\(\int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx\) [704]

Optimal result

Integrand size = 29, antiderivative size = 164 \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\frac {(7+2 n) \sin ^{1+n}(c+d x)}{a^5 d (1+n) (2+n)}-\frac {4 (3+2 n) \operatorname {Hypergeometric2F1}(1,1+n,2+n,-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^5 d (1+n)}-\frac {\sin ^{1+n}(c+d x) (a-a \sin (c+d x))^2}{a^7 d (2+n) (1+\sin (c+d x))}+\frac {4 (5+2 n) \sin ^{1+n}(c+d x)}{d (2+n) \left (a^5+a^5 \sin (c+d x)\right )} \] Output:

(7+2*n)*sin(d*x+c)^(1+n)/a^5/d/(1+n)/(2+n)-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/a^7/d/(2+n)/(1+sin(d*x+c))+4*(5+2*n)*sin(d*x+c)^(1+n)/d/(2+n)/(a^5 
+a^5*sin(d*x+c))
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.66 \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\frac {\sin ^{1+n}(c+d x) \left (26+29 n+8 n^2+(9+4 n) \sin (c+d x)-(1+n) \sin ^2(c+d x)-4 \left (6+7 n+2 n^2\right ) \operatorname {Hypergeometric2F1}(1,1+n,2+n,-\sin (c+d x)) (1+\sin (c+d x))\right )}{a^5 d (1+n) (2+n) (1+\sin (c+d x))} \] Input:

Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^5,x]
 

Output:

(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]))
 

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.95, 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

\(\displaystyle \frac {\frac {a \left (\frac {a \sin ^{n+1}(c+d x) \left (a (2 n+7) \sin (c+d x)+a \left (8 n^2+30 n+27\right )\right )}{(n+1) (a \sin (c+d x)+a)}-\frac {4 a (n+2) (2 n+3) \sin ^{n+1}(c+d x) \operatorname {Hypergeometric2F1}(1,n+1,n+2,-\sin (c+d x))}{n+1}\right )}{n+2}-\frac {a (a-a \sin (c+d x))^2 \sin ^{n+1}(c+d x)}{(n+2) (a \sin (c+d x)+a)}}{a^7 d}\)

Input:

Int[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^5,x]
 

Output:

(-((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)
 

Defintions of rubi rules used

rule 74

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])))
 

rule 111

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]
 

rule 163

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3315

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]
 

Maple [F]

\[\int \frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{n}}{\left (a +a \sin \left (d x +c \right )\right )^{5}}d x\]

Input:

int(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^5,x)
 

Output:

int(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^5,x)
 

Fricas [F]

\[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{5}} \,d x } \] Input:

integrate(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^5,x, algorithm="frica 
s")
 

Output:

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)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**7*sin(d*x+c)**n/(a+a*sin(d*x+c))**5,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{5}} \,d x } \] Input:

integrate(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^5,x, algorithm="maxim 
a")
 

Output:

integrate(sin(d*x + c)^n*cos(d*x + c)^7/(a*sin(d*x + c) + a)^5, x)
 

Giac [F]

\[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{5}} \,d x } \] Input:

integrate(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^5,x, algorithm="giac" 
)
 

Output:

integrate(sin(d*x + c)^n*cos(d*x + c)^7/(a*sin(d*x + c) + a)^5, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\text {Hanged} \] Input:

int((cos(c + d*x)^7*sin(c + d*x)^n)/(a + a*sin(c + d*x))^5,x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\int \frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{n}}{\left (\sin \left (d x +c \right ) a +a \right )^{5}}d x \] Input:

int(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^5,x)
 

Output:

int(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^5,x)