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

Optimal. Leaf size=137 \[ \frac{\sin ^{n+1}(c+d x)}{a d (n+1)}-\frac{\sin ^{n+2}(c+d x)}{a d (n+2)}-\frac{2 \sin ^{n+3}(c+d x)}{a d (n+3)}+\frac{2 \sin ^{n+4}(c+d x)}{a d (n+4)}+\frac{\sin ^{n+5}(c+d x)}{a d (n+5)}-\frac{\sin ^{n+6}(c+d x)}{a d (n+6)} \]

[Out]

Sin[c + d*x]^(1 + n)/(a*d*(1 + n)) - Sin[c + d*x]^(2 + n)/(a*d*(2 + n)) - (2*Sin[c + d*x]^(3 + n))/(a*d*(3 + n
)) + (2*Sin[c + d*x]^(4 + n))/(a*d*(4 + n)) + Sin[c + d*x]^(5 + n)/(a*d*(5 + n)) - Sin[c + d*x]^(6 + n)/(a*d*(
6 + n))

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Rubi [A]  time = 0.164255, antiderivative size = 137, 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, 88} \[ \frac{\sin ^{n+1}(c+d x)}{a d (n+1)}-\frac{\sin ^{n+2}(c+d x)}{a d (n+2)}-\frac{2 \sin ^{n+3}(c+d x)}{a d (n+3)}+\frac{2 \sin ^{n+4}(c+d x)}{a d (n+4)}+\frac{\sin ^{n+5}(c+d x)}{a d (n+5)}-\frac{\sin ^{n+6}(c+d x)}{a d (n+6)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sin[c + d*x]^(1 + n)/(a*d*(1 + n)) - Sin[c + d*x]^(2 + n)/(a*d*(2 + n)) - (2*Sin[c + d*x]^(3 + n))/(a*d*(3 + n
)) + (2*Sin[c + d*x]^(4 + n))/(a*d*(4 + n)) + Sin[c + d*x]^(5 + n)/(a*d*(5 + n)) - Sin[c + d*x]^(6 + n)/(a*d*(
6 + 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]))

Rubi steps

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

Mathematica [A]  time = 0.307137, size = 95, normalized size = 0.69 \[ \frac{\sin ^{n+1}(c+d x) \left (-\frac{\sin ^5(c+d x)}{n+6}+\frac{\sin ^4(c+d x)}{n+5}+\frac{2 \sin ^3(c+d x)}{n+4}-\frac{2 \sin ^2(c+d x)}{n+3}-\frac{\sin (c+d x)}{n+2}+\frac{1}{n+1}\right )}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sin[c + d*x]^(1 + n)*((1 + n)^(-1) - Sin[c + d*x]/(2 + n) - (2*Sin[c + d*x]^2)/(3 + n) + (2*Sin[c + d*x]^3)/(
4 + n) + Sin[c + d*x]^4/(5 + n) - Sin[c + d*x]^5/(6 + n)))/(a*d)

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Maple [F]  time = 3.677, 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}}{a+a\sin \left ( dx+c \right ) }}\, 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)),x)

[Out]

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

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Maxima [A]  time = 1.20259, size = 325, normalized size = 2.37 \begin{align*} -\frac{{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} \sin \left (d x + c\right )^{6} -{\left (n^{5} + 16 \, n^{4} + 95 \, n^{3} + 260 \, n^{2} + 324 \, n + 144\right )} \sin \left (d x + c\right )^{5} - 2 \,{\left (n^{5} + 17 \, n^{4} + 107 \, n^{3} + 307 \, n^{2} + 396 \, n + 180\right )} \sin \left (d x + c\right )^{4} + 2 \,{\left (n^{5} + 18 \, n^{4} + 121 \, n^{3} + 372 \, n^{2} + 508 \, n + 240\right )} \sin \left (d x + c\right )^{3} +{\left (n^{5} + 19 \, n^{4} + 137 \, n^{3} + 461 \, n^{2} + 702 \, n + 360\right )} \sin \left (d x + c\right )^{2} -{\left (n^{5} + 20 \, n^{4} + 155 \, n^{3} + 580 \, n^{2} + 1044 \, n + 720\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} a 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)),x, algorithm="maxima")

[Out]

-((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*sin(d*x + c)^6 - (n^5 + 16*n^4 + 95*n^3 + 260*n^2 + 324*n +
144)*sin(d*x + c)^5 - 2*(n^5 + 17*n^4 + 107*n^3 + 307*n^2 + 396*n + 180)*sin(d*x + c)^4 + 2*(n^5 + 18*n^4 + 12
1*n^3 + 372*n^2 + 508*n + 240)*sin(d*x + c)^3 + (n^5 + 19*n^4 + 137*n^3 + 461*n^2 + 702*n + 360)*sin(d*x + c)^
2 - (n^5 + 20*n^4 + 155*n^3 + 580*n^2 + 1044*n + 720)*sin(d*x + c))*sin(d*x + c)^n/((n^6 + 21*n^5 + 175*n^4 +
735*n^3 + 1624*n^2 + 1764*n + 720)*a*d)

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Fricas [A]  time = 1.29196, size = 639, normalized size = 4.66 \begin{align*} \frac{{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} \cos \left (d x + c\right )^{6} -{\left (n^{5} + 11 \, n^{4} + 41 \, n^{3} + 61 \, n^{2} + 30 \, n\right )} \cos \left (d x + c\right )^{4} - 8 \, n^{3} - 4 \,{\left (n^{4} + 9 \, n^{3} + 23 \, n^{2} + 15 \, n\right )} \cos \left (d x + c\right )^{2} - 72 \, n^{2} +{\left ({\left (n^{5} + 16 \, n^{4} + 95 \, n^{3} + 260 \, n^{2} + 324 \, n + 144\right )} \cos \left (d x + c\right )^{4} + 8 \, n^{3} + 4 \,{\left (n^{4} + 13 \, n^{3} + 56 \, n^{2} + 92 \, n + 48\right )} \cos \left (d x + c\right )^{2} + 96 \, n^{2} + 352 \, n + 384\right )} \sin \left (d x + c\right ) - 184 \, n - 120\right )} \sin \left (d x + c\right )^{n}}{a d n^{6} + 21 \, a d n^{5} + 175 \, a d n^{4} + 735 \, a d n^{3} + 1624 \, a d n^{2} + 1764 \, a d n + 720 \, a 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)),x, algorithm="fricas")

[Out]

((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*cos(d*x + c)^6 - (n^5 + 11*n^4 + 41*n^3 + 61*n^2 + 30*n)*cos(
d*x + c)^4 - 8*n^3 - 4*(n^4 + 9*n^3 + 23*n^2 + 15*n)*cos(d*x + c)^2 - 72*n^2 + ((n^5 + 16*n^4 + 95*n^3 + 260*n
^2 + 324*n + 144)*cos(d*x + c)^4 + 8*n^3 + 4*(n^4 + 13*n^3 + 56*n^2 + 92*n + 48)*cos(d*x + c)^2 + 96*n^2 + 352
*n + 384)*sin(d*x + c) - 184*n - 120)*sin(d*x + c)^n/(a*d*n^6 + 21*a*d*n^5 + 175*a*d*n^4 + 735*a*d*n^3 + 1624*
a*d*n^2 + 1764*a*d*n + 720*a*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)),x)

[Out]

Timed out

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Giac [A]  time = 1.52594, size = 188, normalized size = 1.37 \begin{align*} -\frac{\frac{\sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6}}{n + 6} - \frac{\sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5}}{n + 5} - \frac{2 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4}}{n + 4} + \frac{2 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3}}{n + 3} + \frac{\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 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)),x, algorithm="giac")

[Out]

-(sin(d*x + c)^n*sin(d*x + c)^6/(n + 6) - sin(d*x + c)^n*sin(d*x + c)^5/(n + 5) - 2*sin(d*x + c)^n*sin(d*x + c
)^4/(n + 4) + 2*sin(d*x + c)^n*sin(d*x + c)^3/(n + 3) + sin(d*x + c)^n*sin(d*x + c)^2/(n + 2) - sin(d*x + c)^(
n + 1)/(n + 1))/(a*d)