Integrand size = 21, antiderivative size = 141 \[ \int \frac {\cos ^5(c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {\left (a^2-b^2\right )^2}{7 b^5 d (a+b \sin (c+d x))^7}+\frac {2 a \left (a^2-b^2\right )}{3 b^5 d (a+b \sin (c+d x))^6}-\frac {2 \left (3 a^2-b^2\right )}{5 b^5 d (a+b \sin (c+d x))^5}+\frac {a}{b^5 d (a+b \sin (c+d x))^4}-\frac {1}{3 b^5 d (a+b \sin (c+d x))^3} \]
-1/7*(a^2-b^2)^2/b^5/d/(a+b*sin(d*x+c))^7+2/3*a*(a^2-b^2)/b^5/d/(a+b*sin(d *x+c))^6-2/5*(3*a^2-b^2)/b^5/d/(a+b*sin(d*x+c))^5+a/b^5/d/(a+b*sin(d*x+c)) ^4-1/3/b^5/d/(a+b*sin(d*x+c))^3
Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.76 \[ \int \frac {\cos ^5(c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {a^4-2 a^2 b^2+15 b^4+7 a b \left (a^2-2 b^2\right ) \sin (c+d x)+21 b^2 \left (a^2-2 b^2\right ) \sin ^2(c+d x)+35 a b^3 \sin ^3(c+d x)+35 b^4 \sin ^4(c+d x)}{105 b^5 d (a+b \sin (c+d x))^7} \]
-1/105*(a^4 - 2*a^2*b^2 + 15*b^4 + 7*a*b*(a^2 - 2*b^2)*Sin[c + d*x] + 21*b ^2*(a^2 - 2*b^2)*Sin[c + d*x]^2 + 35*a*b^3*Sin[c + d*x]^3 + 35*b^4*Sin[c + d*x]^4)/(b^5*d*(a + b*Sin[c + d*x])^7)
Time = 0.30 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.84, 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.
| \(\displaystyle \int \frac {\cos ^5(c+d x)}{(a+b \sin (c+d x))^8} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (c+d x)^5}{(a+b \sin (c+d x))^8}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {\int \frac {\left (b^2-b^2 \sin ^2(c+d x)\right )^2}{(a+b \sin (c+d x))^8}d(b \sin (c+d x))}{b^5 d}\) |
| \(\Big \downarrow \) 476 |
| \(\displaystyle \frac {\int \left (\frac {\left (a^2-b^2\right )^2}{(a+b \sin (c+d x))^8}+\frac {1}{(a+b \sin (c+d x))^4}-\frac {4 a}{(a+b \sin (c+d x))^5}+\frac {2 \left (3 a^2-b^2\right )}{(a+b \sin (c+d x))^6}-\frac {4 \left (a^3-a b^2\right )}{(a+b \sin (c+d x))^7}\right )d(b \sin (c+d x))}{b^5 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\left (a^2-b^2\right )^2}{7 (a+b \sin (c+d x))^7}+\frac {2 a \left (a^2-b^2\right )}{3 (a+b \sin (c+d x))^6}-\frac {2 \left (3 a^2-b^2\right )}{5 (a+b \sin (c+d x))^5}-\frac {1}{3 (a+b \sin (c+d x))^3}+\frac {a}{(a+b \sin (c+d x))^4}}{b^5 d}\) |
(-1/7*(a^2 - b^2)^2/(a + b*Sin[c + d*x])^7 + (2*a*(a^2 - b^2))/(3*(a + b*S in[c + d*x])^6) - (2*(3*a^2 - b^2))/(5*(a + b*Sin[c + d*x])^5) + a/(a + b* Sin[c + d*x])^4 - 1/(3*(a + b*Sin[c + d*x])^3))/(b^5*d)
3.5.62.3.1 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 = 7.48 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.90
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{3 b^{5} \left (a +b \sin \left (d x +c \right )\right )^{3}}-\frac {a^{4}-2 a^{2} b^{2}+b^{4}}{7 b^{5} \left (a +b \sin \left (d x +c \right )\right )^{7}}-\frac {6 a^{2}-2 b^{2}}{5 b^{5} \left (a +b \sin \left (d x +c \right )\right )^{5}}+\frac {a}{b^{5} \left (a +b \sin \left (d x +c \right )\right )^{4}}+\frac {2 a \left (a^{2}-b^{2}\right )}{3 b^{5} \left (a +b \sin \left (d x +c \right )\right )^{6}}}{d}\) | \(127\) |
| default | \(\frac {-\frac {1}{3 b^{5} \left (a +b \sin \left (d x +c \right )\right )^{3}}-\frac {a^{4}-2 a^{2} b^{2}+b^{4}}{7 b^{5} \left (a +b \sin \left (d x +c \right )\right )^{7}}-\frac {6 a^{2}-2 b^{2}}{5 b^{5} \left (a +b \sin \left (d x +c \right )\right )^{5}}+\frac {a}{b^{5} \left (a +b \sin \left (d x +c \right )\right )^{4}}+\frac {2 a \left (a^{2}-b^{2}\right )}{3 b^{5} \left (a +b \sin \left (d x +c \right )\right )^{6}}}{d}\) | \(127\) |
| risch | \(\frac {8 i {\mathrm e}^{3 i \left (d x +c \right )} \left (70 i b^{3} {\mathrm e}^{7 i \left (d x +c \right )} a +35 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-56 i b \,{\mathrm e}^{5 i \left (d x +c \right )} a^{3}-98 i b^{3} {\mathrm e}^{5 i \left (d x +c \right )} a -84 b^{2} {\mathrm e}^{6 i \left (d x +c \right )} a^{2}+28 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+56 i a^{3} {\mathrm e}^{3 i \left (d x +c \right )} b +98 i b^{3} {\mathrm e}^{3 i \left (d x +c \right )} a +16 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+136 b^{2} {\mathrm e}^{4 i \left (d x +c \right )} a^{2}+114 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-70 i b^{3} a \,{\mathrm e}^{i \left (d x +c \right )}-84 b^{2} a^{2} {\mathrm e}^{2 i \left (d x +c \right )}+28 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+35 b^{4}\right )}{105 \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{7} d \,b^{5}}\) | \(302\) |
| parallelrisch | \(\frac {2 \left (\left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{6}+6 \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{5} b +\left (20 a^{4} b^{2}+\frac {10}{3} a^{6}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {74}{3} a^{5} b +40 a^{3} b^{3}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (48 a^{2} b^{4}+\frac {368}{5} a^{4} b^{2}+\frac {113}{15} a^{6}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {692}{15} a^{5} b +\frac {1736}{15} a^{3} b^{3}+32 a \,b^{5}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {2264}{21} a^{4} b^{2}+\frac {9952}{105} a^{2} b^{4}+\frac {52}{5} a^{6}+\frac {64}{7} b^{6}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {692}{15} a^{5} b +\frac {1736}{15} a^{3} b^{3}+32 a \,b^{5}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (48 a^{2} b^{4}+\frac {368}{5} a^{4} b^{2}+\frac {113}{15} a^{6}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {74}{3} a^{5} b +40 a^{3} b^{3}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (20 a^{4} b^{2}+\frac {10}{3} a^{6}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{5} b +a^{6}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{a^{7} d {\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}^{7}}\) | \(377\) |
1/d*(-1/3/b^5/(a+b*sin(d*x+c))^3-1/7*(a^4-2*a^2*b^2+b^4)/b^5/(a+b*sin(d*x+ c))^7-1/5*(6*a^2-2*b^2)/b^5/(a+b*sin(d*x+c))^5+a/b^5/(a+b*sin(d*x+c))^4+2/ 3*a*(a^2-b^2)/b^5/(a+b*sin(d*x+c))^6)
Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (133) = 266\).
Time = 0.34 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.19 \[ \int \frac {\cos ^5(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\frac {35 \, b^{4} \cos \left (d x + c\right )^{4} + a^{4} + 19 \, a^{2} b^{2} + 8 \, b^{4} - 7 \, {\left (3 \, a^{2} b^{2} + 4 \, b^{4}\right )} \cos \left (d x + c\right )^{2} - 7 \, {\left (5 \, a b^{3} \cos \left (d x + c\right )^{2} - a^{3} b - 3 \, a b^{3}\right )} \sin \left (d x + c\right )}{105 \, {\left (7 \, a b^{11} d \cos \left (d x + c\right )^{6} - 7 \, {\left (5 \, a^{3} b^{9} + 3 \, a b^{11}\right )} d \cos \left (d x + c\right )^{4} + 7 \, {\left (3 \, a^{5} b^{7} + 10 \, a^{3} b^{9} + 3 \, a b^{11}\right )} d \cos \left (d x + c\right )^{2} - {\left (a^{7} b^{5} + 21 \, a^{5} b^{7} + 35 \, a^{3} b^{9} + 7 \, a b^{11}\right )} d + {\left (b^{12} d \cos \left (d x + c\right )^{6} - 3 \, {\left (7 \, a^{2} b^{10} + b^{12}\right )} d \cos \left (d x + c\right )^{4} + {\left (35 \, a^{4} b^{8} + 42 \, a^{2} b^{10} + 3 \, b^{12}\right )} d \cos \left (d x + c\right )^{2} - {\left (7 \, a^{6} b^{6} + 35 \, a^{4} b^{8} + 21 \, a^{2} b^{10} + b^{12}\right )} d\right )} \sin \left (d x + c\right )\right )}} \]
1/105*(35*b^4*cos(d*x + c)^4 + a^4 + 19*a^2*b^2 + 8*b^4 - 7*(3*a^2*b^2 + 4 *b^4)*cos(d*x + c)^2 - 7*(5*a*b^3*cos(d*x + c)^2 - a^3*b - 3*a*b^3)*sin(d* x + c))/(7*a*b^11*d*cos(d*x + c)^6 - 7*(5*a^3*b^9 + 3*a*b^11)*d*cos(d*x + c)^4 + 7*(3*a^5*b^7 + 10*a^3*b^9 + 3*a*b^11)*d*cos(d*x + c)^2 - (a^7*b^5 + 21*a^5*b^7 + 35*a^3*b^9 + 7*a*b^11)*d + (b^12*d*cos(d*x + c)^6 - 3*(7*a^2 *b^10 + b^12)*d*cos(d*x + c)^4 + (35*a^4*b^8 + 42*a^2*b^10 + 3*b^12)*d*cos (d*x + c)^2 - (7*a^6*b^6 + 35*a^4*b^8 + 21*a^2*b^10 + b^12)*d)*sin(d*x + c ))
Leaf count of result is larger than twice the leaf count of optimal. 1425 vs. \(2 (124) = 248\).
Time = 9.69 (sec) , antiderivative size = 1425, normalized size of antiderivative = 10.11 \[ \int \frac {\cos ^5(c+d x)}{(a+b \sin (c+d x))^8} \, dx=\text {Too large to display} \]
Piecewise((x*cos(c)**5/a**8, Eq(b, 0) & Eq(d, 0)), ((8*sin(c + d*x)**5/(15 *d) + 4*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + sin(c + d*x)*cos(c + d*x)* *4/d)/a**8, Eq(b, 0)), (x*cos(c)**5/(a + b*sin(c))**8, Eq(d, 0)), (-a**4/( 105*a**7*b**5*d + 735*a**6*b**6*d*sin(c + d*x) + 2205*a**5*b**7*d*sin(c + d*x)**2 + 3675*a**4*b**8*d*sin(c + d*x)**3 + 3675*a**3*b**9*d*sin(c + d*x) **4 + 2205*a**2*b**10*d*sin(c + d*x)**5 + 735*a*b**11*d*sin(c + d*x)**6 + 105*b**12*d*sin(c + d*x)**7) - 7*a**3*b*sin(c + d*x)/(105*a**7*b**5*d + 73 5*a**6*b**6*d*sin(c + d*x) + 2205*a**5*b**7*d*sin(c + d*x)**2 + 3675*a**4* b**8*d*sin(c + d*x)**3 + 3675*a**3*b**9*d*sin(c + d*x)**4 + 2205*a**2*b**1 0*d*sin(c + d*x)**5 + 735*a*b**11*d*sin(c + d*x)**6 + 105*b**12*d*sin(c + d*x)**7) - 19*a**2*b**2*sin(c + d*x)**2/(105*a**7*b**5*d + 735*a**6*b**6*d *sin(c + d*x) + 2205*a**5*b**7*d*sin(c + d*x)**2 + 3675*a**4*b**8*d*sin(c + d*x)**3 + 3675*a**3*b**9*d*sin(c + d*x)**4 + 2205*a**2*b**10*d*sin(c + d *x)**5 + 735*a*b**11*d*sin(c + d*x)**6 + 105*b**12*d*sin(c + d*x)**7) + 2* a**2*b**2*cos(c + d*x)**2/(105*a**7*b**5*d + 735*a**6*b**6*d*sin(c + d*x) + 2205*a**5*b**7*d*sin(c + d*x)**2 + 3675*a**4*b**8*d*sin(c + d*x)**3 + 36 75*a**3*b**9*d*sin(c + d*x)**4 + 2205*a**2*b**10*d*sin(c + d*x)**5 + 735*a *b**11*d*sin(c + d*x)**6 + 105*b**12*d*sin(c + d*x)**7) - 21*a*b**3*sin(c + d*x)**3/(105*a**7*b**5*d + 735*a**6*b**6*d*sin(c + d*x) + 2205*a**5*b**7 *d*sin(c + d*x)**2 + 3675*a**4*b**8*d*sin(c + d*x)**3 + 3675*a**3*b**9*...
Time = 0.19 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.46 \[ \int \frac {\cos ^5(c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {35 \, b^{4} \sin \left (d x + c\right )^{4} + 35 \, a b^{3} \sin \left (d x + c\right )^{3} + a^{4} - 2 \, a^{2} b^{2} + 15 \, b^{4} + 21 \, {\left (a^{2} b^{2} - 2 \, b^{4}\right )} \sin \left (d x + c\right )^{2} + 7 \, {\left (a^{3} b - 2 \, a b^{3}\right )} \sin \left (d x + c\right )}{105 \, {\left (b^{12} \sin \left (d x + c\right )^{7} + 7 \, a b^{11} \sin \left (d x + c\right )^{6} + 21 \, a^{2} b^{10} \sin \left (d x + c\right )^{5} + 35 \, a^{3} b^{9} \sin \left (d x + c\right )^{4} + 35 \, a^{4} b^{8} \sin \left (d x + c\right )^{3} + 21 \, a^{5} b^{7} \sin \left (d x + c\right )^{2} + 7 \, a^{6} b^{6} \sin \left (d x + c\right ) + a^{7} b^{5}\right )} d} \]
-1/105*(35*b^4*sin(d*x + c)^4 + 35*a*b^3*sin(d*x + c)^3 + a^4 - 2*a^2*b^2 + 15*b^4 + 21*(a^2*b^2 - 2*b^4)*sin(d*x + c)^2 + 7*(a^3*b - 2*a*b^3)*sin(d *x + c))/((b^12*sin(d*x + c)^7 + 7*a*b^11*sin(d*x + c)^6 + 21*a^2*b^10*sin (d*x + c)^5 + 35*a^3*b^9*sin(d*x + c)^4 + 35*a^4*b^8*sin(d*x + c)^3 + 21*a ^5*b^7*sin(d*x + c)^2 + 7*a^6*b^6*sin(d*x + c) + a^7*b^5)*d)
Time = 0.49 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.83 \[ \int \frac {\cos ^5(c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {35 \, b^{4} \sin \left (d x + c\right )^{4} + 35 \, a b^{3} \sin \left (d x + c\right )^{3} + 21 \, a^{2} b^{2} \sin \left (d x + c\right )^{2} - 42 \, b^{4} \sin \left (d x + c\right )^{2} + 7 \, a^{3} b \sin \left (d x + c\right ) - 14 \, a b^{3} \sin \left (d x + c\right ) + a^{4} - 2 \, a^{2} b^{2} + 15 \, b^{4}}{105 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{7} b^{5} d} \]
-1/105*(35*b^4*sin(d*x + c)^4 + 35*a*b^3*sin(d*x + c)^3 + 21*a^2*b^2*sin(d *x + c)^2 - 42*b^4*sin(d*x + c)^2 + 7*a^3*b*sin(d*x + c) - 14*a*b^3*sin(d* x + c) + a^4 - 2*a^2*b^2 + 15*b^4)/((b*sin(d*x + c) + a)^7*b^5*d)
Time = 0.15 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.46 \[ \int \frac {\cos ^5(c+d x)}{(a+b \sin (c+d x))^8} \, dx=-\frac {\frac {a^4-2\,a^2\,b^2+15\,b^4}{105\,b^5}+\frac {{\sin \left (c+d\,x\right )}^4}{3\,b}+\frac {{\sin \left (c+d\,x\right )}^2\,\left (a^2-2\,b^2\right )}{5\,b^3}+\frac {a\,{\sin \left (c+d\,x\right )}^3}{3\,b^2}+\frac {a\,\sin \left (c+d\,x\right )\,\left (a^2-2\,b^2\right )}{15\,b^4}}{d\,\left (a^7+7\,a^6\,b\,\sin \left (c+d\,x\right )+21\,a^5\,b^2\,{\sin \left (c+d\,x\right )}^2+35\,a^4\,b^3\,{\sin \left (c+d\,x\right )}^3+35\,a^3\,b^4\,{\sin \left (c+d\,x\right )}^4+21\,a^2\,b^5\,{\sin \left (c+d\,x\right )}^5+7\,a\,b^6\,{\sin \left (c+d\,x\right )}^6+b^7\,{\sin \left (c+d\,x\right )}^7\right )} \]
-((a^4 + 15*b^4 - 2*a^2*b^2)/(105*b^5) + sin(c + d*x)^4/(3*b) + (sin(c + d *x)^2*(a^2 - 2*b^2))/(5*b^3) + (a*sin(c + d*x)^3)/(3*b^2) + (a*sin(c + d*x )*(a^2 - 2*b^2))/(15*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*sin(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)))