3.702 \(\int \frac{\cos ^7(c+d x) \sin ^n(c+d x)}{(a+a \sin (c+d x))^3} \, dx\)

Optimal. Leaf size=92 \[ \frac{\sin ^{n+1}(c+d x)}{a^3 d (n+1)}-\frac{3 \sin ^{n+2}(c+d x)}{a^3 d (n+2)}+\frac{3 \sin ^{n+3}(c+d x)}{a^3 d (n+3)}-\frac{\sin ^{n+4}(c+d x)}{a^3 d (n+4)} \]

[Out]

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

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Rubi [A]  time = 0.137075, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2836, 43} \[ \frac{\sin ^{n+1}(c+d x)}{a^3 d (n+1)}-\frac{3 \sin ^{n+2}(c+d x)}{a^3 d (n+2)}+\frac{3 \sin ^{n+3}(c+d x)}{a^3 d (n+3)}-\frac{\sin ^{n+4}(c+d x)}{a^3 d (n+4)} \]

Antiderivative was successfully verified.

[In]

Int[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^3,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.188185, size = 66, normalized size = 0.72 \[ \frac{\sin ^{n+1}(c+d x) \left (-\frac{\sin ^3(c+d x)}{n+4}+\frac{3 \sin ^2(c+d x)}{n+3}-\frac{3 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right )}{a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[(Cos[c + d*x]^7*Sin[c + d*x]^n)/(a + a*Sin[c + d*x])^3,x]

[Out]

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

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Maple [F]  time = 2.116, 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 ) ^{3}}}\, 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))^3,x)

[Out]

int(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c))^3,x)

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Maxima [A]  time = 1.21105, size = 170, normalized size = 1.85 \begin{align*} -\frac{{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} \sin \left (d x + c\right )^{4} - 3 \,{\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} \sin \left (d x + c\right )^{3} + 3 \,{\left (n^{3} + 8 \, n^{2} + 19 \, n + 12\right )} \sin \left (d x + c\right )^{2} -{\left (n^{3} + 9 \, n^{2} + 26 \, n + 24\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} a^{3} d} \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))^3,x, algorithm="maxima")

[Out]

-((n^3 + 6*n^2 + 11*n + 6)*sin(d*x + c)^4 - 3*(n^3 + 7*n^2 + 14*n + 8)*sin(d*x + c)^3 + 3*(n^3 + 8*n^2 + 19*n
+ 12)*sin(d*x + c)^2 - (n^3 + 9*n^2 + 26*n + 24)*sin(d*x + c))*sin(d*x + c)^n/((n^4 + 10*n^3 + 35*n^2 + 50*n +
24)*a^3*d)

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Fricas [A]  time = 1.18212, size = 389, normalized size = 4.23 \begin{align*} -\frac{{\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} \cos \left (d x + c\right )^{4} + 4 \, n^{3} -{\left (5 \, n^{3} + 36 \, n^{2} + 79 \, n + 48\right )} \cos \left (d x + c\right )^{2} + 30 \, n^{2} -{\left (4 \, n^{3} - 3 \,{\left (n^{3} + 7 \, n^{2} + 14 \, n + 8\right )} \cos \left (d x + c\right )^{2} + 30 \, n^{2} + 68 \, n + 48\right )} \sin \left (d x + c\right ) + 68 \, n + 42\right )} \sin \left (d x + c\right )^{n}}{a^{3} d n^{4} + 10 \, a^{3} d n^{3} + 35 \, a^{3} d n^{2} + 50 \, a^{3} d n + 24 \, a^{3} d} \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))^3,x, algorithm="fricas")

[Out]

-((n^3 + 6*n^2 + 11*n + 6)*cos(d*x + c)^4 + 4*n^3 - (5*n^3 + 36*n^2 + 79*n + 48)*cos(d*x + c)^2 + 30*n^2 - (4*
n^3 - 3*(n^3 + 7*n^2 + 14*n + 8)*cos(d*x + c)^2 + 30*n^2 + 68*n + 48)*sin(d*x + c) + 68*n + 42)*sin(d*x + c)^n
/(a^3*d*n^4 + 10*a^3*d*n^3 + 35*a^3*d*n^2 + 50*a^3*d*n + 24*a^3*d)

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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))**3,x)

[Out]

Timed out

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Giac [A]  time = 1.64291, size = 127, normalized size = 1.38 \begin{align*} -\frac{\frac{\sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4}}{n + 4} - \frac{3 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3}}{n + 3} + \frac{3 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2}}{n + 2} - \frac{\sin \left (d x + c\right )^{n + 1}}{n + 1}}{a^{3} d} \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))^3,x, algorithm="giac")

[Out]

-(sin(d*x + c)^n*sin(d*x + c)^4/(n + 4) - 3*sin(d*x + c)^n*sin(d*x + c)^3/(n + 3) + 3*sin(d*x + c)^n*sin(d*x +
c)^2/(n + 2) - sin(d*x + c)^(n + 1)/(n + 1))/(a^3*d)