Integrand size = 29, antiderivative size = 160 \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=-\frac {4 (3+2 n) \operatorname {Hypergeometric2F1}(1,1+n,2+n,-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^5 d (1+n)}-\frac {\sin ^{1+n}(c+d x) (a-a \sin (c+d x))^2}{d (2+n) \left (a^7+a^7 \sin (c+d x)\right )}+\frac {\sin ^{1+n}(c+d x) \left (a \left (27+30 n+8 n^2\right )+a (7+2 n) \sin (c+d x)\right )}{d \left (2+3 n+n^2\right ) \left (a^6+a^6 \sin (c+d x)\right )} \]
-4*(3+2*n)*hypergeom([1, 1+n],[2+n],-sin(d*x+c))*sin(d*x+c)^(1+n)/a^5/d/(1 +n)-sin(d*x+c)^(1+n)*(a-a*sin(d*x+c))^2/d/(2+n)/(a^7+a^7*sin(d*x+c))+sin(d *x+c)^(1+n)*(a*(8*n^2+30*n+27)+a*(7+2*n)*sin(d*x+c))/d/(n^2+3*n+2)/(a^6+a^ 6*sin(d*x+c))
Time = 0.15 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\frac {\sin ^{1+n}(c+d x) \left (26+29 n+8 n^2+(9+4 n) \sin (c+d x)-(1+n) \sin ^2(c+d x)-4 \left (6+7 n+2 n^2\right ) \operatorname {Hypergeometric2F1}(1,1+n,2+n,-\sin (c+d x)) (1+\sin (c+d x))\right )}{a^5 d (1+n) (2+n) (1+\sin (c+d x))} \]
(Sin[c + d*x]^(1 + n)*(26 + 29*n + 8*n^2 + (9 + 4*n)*Sin[c + d*x] - (1 + n )*Sin[c + d*x]^2 - 4*(6 + 7*n + 2*n^2)*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*(1 + Sin[c + d*x])))/(a^5*d*(1 + n)*(2 + n)*(1 + Sin[c + d* x]))
Time = 0.37 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.98, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {3042, 3315, 111, 163, 74}
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.
\(\displaystyle \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a \sin (c+d x)+a)^5} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos (c+d x)^7 \sin (c+d x)^n}{(a \sin (c+d x)+a)^5}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle \frac {\int \frac {\sin ^n(c+d x) (a-a \sin (c+d x))^3}{(\sin (c+d x) a+a)^2}d(a \sin (c+d x))}{a^7 d}\) |
\(\Big \downarrow \) 111 |
\(\displaystyle \frac {\frac {a \int \frac {\sin ^n(c+d x) (a-a \sin (c+d x)) (a (2 n+3)-a (2 n+7) \sin (c+d x))}{(\sin (c+d x) a+a)^2}d(a \sin (c+d x))}{n+2}-\frac {a (a-a \sin (c+d x))^2 \sin ^{n+1}(c+d x)}{(n+2) (a \sin (c+d x)+a)}}{a^7 d}\) |
\(\Big \downarrow \) 163 |
\(\displaystyle \frac {\frac {a \left (\frac {a \sin ^{n+1}(c+d x) \left (a (2 n+7) \sin (c+d x)+a \left (8 n^2+30 n+27\right )\right )}{(n+1) (a \sin (c+d x)+a)}-4 a (n+2) (2 n+3) \int \frac {\sin ^n(c+d x)}{\sin (c+d x) a+a}d(a \sin (c+d x))\right )}{n+2}-\frac {a (a-a \sin (c+d x))^2 \sin ^{n+1}(c+d x)}{(n+2) (a \sin (c+d x)+a)}}{a^7 d}\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {\frac {a \left (\frac {a \sin ^{n+1}(c+d x) \left (a (2 n+7) \sin (c+d x)+a \left (8 n^2+30 n+27\right )\right )}{(n+1) (a \sin (c+d x)+a)}-\frac {4 a (n+2) (2 n+3) \sin ^{n+1}(c+d x) \operatorname {Hypergeometric2F1}(1,n+1,n+2,-\sin (c+d x))}{n+1}\right )}{n+2}-\frac {a (a-a \sin (c+d x))^2 \sin ^{n+1}(c+d x)}{(n+2) (a \sin (c+d x)+a)}}{a^7 d}\) |
(-((a*Sin[c + d*x]^(1 + n)*(a - a*Sin[c + d*x])^2)/((2 + n)*(a + a*Sin[c + d*x]))) + (a*((-4*a*(2 + n)*(3 + 2*n)*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(1 + n) + (a*Sin[c + d*x]^(1 + n)*(a* (27 + 30*n + 8*n^2) + a*(7 + 2*n)*Sin[c + d*x]))/((1 + n)*(a + a*Sin[c + d *x]))))/(2 + n))/(a^7*d)
3.8.4.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) )*((g_.) + (h_.)*(x_)), x_] :> Simp[((a^2*d*f*h*(n + 2) + b^2*d*e*g*(m + n + 3) + a*b*(c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b*f*h*(b*c - a*d)* (m + 1)*x)/(b^2*d*(b*c - a*d)*(m + 1)*(m + n + 3)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] - Simp[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f *h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c* d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2* d*(b*c - a*d)*(m + 1)*(m + n + 3)) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && ((GeQ[m, -2] && LtQ[m, - 1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 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 + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]
\[\int \frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{\left (a +a \sin \left (d x +c \right )\right )^{5}}d x\]
\[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{5}} \,d x } \]
integral(sin(d*x + c)^n*cos(d*x + c)^7/(5*a^5*cos(d*x + c)^4 - 20*a^5*cos( d*x + c)^2 + 16*a^5 + (a^5*cos(d*x + c)^4 - 12*a^5*cos(d*x + c)^2 + 16*a^5 )*sin(d*x + c)), x)
Timed out. \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\text {Timed out} \]
\[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{5}} \,d x } \]
\[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{5}} \,d x } \]
Timed out. \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^7\,{\sin \left (c+d\,x\right )}^n}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^5} \,d x \]