e}^{10 i \left (d x +c \right )} a^{2}\right )}{35 \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{7} d \,b^{7}}\) | \(543\) |
-1/d*(a*(5*a^2-3*b^2)/b^7/(a+b*sin(d*x+c))^4+a*(a^4-2*a^2*b^2+b^4)/b^7/(a+ b*sin(d*x+c))^6-1/b^7/(a+b*sin(d*x+c))-1/7*(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/b ^7/(a+b*sin(d*x+c))^7-1/3*(15*a^2-3*b^2)/b^7/(a+b*sin(d*x+c))^3-1/5*(15*a^ 4-18*a^2*b^2+3*b^4)/b^7/(a+b*sin(d*x+c))^5+3*a/b^7/(a+b*sin(d*x+c))^2)
Time = 0.34 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.85 \[ \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {35 \, b^{6} \cos \left (d x + c\right )^{6} - 5 \, a^{6} - 104 \, a^{4} b^{2} - 155 \, a^{2} b^{4} - 16 \, b^{6} - 35 \, {\left (5 \, a^{2} b^{4} + 2 \, b^{6}\right )} \cos \left (d x + c\right )^{4} + 7 \, {\left (15 \, a^{4} b^{2} + 47 \, a^{2} b^{4} + 8 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - 7 \, {\left (15 \, a b^{5} \cos \left (d x + c\right )^{4} + 5 \, a^{5} b + 24 \, a^{3} b^{3} + 11 \, a b^{5} - 25 \, {\left (a^{3} b^{3} + a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{35 \, {\left (7 \, a b^{13} d \cos \left (d x + c\right )^{6} - 7 \, {\left (5 \, a^{3} b^{11} + 3 \, a b^{13}\right )} d \cos \left (d x + c\right )^{4} + 7 \, {\left (3 \, a^{5} b^{9} + 10 \, a^{3} b^{11} + 3 \, a b^{13}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{7} b^{7} + 21 \, a^{5} b^{9} + 35 \, a^{3} b^{11} + 7 \, a b^{13}\right )} d + {\left (b^{14} d \cos \left (d x + c\right )^{6} - 3 \, {\left (7 \, a^{2} b^{12} + b^{14}\right )} d \cos \left (d x + c\right )^{4} + {\left (35 \, a^{4} b^{10} + 42 \, a^{2} b^{12} + 3 \, b^{14}\right )} d \cos \left (d x + c\right )^{2} - {\left (7 \, a^{6} b^{8} + 35 \, a^{4} b^{10} + 21 \, a^{2} b^{12} + b^{14}\right )} d\right )} \sin \left (d x + c\right )\right )}} \]
1/35*(35*b^6*cos(d*x + c)^6 - 5*a^6 - 104*a^4*b^2 - 155*a^2*b^4 - 16*b^6 - 35*(5*a^2*b^4 + 2*b^6)*cos(d*x + c)^4 + 7*(15*a^4*b^2 + 47*a^2*b^4 + 8*b^ 6)*cos(d*x + c)^2 - 7*(15*a*b^5*cos(d*x + c)^4 + 5*a^5*b + 24*a^3*b^3 + 11 *a*b^5 - 25*(a^3*b^3 + a*b^5)*cos(d*x + c)^2)*sin(d*x + c))/(7*a*b^13*d*co s(d*x + c)^6 - 7*(5*a^3*b^11 + 3*a*b^13)*d*cos(d*x + c)^4 + 7*(3*a^5*b^9 + 10*a^3*b^11 + 3*a*b^13)*d*cos(d*x + c)^2 - (a^7*b^7 + 21*a^5*b^9 + 35*a^3 *b^11 + 7*a*b^13)*d + (b^14*d*cos(d*x + c)^6 - 3*(7*a^2*b^12 + b^14)*d*cos (d*x + c)^4 + (35*a^4*b^10 + 42*a^2*b^12 + 3*b^14)*d*cos(d*x + c)^2 - (7*a ^6*b^8 + 35*a^4*b^10 + 21*a^2*b^12 + b^14)*d)*sin(d*x + c))
Leaf count of result is larger than twice the leaf count of optimal. 2530 vs. \(2 (184) = 368\).
Time = 9.85 (sec) , antiderivative size = 2530, normalized size of antiderivative = 12.22 \[ \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
Piecewise((x*cos(c)**7/a**8, Eq(b, 0) & Eq(d, 0)), ((16*sin(c + d*x)**7/(3 5*d) + 8*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 2*sin(c + d*x)**3*cos(c + d*x)**4/d + sin(c + d*x)*cos(c + d*x)**6/d)/a**8, Eq(b, 0)), (x*cos(c)**7 /(a + b*sin(c))**8, Eq(d, 0)), (5*a**6/(35*a**7*b**7*d + 245*a**6*b**8*d*s in(c + d*x) + 735*a**5*b**9*d*sin(c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a**2*b**12*d*sin(c + d*x )**5 + 245*a*b**13*d*sin(c + d*x)**6 + 35*b**14*d*sin(c + d*x)**7) + 35*a* *5*b*sin(c + d*x)/(35*a**7*b**7*d + 245*a**6*b**8*d*sin(c + d*x) + 735*a** 5*b**9*d*sin(c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b **11*d*sin(c + d*x)**4 + 735*a**2*b**12*d*sin(c + d*x)**5 + 245*a*b**13*d* sin(c + d*x)**6 + 35*b**14*d*sin(c + d*x)**7) + 104*a**4*b**2*sin(c + d*x) **2/(35*a**7*b**7*d + 245*a**6*b**8*d*sin(c + d*x) + 735*a**5*b**9*d*sin(c + d*x)**2 + 1225*a**4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a**2*b**12*d*sin(c + d*x)**5 + 245*a*b**13*d*sin(c + d*x)** 6 + 35*b**14*d*sin(c + d*x)**7) - a**4*b**2*cos(c + d*x)**2/(35*a**7*b**7* d + 245*a**6*b**8*d*sin(c + d*x) + 735*a**5*b**9*d*sin(c + d*x)**2 + 1225* a**4*b**10*d*sin(c + d*x)**3 + 1225*a**3*b**11*d*sin(c + d*x)**4 + 735*a** 2*b**12*d*sin(c + d*x)**5 + 245*a*b**13*d*sin(c + d*x)**6 + 35*b**14*d*sin (c + d*x)**7) + 168*a**3*b**3*sin(c + d*x)**3/(35*a**7*b**7*d + 245*a**6*b **8*d*sin(c + d*x) + 735*a**5*b**9*d*sin(c + d*x)**2 + 1225*a**4*b**10*...
Time = 0.20 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.35 \[ \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {35 \, b^{6} \sin \left (d x + c\right )^{6} + 105 \, a b^{5} \sin \left (d x + c\right )^{5} + 5 \, a^{6} - a^{4} b^{2} + a^{2} b^{4} - 5 \, b^{6} + 35 \, {\left (5 \, a^{2} b^{4} - b^{6}\right )} \sin \left (d x + c\right )^{4} + 35 \, {\left (5 \, a^{3} b^{3} - a b^{5}\right )} \sin \left (d x + c\right )^{3} + 21 \, {\left (5 \, a^{4} b^{2} - a^{2} b^{4} + b^{6}\right )} \sin \left (d x + c\right )^{2} + 7 \, {\left (5 \, a^{5} b - a^{3} b^{3} + a b^{5}\right )} \sin \left (d x + c\right )}{35 \, {\left (b^{14} \sin \left (d x + c\right )^{7} + 7 \, a b^{13} \sin \left (d x + c\right )^{6} + 21 \, a^{2} b^{12} \sin \left (d x + c\right )^{5} + 35 \, a^{3} b^{11} \sin \left (d x + c\right )^{4} + 35 \, a^{4} b^{10} \sin \left (d x + c\right )^{3} + 21 \, a^{5} b^{9} \sin \left (d x + c\right )^{2} + 7 \, a^{6} b^{8} \sin \left (d x + c\right ) + a^{7} b^{7}\right )} d} \]
1/35*(35*b^6*sin(d*x + c)^6 + 105*a*b^5*sin(d*x + c)^5 + 5*a^6 - a^4*b^2 + a^2*b^4 - 5*b^6 + 35*(5*a^2*b^4 - b^6)*sin(d*x + c)^4 + 35*(5*a^3*b^3 - a *b^5)*sin(d*x + c)^3 + 21*(5*a^4*b^2 - a^2*b^4 + b^6)*sin(d*x + c)^2 + 7*( 5*a^5*b - a^3*b^3 + a*b^5)*sin(d*x + c))/((b^14*sin(d*x + c)^7 + 7*a*b^13* sin(d*x + c)^6 + 21*a^2*b^12*sin(d*x + c)^5 + 35*a^3*b^11*sin(d*x + c)^4 + 35*a^4*b^10*sin(d*x + c)^3 + 21*a^5*b^9*sin(d*x + c)^2 + 7*a^6*b^8*sin(d* x + c) + a^7*b^7)*d)
Time = 0.54 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.04 \[ \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {35 \, b^{6} \sin \left (d x + c\right )^{6} + 105 \, a b^{5} \sin \left (d x + c\right )^{5} + 175 \, a^{2} b^{4} \sin \left (d x + c\right )^{4} - 35 \, b^{6} \sin \left (d x + c\right )^{4} + 175 \, a^{3} b^{3} \sin \left (d x + c\right )^{3} - 35 \, a b^{5} \sin \left (d x + c\right )^{3} + 105 \, a^{4} b^{2} \sin \left (d x + c\right )^{2} - 21 \, a^{2} b^{4} \sin \left (d x + c\right )^{2} + 21 \, b^{6} \sin \left (d x + c\right )^{2} + 35 \, a^{5} b \sin \left (d x + c\right ) - 7 \, a^{3} b^{3} \sin \left (d x + c\right ) + 7 \, a b^{5} \sin \left (d x + c\right ) + 5 \, a^{6} - a^{4} b^{2} + a^{2} b^{4} - 5 \, b^{6}}{35 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{7} b^{7} d} \]
1/35*(35*b^6*sin(d*x + c)^6 + 105*a*b^5*sin(d*x + c)^5 + 175*a^2*b^4*sin(d *x + c)^4 - 35*b^6*sin(d*x + c)^4 + 175*a^3*b^3*sin(d*x + c)^3 - 35*a*b^5* sin(d*x + c)^3 + 105*a^4*b^2*sin(d*x + c)^2 - 21*a^2*b^4*sin(d*x + c)^2 + 21*b^6*sin(d*x + c)^2 + 35*a^5*b*sin(d*x + c) - 7*a^3*b^3*sin(d*x + c) + 7 *a*b^5*sin(d*x + c) + 5*a^6 - a^4*b^2 + a^2*b^4 - 5*b^6)/((b*sin(d*x + c) + a)^7*b^7*d)
Time = 0.26 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.33 \[ \int \frac {\cos ^7(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {\frac {5\,a^6-a^4\,b^2+a^2\,b^4-5\,b^6}{35\,b^7}+\frac {{\sin \left (c+d\,x\right )}^6}{b}+\frac {3\,{\sin \left (c+d\,x\right )}^2\,\left (5\,a^4-a^2\,b^2+b^4\right )}{5\,b^5}+\frac {3\,a\,{\sin \left (c+d\,x\right )}^5}{b^2}+\frac {{\sin \left (c+d\,x\right )}^4\,\left (5\,a^2-b^2\right )}{b^3}+\frac {a\,\sin \left (c+d\,x\right )\,\left (5\,a^4-a^2\,b^2+b^4\right )}{5\,b^6}+\frac {a\,{\sin \left (c+d\,x\right )}^3\,\left (5\,a^2-b^2\right )}{b^4}}{d\,\left (a^7+7\,a^6\,b\,\sin \left (c+d\,x\right )+21\,a^5\,b^2\,{\sin \left (c+d\,x\right )}^2+35\,a^4\,b^3\,{\sin \left (c+d\,x\right )}^3+35\,a^3\,b^4\,{\sin \left (c+d\,x\right )}^4+21\,a^2\,b^5\,{\sin \left (c+d\,x\right )}^5+7\,a\,b^6\,{\sin \left (c+d\,x\right )}^6+b^7\,{\sin \left (c+d\,x\right )}^7\right )} \]
((5*a^6 - 5*b^6 + a^2*b^4 - a^4*b^2)/(35*b^7) + sin(c + d*x)^6/b + (3*sin( c + d*x)^2*(5*a^4 + b^4 - a^2*b^2))/(5*b^5) + (3*a*sin(c + d*x)^5)/b^2 + ( sin(c + d*x)^4*(5*a^2 - b^2))/b^3 + (a*sin(c + d*x)*(5*a^4 + b^4 - a^2*b^2 ))/(5*b^6) + (a*sin(c + d*x)^3*(5*a^2 - b^2))/b^4)/(d*(a^7 + b^7*sin(c + d *x)^7 + 7*a*b^6*sin(c + d*x)^6 + 21*a^5*b^2*sin(c + d*x)^2 + 35*a^4*b^3*si n(c + d*x)^3 + 35*a^3*b^4*sin(c + d*x)^4 + 21*a^2*b^5*sin(c + d*x)^5 + 7*a ^6*b*sin(c + d*x)))