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

Optimal result

Integrand size = 29, antiderivative size = 160 \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=-\frac {4 (3+2 n) \operatorname {Hypergeometric2F1}(1,1+n,2+n,-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^5 d (1+n)}-\frac {\sin ^{1+n}(c+d x) (a-a \sin (c+d x))^2}{d (2+n) \left (a^7+a^7 \sin (c+d x)\right )}+\frac {\sin ^{1+n}(c+d x) \left (a \left (27+30 n+8 n^2\right )+a (7+2 n) \sin (c+d x)\right )}{d \left (2+3 n+n^2\right ) \left (a^6+a^6 \sin (c+d x)\right )} \]

[Out]

-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))

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2915, 102, 151, 66} \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=-\frac {(a-a \sin (c+d x))^2 \sin ^{n+1}(c+d x)}{d (n+2) \left (a^7 \sin (c+d x)+a^7\right )}+\frac {\sin ^{n+1}(c+d x) \left (a (2 n+7) \sin (c+d x)+a \left (8 n^2+30 n+27\right )\right )}{d \left (n^2+3 n+2\right ) \left (a^6 \sin (c+d x)+a^6\right )}-\frac {4 (2 n+3) \sin ^{n+1}(c+d x) \operatorname {Hypergeometric2F1}(1,n+1,n+2,-\sin (c+d x))}{a^5 d (n+1)} \]

[In]

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

[Out]

(-4*(3 + 2*n)*Hypergeometric2F1[1, 1 + n, 2 + n, -Sin[c + d*x]]*Sin[c + d*x]^(1 + n))/(a^5*d*(1 + n)) - (Sin[c
+ d*x]^(1 + n)*(a - a*Sin[c + d*x])^2)/(d*(2 + n)*(a^7 + a^7*Sin[c + d*x])) + (Sin[c + d*x]^(1 + n)*(a*(27 +
30*n + 8*n^2) + a*(7 + 2*n)*Sin[c + d*x]))/(d*(2 + 3*n + n^2)*(a^6 + a^6*Sin[c + d*x]))

Rule 66

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x)^(m + 1)/(b*(m + 1)))*Hypergeometr
ic2F1[-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 102

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(a +
b*x)^(m - 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(m + n + p + 1))), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(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 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol] :
> 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] - Dist[
(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] && ((Ge
Q[m, -2] && LtQ[m, -1]) || SumSimplerQ[m, 1]) && NeQ[m, -1] && NeQ[m + n + 3, 0]

Rule 2915

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[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] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2,
0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a-x)^3 \left (\frac {x}{a}\right )^n}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = -\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 {\text {Subst}\left (\int \frac {(a-x) \left (\frac {x}{a}\right )^n (a (3+2 n)+(-7-2 n) x)}{(a+x)^2} \, dx,x,a \sin (c+d x)\right )}{a^6 d (2+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 (1+n) (2+n) \left (a^6+a^6 \sin (c+d x)\right )}-\frac {(4 (3+2 n)) \text {Subst}\left (\int \frac {\left (\frac {x}{a}\right )^n}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^5 d} \\ & = -\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 (1+n) (2+n) \left (a^6+a^6 \sin (c+d x)\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^5} \, dx=\frac {\sin ^{1+n}(c+d x) \left (26+29 n+8 n^2+(9+4 n) \sin (c+d x)-(1+n) \sin ^2(c+d x)-4 \left (6+7 n+2 n^2\right ) \operatorname {Hypergeometric2F1}(1,1+n,2+n,-\sin (c+d x)) (1+\sin (c+d x))\right )}{a^5 d (1+n) (2+n) (1+\sin (c+d x))} \]

[In]

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

[Out]

(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]))

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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