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

Optimal. Leaf size=109 \[ \frac{8 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^4 d (n+1)}-\frac{7 \sin ^{n+1}(c+d x)}{a^4 d (n+1)}+\frac{4 \sin ^{n+2}(c+d x)}{a^4 d (n+2)}-\frac{\sin ^{n+3}(c+d x)}{a^4 d (n+3)} \]

[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))

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Rubi [A]  time = 0.174651, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {2836, 88, 43, 64} \[ \frac{8 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{a^4 d (n+1)}-\frac{7 \sin ^{n+1}(c+d x)}{a^4 d (n+1)}+\frac{4 \sin ^{n+2}(c+d x)}{a^4 d (n+2)}-\frac{\sin ^{n+3}(c+d x)}{a^4 d (n+3)} \]

Antiderivative was successfully verified.

[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 2836

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*x)/b
)^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]

Rule 88

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 43

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 64

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c^n*(b*x)^(m + 1)*Hypergeometric2F1[-n, m +
1, m + 2, -((d*x)/c)])/(b*(m + 1)), 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])))

Rubi steps

\begin{align*} \int \frac{\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^4} \, dx &=\frac{\operatorname{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{\operatorname{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{\operatorname{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 \operatorname{Subst}\left (\int (a-x) \left (\frac{x}{a}\right )^n \, dx,x,a \sin (c+d x)\right )}{a^6 d}+\frac{8 \operatorname{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 \, _2F_1(1,1+n;2+n;-\sin (c+d x)) \sin ^{1+n}(c+d x)}{a^4 d (1+n)}-\frac{\operatorname{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 \operatorname{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 \, _2F_1(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]  time = 0.254182, size = 104, normalized size = 0.95 \[ \frac{\frac{8 a^3 \sin ^{n+1}(c+d x) \, _2F_1(1,n+1;n+2;-\sin (c+d x))}{n+1}-\frac{7 a^3 \sin ^{n+1}(c+d x)}{n+1}+\frac{4 a^3 \sin ^{n+2}(c+d x)}{n+2}-\frac{a^3 \sin ^{n+3}(c+d x)}{n+3}}{a^7 d} \]

Antiderivative was successfully verified.

[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)

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Maple [F]  time = 1.497, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{7} \left ( \sin \left ( dx+c \right ) \right ) ^{n}}{ \left ( a+a\sin \left ( dx+c \right ) \right ) ^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sin \left (d x + c\right )^{n} \cos \left (d x + c\right )^{7}}{a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \,{\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)