Integrand size = 21, antiderivative size = 45 \[ \int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {1}{3 a^2 d (a+a \sin (c+d x))^6}+\frac {1}{5 a^3 d (a+a \sin (c+d x))^5} \] Output:
-1/3/a^2/d/(a+a*sin(d*x+c))^6+1/5/a^3/d/(a+a*sin(d*x+c))^5
Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.96 \[ \int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {-2+3 \sin (c+d x)}{15 a^8 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^{12}} \] Input:
Integrate[Cos[c + d*x]^3/(a + a*Sin[c + d*x])^8,x]
Output:
(-2 + 3*Sin[c + d*x])/(15*a^8*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^12)
Time = 0.24 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3042, 3146, 53, 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.
\(\displaystyle \int \frac {\cos ^3(c+d x)}{(a \sin (c+d x)+a)^8} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^3}{(a \sin (c+d x)+a)^8}dx\) |
\(\Big \downarrow \) 3146 |
\(\displaystyle \frac {\int \frac {a-a \sin (c+d x)}{(\sin (c+d x) a+a)^7}d(a \sin (c+d x))}{a^3 d}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {\int \left (\frac {2 a}{(\sin (c+d x) a+a)^7}-\frac {1}{(\sin (c+d x) a+a)^6}\right )d(a \sin (c+d x))}{a^3 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\frac {1}{5 (a \sin (c+d x)+a)^5}-\frac {a}{3 (a \sin (c+d x)+a)^6}}{a^3 d}\) |
Input:
Int[Cos[c + d*x]^3/(a + a*Sin[c + d*x])^8,x]
Output:
(-1/3*a/(a + a*Sin[c + d*x])^6 + 1/(5*(a + a*Sin[c + d*x])^5))/(a^3*d)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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 + (p - 1)/2)*(a - x )^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && I ntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && (GeQ[p, -1] || !IntegerQ[m + 1/ 2])
Time = 0.95 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.73
method | result | size |
derivativedivides | \(\frac {\frac {1}{5 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{6}}}{d \,a^{8}}\) | \(33\) |
default | \(\frac {\frac {1}{5 \left (1+\sin \left (d x +c \right )\right )^{5}}-\frac {1}{3 \left (1+\sin \left (d x +c \right )\right )^{6}}}{d \,a^{8}}\) | \(33\) |
risch | \(\frac {32 i \left (-4 i {\mathrm e}^{6 i \left (d x +c \right )}+3 \,{\mathrm e}^{7 i \left (d x +c \right )}-3 \,{\mathrm e}^{5 i \left (d x +c \right )}\right )}{15 d \,a^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{12}}\) | \(59\) |
parallelrisch | \(\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (15 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+60 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+235 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+480 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+822 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+904 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+822 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+480 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}+235 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+60 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+15\right )}{15 d \,a^{8} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{12}}\) | \(161\) |
Input:
int(cos(d*x+c)^3/(a+a*sin(d*x+c))^8,x,method=_RETURNVERBOSE)
Output:
1/d/a^8*(1/5/(1+sin(d*x+c))^5-1/3/(1+sin(d*x+c))^6)
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (41) = 82\).
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.33 \[ \int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=-\frac {3 \, \sin \left (d x + c\right ) - 2}{15 \, {\left (a^{8} d \cos \left (d x + c\right )^{6} - 18 \, a^{8} d \cos \left (d x + c\right )^{4} + 48 \, a^{8} d \cos \left (d x + c\right )^{2} - 32 \, a^{8} d - 2 \, {\left (3 \, a^{8} d \cos \left (d x + c\right )^{4} - 16 \, a^{8} d \cos \left (d x + c\right )^{2} + 16 \, a^{8} d\right )} \sin \left (d x + c\right )\right )}} \] Input:
integrate(cos(d*x+c)^3/(a+a*sin(d*x+c))^8,x, algorithm="fricas")
Output:
-1/15*(3*sin(d*x + c) - 2)/(a^8*d*cos(d*x + c)^6 - 18*a^8*d*cos(d*x + c)^4 + 48*a^8*d*cos(d*x + c)^2 - 32*a^8*d - 2*(3*a^8*d*cos(d*x + c)^4 - 16*a^8 *d*cos(d*x + c)^2 + 16*a^8*d)*sin(d*x + c))
Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (39) = 78\).
Time = 9.28 (sec) , antiderivative size = 493, normalized size of antiderivative = 10.96 \[ \int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\begin {cases} \frac {6 \sin ^{2}{\left (c + d x \right )}}{105 a^{8} d \sin ^{7}{\left (c + d x \right )} + 735 a^{8} d \sin ^{6}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{5}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{4}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{3}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{2}{\left (c + d x \right )} + 735 a^{8} d \sin {\left (c + d x \right )} + 105 a^{8} d} + \frac {7 \sin {\left (c + d x \right )}}{105 a^{8} d \sin ^{7}{\left (c + d x \right )} + 735 a^{8} d \sin ^{6}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{5}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{4}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{3}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{2}{\left (c + d x \right )} + 735 a^{8} d \sin {\left (c + d x \right )} + 105 a^{8} d} - \frac {15 \cos ^{2}{\left (c + d x \right )}}{105 a^{8} d \sin ^{7}{\left (c + d x \right )} + 735 a^{8} d \sin ^{6}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{5}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{4}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{3}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{2}{\left (c + d x \right )} + 735 a^{8} d \sin {\left (c + d x \right )} + 105 a^{8} d} + \frac {1}{105 a^{8} d \sin ^{7}{\left (c + d x \right )} + 735 a^{8} d \sin ^{6}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{5}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{4}{\left (c + d x \right )} + 3675 a^{8} d \sin ^{3}{\left (c + d x \right )} + 2205 a^{8} d \sin ^{2}{\left (c + d x \right )} + 735 a^{8} d \sin {\left (c + d x \right )} + 105 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \cos ^{3}{\left (c \right )}}{\left (a \sin {\left (c \right )} + a\right )^{8}} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**3/(a+a*sin(d*x+c))**8,x)
Output:
Piecewise((6*sin(c + d*x)**2/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d*sin( c + d*x)**6 + 2205*a**8*d*sin(c + d*x)**5 + 3675*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 + 2205*a**8*d*sin(c + d*x)**2 + 735*a**8*d*sin (c + d*x) + 105*a**8*d) + 7*sin(c + d*x)/(105*a**8*d*sin(c + d*x)**7 + 735 *a**8*d*sin(c + d*x)**6 + 2205*a**8*d*sin(c + d*x)**5 + 3675*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 + 2205*a**8*d*sin(c + d*x)**2 + 73 5*a**8*d*sin(c + d*x) + 105*a**8*d) - 15*cos(c + d*x)**2/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d*sin(c + d*x)**6 + 2205*a**8*d*sin(c + d*x)**5 + 36 75*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 + 2205*a**8*d*sin( c + d*x)**2 + 735*a**8*d*sin(c + d*x) + 105*a**8*d) + 1/(105*a**8*d*sin(c + d*x)**7 + 735*a**8*d*sin(c + d*x)**6 + 2205*a**8*d*sin(c + d*x)**5 + 367 5*a**8*d*sin(c + d*x)**4 + 3675*a**8*d*sin(c + d*x)**3 + 2205*a**8*d*sin(c + d*x)**2 + 735*a**8*d*sin(c + d*x) + 105*a**8*d), Ne(d, 0)), (x*cos(c)** 3/(a*sin(c) + a)**8, True))
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (41) = 82\).
Time = 0.03 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.13 \[ \int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {3 \, \sin \left (d x + c\right ) - 2}{15 \, {\left (a^{8} \sin \left (d x + c\right )^{6} + 6 \, a^{8} \sin \left (d x + c\right )^{5} + 15 \, a^{8} \sin \left (d x + c\right )^{4} + 20 \, a^{8} \sin \left (d x + c\right )^{3} + 15 \, a^{8} \sin \left (d x + c\right )^{2} + 6 \, a^{8} \sin \left (d x + c\right ) + a^{8}\right )} d} \] Input:
integrate(cos(d*x+c)^3/(a+a*sin(d*x+c))^8,x, algorithm="maxima")
Output:
1/15*(3*sin(d*x + c) - 2)/((a^8*sin(d*x + c)^6 + 6*a^8*sin(d*x + c)^5 + 15 *a^8*sin(d*x + c)^4 + 20*a^8*sin(d*x + c)^3 + 15*a^8*sin(d*x + c)^2 + 6*a^ 8*sin(d*x + c) + a^8)*d)
Time = 0.13 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.62 \[ \int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {3 \, \sin \left (d x + c\right ) - 2}{15 \, a^{8} d {\left (\sin \left (d x + c\right ) + 1\right )}^{6}} \] Input:
integrate(cos(d*x+c)^3/(a+a*sin(d*x+c))^8,x, algorithm="giac")
Output:
1/15*(3*sin(d*x + c) - 2)/(a^8*d*(sin(d*x + c) + 1)^6)
Time = 25.55 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.62 \[ \int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {3\,\sin \left (c+d\,x\right )-2}{15\,a^8\,d\,{\left (\sin \left (c+d\,x\right )+1\right )}^6} \] Input:
int(cos(c + d*x)^3/(a + a*sin(c + d*x))^8,x)
Output:
(3*sin(c + d*x) - 2)/(15*a^8*d*(sin(c + d*x) + 1)^6)
Time = 0.17 (sec) , antiderivative size = 108, normalized size of antiderivative = 2.40 \[ \int \frac {\cos ^3(c+d x)}{(a+a \sin (c+d x))^8} \, dx=\frac {-15 \cos \left (d x +c \right )^{2}+6 \sin \left (d x +c \right )^{2}+7 \sin \left (d x +c \right )+1}{105 a^{8} d \left (\sin \left (d x +c \right )^{7}+7 \sin \left (d x +c \right )^{6}+21 \sin \left (d x +c \right )^{5}+35 \sin \left (d x +c \right )^{4}+35 \sin \left (d x +c \right )^{3}+21 \sin \left (d x +c \right )^{2}+7 \sin \left (d x +c \right )+1\right )} \] Input:
int(cos(d*x+c)^3/(a+a*sin(d*x+c))^8,x)
Output:
( - 15*cos(c + d*x)**2 + 6*sin(c + d*x)**2 + 7*sin(c + d*x) + 1)/(105*a**8 *d*(sin(c + d*x)**7 + 7*sin(c + d*x)**6 + 21*sin(c + d*x)**5 + 35*sin(c + d*x)**4 + 35*sin(c + d*x)**3 + 21*sin(c + d*x)**2 + 7*sin(c + d*x) + 1))