3.699 \(\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x)) \, dx\)

Optimal. Leaf size=167 \[ \frac{a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{a \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{3 a \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{3 a \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{3 a \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{3 a \sin ^{n+6}(c+d x)}{d (n+6)}-\frac{a \sin ^{n+7}(c+d x)}{d (n+7)}-\frac{a \sin ^{n+8}(c+d x)}{d (n+8)} \]

[Out]

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

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Rubi [A]  time = 0.137666, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2836, 88} \[ \frac{a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{a \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{3 a \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{3 a \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{3 a \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{3 a \sin ^{n+6}(c+d x)}{d (n+6)}-\frac{a \sin ^{n+7}(c+d x)}{d (n+7)}-\frac{a \sin ^{n+8}(c+d x)}{d (n+8)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (a*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (3*a*Sin[c + d*x]^(3 + n))/(d*(3
+ n)) - (3*a*Sin[c + d*x]^(4 + n))/(d*(4 + n)) + (3*a*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (3*a*Sin[c + d*x]^(
6 + n))/(d*(6 + n)) - (a*Sin[c + d*x]^(7 + n))/(d*(7 + n)) - (a*Sin[c + d*x]^(8 + n))/(d*(8 + 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 \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)^4 \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^7 \left (\frac{x}{a}\right )^n+a^7 \left (\frac{x}{a}\right )^{1+n}-3 a^7 \left (\frac{x}{a}\right )^{2+n}-3 a^7 \left (\frac{x}{a}\right )^{3+n}+3 a^7 \left (\frac{x}{a}\right )^{4+n}+3 a^7 \left (\frac{x}{a}\right )^{5+n}-a^7 \left (\frac{x}{a}\right )^{6+n}-a^7 \left (\frac{x}{a}\right )^{7+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{a \sin ^{1+n}(c+d x)}{d (1+n)}+\frac{a \sin ^{2+n}(c+d x)}{d (2+n)}-\frac{3 a \sin ^{3+n}(c+d x)}{d (3+n)}-\frac{3 a \sin ^{4+n}(c+d x)}{d (4+n)}+\frac{3 a \sin ^{5+n}(c+d x)}{d (5+n)}+\frac{3 a \sin ^{6+n}(c+d x)}{d (6+n)}-\frac{a \sin ^{7+n}(c+d x)}{d (7+n)}-\frac{a \sin ^{8+n}(c+d x)}{d (8+n)}\\ \end{align*}

Mathematica [B]  time = 3.15482, size = 659, normalized size = 3.95 \[ \frac{a \sin ^{n+1}(c+d x) \left (5 n^7 \sin (c+d x)+9 n^7 \sin (3 (c+d x))+5 n^7 \sin (5 (c+d x))+n^7 \sin (7 (c+d x))+188 n^6 \sin (c+d x)+324 n^6 \sin (3 (c+d x))+164 n^6 \sin (5 (c+d x))+28 n^6 \sin (7 (c+d x))+3050 n^5 \sin (c+d x)+4866 n^5 \sin (3 (c+d x))+2138 n^5 \sin (5 (c+d x))+322 n^5 \sin (7 (c+d x))+28904 n^4 \sin (c+d x)+38232 n^4 \sin (3 (c+d x))+14360 n^4 \sin (5 (c+d x))+1960 n^4 \sin (7 (c+d x))+167669 n^3 \sin (c+d x)+165273 n^3 \sin (3 (c+d x))+53525 n^3 \sin (5 (c+d x))+6769 n^3 \sin (7 (c+d x))+552236 n^2 \sin (c+d x)+384948 n^2 \sin (3 (c+d x))+110036 n^2 \sin (5 (c+d x))+13132 n^2 \sin (7 (c+d x))+2 n^7 \cos (6 (c+d x))+58 n^6 \cos (6 (c+d x))+686 n^5 \cos (6 (c+d x))+4270 n^4 \cos (6 (c+d x))+15008 n^3 \cos (6 (c+d x))+29512 n^2 \cos (6 (c+d x))+6 \left (5 n^7+177 n^6+2611 n^5+20499 n^4+90640 n^3+219828 n^2+262064 n+114816\right ) \cos (2 (c+d x))+12 \left (n^7+33 n^6+439 n^5+3027 n^4+11584 n^3+24372 n^2+25776 n+10368\right ) \cos (4 (c+d x))+879324 n \sin (c+d x)+439836 n \sin (3 (c+d x))+114252 n \sin (5 (c+d x))+13068 n \sin (7 (c+d x))+29664 n \cos (6 (c+d x))+468720 \sin (c+d x)+186480 \sin (3 (c+d x))+45360 \sin (5 (c+d x))+5040 \sin (7 (c+d x))+11520 \cos (6 (c+d x))+20 n^7+724 n^6+11084 n^5+94012 n^4+481280 n^3+1486096 n^2+2521536 n+1755648\right )}{64 d (n+1) (n+2) (n+3) (n+4) (n+5) (n+6) (n+7) (n+8)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*Sin[c + d*x]^(1 + n)*(1755648 + 2521536*n + 1486096*n^2 + 481280*n^3 + 94012*n^4 + 11084*n^5 + 724*n^6 + 20
*n^7 + 6*(114816 + 262064*n + 219828*n^2 + 90640*n^3 + 20499*n^4 + 2611*n^5 + 177*n^6 + 5*n^7)*Cos[2*(c + d*x)
] + 12*(10368 + 25776*n + 24372*n^2 + 11584*n^3 + 3027*n^4 + 439*n^5 + 33*n^6 + n^7)*Cos[4*(c + d*x)] + 11520*
Cos[6*(c + d*x)] + 29664*n*Cos[6*(c + d*x)] + 29512*n^2*Cos[6*(c + d*x)] + 15008*n^3*Cos[6*(c + d*x)] + 4270*n
^4*Cos[6*(c + d*x)] + 686*n^5*Cos[6*(c + d*x)] + 58*n^6*Cos[6*(c + d*x)] + 2*n^7*Cos[6*(c + d*x)] + 468720*Sin
[c + d*x] + 879324*n*Sin[c + d*x] + 552236*n^2*Sin[c + d*x] + 167669*n^3*Sin[c + d*x] + 28904*n^4*Sin[c + d*x]
+ 3050*n^5*Sin[c + d*x] + 188*n^6*Sin[c + d*x] + 5*n^7*Sin[c + d*x] + 186480*Sin[3*(c + d*x)] + 439836*n*Sin[
3*(c + d*x)] + 384948*n^2*Sin[3*(c + d*x)] + 165273*n^3*Sin[3*(c + d*x)] + 38232*n^4*Sin[3*(c + d*x)] + 4866*n
^5*Sin[3*(c + d*x)] + 324*n^6*Sin[3*(c + d*x)] + 9*n^7*Sin[3*(c + d*x)] + 45360*Sin[5*(c + d*x)] + 114252*n*Si
n[5*(c + d*x)] + 110036*n^2*Sin[5*(c + d*x)] + 53525*n^3*Sin[5*(c + d*x)] + 14360*n^4*Sin[5*(c + d*x)] + 2138*
n^5*Sin[5*(c + d*x)] + 164*n^6*Sin[5*(c + d*x)] + 5*n^7*Sin[5*(c + d*x)] + 5040*Sin[7*(c + d*x)] + 13068*n*Sin
[7*(c + d*x)] + 13132*n^2*Sin[7*(c + d*x)] + 6769*n^3*Sin[7*(c + d*x)] + 1960*n^4*Sin[7*(c + d*x)] + 322*n^5*S
in[7*(c + d*x)] + 28*n^6*Sin[7*(c + d*x)] + n^7*Sin[7*(c + d*x)]))/(64*d*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 +
n)*(6 + n)*(7 + n)*(8 + n))

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Maple [F]  time = 8.074, size = 0, normalized size = 0. \begin{align*} \int \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 ) \, 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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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]

Exception raised: ValueError

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Fricas [B]  time = 1.57113, size = 1219, normalized size = 7.3 \begin{align*} -\frac{{\left ({\left (a n^{7} + 28 \, a n^{6} + 322 \, a n^{5} + 1960 \, a n^{4} + 6769 \, a n^{3} + 13132 \, a n^{2} + 13068 \, a n + 5040 \, a\right )} \cos \left (d x + c\right )^{8} -{\left (a n^{7} + 22 \, a n^{6} + 190 \, a n^{5} + 820 \, a n^{4} + 1849 \, a n^{3} + 2038 \, a n^{2} + 840 \, a n\right )} \cos \left (d x + c\right )^{6} - 48 \, a n^{4} - 6 \,{\left (a n^{6} + 18 \, a n^{5} + 118 \, a n^{4} + 348 \, a n^{3} + 457 \, a n^{2} + 210 \, a n\right )} \cos \left (d x + c\right )^{4} - 768 \, a n^{3} - 4128 \, a n^{2} - 24 \,{\left (a n^{5} + 16 \, a n^{4} + 86 \, a n^{3} + 176 \, a n^{2} + 105 \, a n\right )} \cos \left (d x + c\right )^{2} - 8448 \, a n -{\left ({\left (a n^{7} + 29 \, a n^{6} + 343 \, a n^{5} + 2135 \, a n^{4} + 7504 \, a n^{3} + 14756 \, a n^{2} + 14832 \, a n + 5760 \, a\right )} \cos \left (d x + c\right )^{6} + 48 \, a n^{4} + 6 \,{\left (a n^{6} + 24 \, a n^{5} + 223 \, a n^{4} + 1020 \, a n^{3} + 2404 \, a n^{2} + 2736 \, a n + 1152 \, a\right )} \cos \left (d x + c\right )^{4} + 960 \, a n^{3} + 6720 \, a n^{2} + 24 \,{\left (a n^{5} + 21 \, a n^{4} + 160 \, a n^{3} + 540 \, a n^{2} + 784 \, a n + 384 \, a\right )} \cos \left (d x + c\right )^{2} + 19200 \, a n + 18432 \, a\right )} \sin \left (d x + c\right ) - 5040 \, a\right )} \sin \left (d x + c\right )^{n}}{d n^{8} + 36 \, d n^{7} + 546 \, d n^{6} + 4536 \, d n^{5} + 22449 \, d n^{4} + 67284 \, d n^{3} + 118124 \, d n^{2} + 109584 \, d n + 40320 \, 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]

-((a*n^7 + 28*a*n^6 + 322*a*n^5 + 1960*a*n^4 + 6769*a*n^3 + 13132*a*n^2 + 13068*a*n + 5040*a)*cos(d*x + c)^8 -
(a*n^7 + 22*a*n^6 + 190*a*n^5 + 820*a*n^4 + 1849*a*n^3 + 2038*a*n^2 + 840*a*n)*cos(d*x + c)^6 - 48*a*n^4 - 6*
(a*n^6 + 18*a*n^5 + 118*a*n^4 + 348*a*n^3 + 457*a*n^2 + 210*a*n)*cos(d*x + c)^4 - 768*a*n^3 - 4128*a*n^2 - 24*
(a*n^5 + 16*a*n^4 + 86*a*n^3 + 176*a*n^2 + 105*a*n)*cos(d*x + c)^2 - 8448*a*n - ((a*n^7 + 29*a*n^6 + 343*a*n^5
+ 2135*a*n^4 + 7504*a*n^3 + 14756*a*n^2 + 14832*a*n + 5760*a)*cos(d*x + c)^6 + 48*a*n^4 + 6*(a*n^6 + 24*a*n^5
+ 223*a*n^4 + 1020*a*n^3 + 2404*a*n^2 + 2736*a*n + 1152*a)*cos(d*x + c)^4 + 960*a*n^3 + 6720*a*n^2 + 24*(a*n^
5 + 21*a*n^4 + 160*a*n^3 + 540*a*n^2 + 784*a*n + 384*a)*cos(d*x + c)^2 + 19200*a*n + 18432*a)*sin(d*x + c) - 5
040*a)*sin(d*x + c)^n/(d*n^8 + 36*d*n^7 + 546*d*n^6 + 4536*d*n^5 + 22449*d*n^4 + 67284*d*n^3 + 118124*d*n^2 +
109584*d*n + 40320*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 [B]  time = 1.20704, size = 910, normalized size = 5.45 \begin{align*} \text{result too large to display} \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]

-((n^3*sin(d*x + c)^n*sin(d*x + c)^8 + 12*n^2*sin(d*x + c)^n*sin(d*x + c)^8 - 3*n^3*sin(d*x + c)^n*sin(d*x + c
)^6 + 44*n*sin(d*x + c)^n*sin(d*x + c)^8 - 42*n^2*sin(d*x + c)^n*sin(d*x + c)^6 + 48*sin(d*x + c)^n*sin(d*x +
c)^8 + 3*n^3*sin(d*x + c)^n*sin(d*x + c)^4 - 168*n*sin(d*x + c)^n*sin(d*x + c)^6 + 48*n^2*sin(d*x + c)^n*sin(d
*x + c)^4 - 192*sin(d*x + c)^n*sin(d*x + c)^6 - n^3*sin(d*x + c)^n*sin(d*x + c)^2 + 228*n*sin(d*x + c)^n*sin(d
*x + c)^4 - 18*n^2*sin(d*x + c)^n*sin(d*x + c)^2 + 288*sin(d*x + c)^n*sin(d*x + c)^4 - 104*n*sin(d*x + c)^n*si
n(d*x + c)^2 - 192*sin(d*x + c)^n*sin(d*x + c)^2)*a/(n^4 + 20*n^3 + 140*n^2 + 400*n + 384) + (n^3*sin(d*x + c)
^n*sin(d*x + c)^7 + 9*n^2*sin(d*x + c)^n*sin(d*x + c)^7 - 3*n^3*sin(d*x + c)^n*sin(d*x + c)^5 + 23*n*sin(d*x +
c)^n*sin(d*x + c)^7 - 33*n^2*sin(d*x + c)^n*sin(d*x + c)^5 + 15*sin(d*x + c)^n*sin(d*x + c)^7 + 3*n^3*sin(d*x
+ c)^n*sin(d*x + c)^3 - 93*n*sin(d*x + c)^n*sin(d*x + c)^5 + 39*n^2*sin(d*x + c)^n*sin(d*x + c)^3 - 63*sin(d*
x + c)^n*sin(d*x + c)^5 - n^3*sin(d*x + c)^n*sin(d*x + c) + 141*n*sin(d*x + c)^n*sin(d*x + c)^3 - 15*n^2*sin(d
*x + c)^n*sin(d*x + c) + 105*sin(d*x + c)^n*sin(d*x + c)^3 - 71*n*sin(d*x + c)^n*sin(d*x + c) - 105*sin(d*x +
c)^n*sin(d*x + c))*a/(n^4 + 16*n^3 + 86*n^2 + 176*n + 105))/d