\(\int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx\) [703]

Optimal result

Integrand size = 29, antiderivative size = 109 \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=-\frac {7 \sin ^{1+n}(c+d x)}{a^4 d (1+n)}+\frac {8 \operatorname {Hypergeometric2F1}(1,1+n,2+n,-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^4 d (1+n)}+\frac {4 \sin ^{2+n}(c+d x)}{a^4 d (2+n)}-\frac {\sin ^{3+n}(c+d x)}{a^4 d (3+n)} \]

[Out]

-7*sin(d*x+c)^(1+n)/a^4/d/(1+n)+8*hypergeom([1, 1+n],[2+n],-sin(d*x+c))*sin(d*x+c)^(1+n)/a^4/d/(1+n)+4*sin(d*x
+c)^(2+n)/a^4/d/(2+n)-sin(d*x+c)^(3+n)/a^4/d/(3+n)

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2915, 90, 45, 66} \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {8 \sin ^{n+1}(c+d x) \operatorname {Hypergeometric2F1}(1,n+1,n+2,-\sin (c+d x))}{a^4 d (n+1)}-\frac {7 \sin ^{n+1}(c+d x)}{a^4 d (n+1)}+\frac {4 \sin ^{n+2}(c+d x)}{a^4 d (n+2)}-\frac {\sin ^{n+3}(c+d x)}{a^4 d (n+3)} \]

[In]

Int[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^4,x]

[Out]

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

Rule 45

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] && NeQ[b*c - a*d, 0] && 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])

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 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

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} \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = \frac {\text {Subst}\left (\int \left (-4 a^2 \left (\frac {x}{a}\right )^n-2 a (a-x) \left (\frac {x}{a}\right )^n-(a-x)^2 \left (\frac {x}{a}\right )^n+\frac {8 a^3 \left (\frac {x}{a}\right )^n}{a+x}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d} \\ & = -\frac {4 \sin ^{1+n}(c+d x)}{a^4 d (1+n)}-\frac {\text {Subst}\left (\int (a-x)^2 \left (\frac {x}{a}\right )^n \, dx,x,a \sin (c+d x)\right )}{a^7 d}-\frac {2 \text {Subst}\left (\int (a-x) \left (\frac {x}{a}\right )^n \, dx,x,a \sin (c+d x)\right )}{a^6 d}+\frac {8 \text {Subst}\left (\int \frac {\left (\frac {x}{a}\right )^n}{a+x} \, dx,x,a \sin (c+d x)\right )}{a^4 d} \\ & = -\frac {4 \sin ^{1+n}(c+d x)}{a^4 d (1+n)}+\frac {8 \operatorname {Hypergeometric2F1}(1,1+n,2+n,-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^4 d (1+n)}-\frac {\text {Subst}\left (\int \left (a^2 \left (\frac {x}{a}\right )^n-2 a^2 \left (\frac {x}{a}\right )^{1+n}+a^2 \left (\frac {x}{a}\right )^{2+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}-\frac {2 \text {Subst}\left (\int \left (a \left (\frac {x}{a}\right )^n-a \left (\frac {x}{a}\right )^{1+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a^6 d} \\ & = -\frac {7 \sin ^{1+n}(c+d x)}{a^4 d (1+n)}+\frac {8 \operatorname {Hypergeometric2F1}(1,1+n,2+n,-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^4 d (1+n)}+\frac {4 \sin ^{2+n}(c+d x)}{a^4 d (2+n)}-\frac {\sin ^{3+n}(c+d x)}{a^4 d (3+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\frac {-\frac {7 a^3 \sin ^{1+n}(c+d x)}{1+n}+\frac {8 a^3 \operatorname {Hypergeometric2F1}(1,1+n,2+n,-\sin (c+d x)) \sin ^{1+n}(c+d x)}{1+n}+\frac {4 a^3 \sin ^{2+n}(c+d x)}{2+n}-\frac {a^3 \sin ^{3+n}(c+d x)}{3+n}}{a^7 d} \]

[In]

Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^4,x]

[Out]

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

Maple [F]

\[\int \frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{\left (a +a \sin \left (d x +c \right )\right )^{4}}d x\]

[In]

int(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^4,x)

[Out]

int(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^4,x)

Fricas [F]

\[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

integrate(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

integral(sin(d*x + c)^n*cos(d*x + c)^7/(a^4*cos(d*x + c)^4 - 8*a^4*cos(d*x + c)^2 + 8*a^4 - 4*(a^4*cos(d*x + c
)^2 - 2*a^4)*sin(d*x + c)), x)

Sympy [F(-2)]

Exception generated. \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate(cos(d*x+c)**7*sin(d*x+c)**n/(a+a*sin(d*x+c))**4,x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F]

\[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

integrate(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

integrate(sin(d*x + c)^n*cos(d*x + c)^7/(a*sin(d*x + c) + a)^4, x)

Giac [F]

\[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\int { \frac {\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{4}} \,d x } \]

[In]

integrate(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

integrate(sin(d*x + c)^n*cos(d*x + c)^7/(a*sin(d*x + c) + a)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^7\,{\sin \left (c+d\,x\right )}^n}{{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4} \,d x \]

[In]

int((cos(c + d*x)^7*sin(c + d*x)^n)/(a + a*sin(c + d*x))^4,x)

[Out]

int((cos(c + d*x)^7*sin(c + d*x)^n)/(a + a*sin(c + d*x))^4, x)