3.697 \(\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=184 \[ \frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{8 a^3 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac{6 a^3 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{6 a^3 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{8 a^3 \sin ^{n+7}(c+d x)}{d (n+7)}-\frac{3 a^3 \sin ^{n+9}(c+d x)}{d (n+9)}-\frac{a^3 \sin ^{n+10}(c+d x)}{d (n+10)} \]

[Out]

(a^3*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (3*a^3*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (8*a^3*Sin[c + d*x]^(4 + n
))/(d*(4 + n)) - (6*a^3*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (6*a^3*Sin[c + d*x]^(6 + n))/(d*(6 + n)) + (8*a^3*
Sin[c + d*x]^(7 + n))/(d*(7 + n)) - (3*a^3*Sin[c + d*x]^(9 + n))/(d*(9 + n)) - (a^3*Sin[c + d*x]^(10 + n))/(d*
(10 + n))

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Rubi [A]  time = 0.177537, antiderivative size = 184, 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} \[ \frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{8 a^3 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac{6 a^3 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{6 a^3 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{8 a^3 \sin ^{n+7}(c+d x)}{d (n+7)}-\frac{3 a^3 \sin ^{n+9}(c+d x)}{d (n+9)}-\frac{a^3 \sin ^{n+10}(c+d x)}{d (n+10)} \]

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]

(a^3*Sin[c + d*x]^(1 + n))/(d*(1 + n)) + (3*a^3*Sin[c + d*x]^(2 + n))/(d*(2 + n)) - (8*a^3*Sin[c + d*x]^(4 + n
))/(d*(4 + n)) - (6*a^3*Sin[c + d*x]^(5 + n))/(d*(5 + n)) + (6*a^3*Sin[c + d*x]^(6 + n))/(d*(6 + n)) + (8*a^3*
Sin[c + d*x]^(7 + n))/(d*(7 + n)) - (3*a^3*Sin[c + d*x]^(9 + n))/(d*(9 + n)) - (a^3*Sin[c + d*x]^(10 + n))/(d*
(10 + 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))^3 \, dx &=\frac{\operatorname{Subst}\left (\int (a-x)^3 \left (\frac{x}{a}\right )^n (a+x)^6 \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a^9 \left (\frac{x}{a}\right )^n+3 a^9 \left (\frac{x}{a}\right )^{1+n}-8 a^9 \left (\frac{x}{a}\right )^{3+n}-6 a^9 \left (\frac{x}{a}\right )^{4+n}+6 a^9 \left (\frac{x}{a}\right )^{5+n}+8 a^9 \left (\frac{x}{a}\right )^{6+n}-3 a^9 \left (\frac{x}{a}\right )^{8+n}-a^9 \left (\frac{x}{a}\right )^{9+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac{a^3 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac{3 a^3 \sin ^{2+n}(c+d x)}{d (2+n)}-\frac{8 a^3 \sin ^{4+n}(c+d x)}{d (4+n)}-\frac{6 a^3 \sin ^{5+n}(c+d x)}{d (5+n)}+\frac{6 a^3 \sin ^{6+n}(c+d x)}{d (6+n)}+\frac{8 a^3 \sin ^{7+n}(c+d x)}{d (7+n)}-\frac{3 a^3 \sin ^{9+n}(c+d x)}{d (9+n)}-\frac{a^3 \sin ^{10+n}(c+d x)}{d (10+n)}\\ \end{align*}

Mathematica [A]  time = 0.904192, size = 126, normalized size = 0.68 \[ \frac{a^3 \sin ^{n+1}(c+d x) \left (-\frac{\sin ^9(c+d x)}{n+10}-\frac{3 \sin ^8(c+d x)}{n+9}+\frac{8 \sin ^6(c+d x)}{n+7}+\frac{6 \sin ^5(c+d x)}{n+6}-\frac{6 \sin ^4(c+d x)}{n+5}-\frac{8 \sin ^3(c+d x)}{n+4}+\frac{3 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right )}{d} \]

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]

(a^3*Sin[c + d*x]^(1 + n)*((1 + n)^(-1) + (3*Sin[c + d*x])/(2 + n) - (8*Sin[c + d*x]^3)/(4 + n) - (6*Sin[c + d
*x]^4)/(5 + n) + (6*Sin[c + d*x]^5)/(6 + n) + (8*Sin[c + d*x]^6)/(7 + n) - (3*Sin[c + d*x]^8)/(9 + n) - Sin[c
+ d*x]^9/(10 + n)))/d

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Maple [F]  time = 20.262, 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 ) ^{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 [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))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.71032, size = 1751, normalized size = 9.52 \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))^3,x, algorithm="fricas")

[Out]

((a^3*n^7 + 34*a^3*n^6 + 472*a^3*n^5 + 3442*a^3*n^4 + 14083*a^3*n^3 + 31804*a^3*n^2 + 35844*a^3*n + 15120*a^3)
*cos(d*x + c)^10 - 5*(a^3*n^7 + 34*a^3*n^6 + 472*a^3*n^5 + 3442*a^3*n^4 + 14083*a^3*n^3 + 31804*a^3*n^2 + 3584
4*a^3*n + 15120*a^3)*cos(d*x + c)^8 + 192*a^3*n^4 + 4*(a^3*n^7 + 28*a^3*n^6 + 304*a^3*n^5 + 1618*a^3*n^4 + 437
5*a^3*n^3 + 5554*a^3*n^2 + 2520*a^3*n)*cos(d*x + c)^6 + 4224*a^3*n^3 + 31488*a^3*n^2 + 24*(a^3*n^6 + 24*a^3*n^
5 + 208*a^3*n^4 + 786*a^3*n^3 + 1231*a^3*n^2 + 630*a^3*n)*cos(d*x + c)^4 + 87936*a^3*n + 60480*a^3 + 96*(a^3*n
^5 + 22*a^3*n^4 + 164*a^3*n^3 + 458*a^3*n^2 + 315*a^3*n)*cos(d*x + c)^2 - (3*(a^3*n^7 + 35*a^3*n^6 + 497*a^3*n
^5 + 3689*a^3*n^4 + 15302*a^3*n^3 + 34916*a^3*n^2 + 39640*a^3*n + 16800*a^3)*cos(d*x + c)^8 - 192*a^3*n^4 - 4*
(a^3*n^7 + 31*a^3*n^6 + 385*a^3*n^5 + 2485*a^3*n^4 + 8974*a^3*n^3 + 18004*a^3*n^2 + 18360*a^3*n + 7200*a^3)*co
s(d*x + c)^6 - 4224*a^3*n^3 - 31488*a^3*n^2 - 24*(a^3*n^6 + 26*a^3*n^5 + 255*a^3*n^4 + 1210*a^3*n^3 + 2924*a^3
*n^2 + 3384*a^3*n + 1440*a^3)*cos(d*x + c)^4 - 93696*a^3*n - 92160*a^3 - 96*(a^3*n^5 + 23*a^3*n^4 + 186*a^3*n^
3 + 652*a^3*n^2 + 968*a^3*n + 480*a^3)*cos(d*x + c)^2)*sin(d*x + c))*sin(d*x + c)^n/(d*n^8 + 44*d*n^7 + 812*d*
n^6 + 8162*d*n^5 + 48503*d*n^4 + 172634*d*n^3 + 353884*d*n^2 + 373560*d*n + 151200*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 [B]  time = 1.36572, size = 1836, normalized size = 9.98 \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))^3,x, algorithm="giac")

[Out]

-((n^3*sin(d*x + c)^n*sin(d*x + c)^10 + 18*n^2*sin(d*x + c)^n*sin(d*x + c)^10 - 3*n^3*sin(d*x + c)^n*sin(d*x +
c)^8 + 104*n*sin(d*x + c)^n*sin(d*x + c)^10 - 60*n^2*sin(d*x + c)^n*sin(d*x + c)^8 + 192*sin(d*x + c)^n*sin(d
*x + c)^10 + 3*n^3*sin(d*x + c)^n*sin(d*x + c)^6 - 372*n*sin(d*x + c)^n*sin(d*x + c)^8 + 66*n^2*sin(d*x + c)^n
*sin(d*x + c)^6 - 720*sin(d*x + c)^n*sin(d*x + c)^8 - n^3*sin(d*x + c)^n*sin(d*x + c)^4 + 456*n*sin(d*x + c)^n
*sin(d*x + c)^6 - 24*n^2*sin(d*x + c)^n*sin(d*x + c)^4 + 960*sin(d*x + c)^n*sin(d*x + c)^6 - 188*n*sin(d*x + c
)^n*sin(d*x + c)^4 - 480*sin(d*x + c)^n*sin(d*x + c)^4)*a^3/(n^4 + 28*n^3 + 284*n^2 + 1232*n + 1920) + 3*(n^3*
sin(d*x + c)^n*sin(d*x + c)^9 + 15*n^2*sin(d*x + c)^n*sin(d*x + c)^9 - 3*n^3*sin(d*x + c)^n*sin(d*x + c)^7 + 7
1*n*sin(d*x + c)^n*sin(d*x + c)^9 - 51*n^2*sin(d*x + c)^n*sin(d*x + c)^7 + 105*sin(d*x + c)^n*sin(d*x + c)^9 +
3*n^3*sin(d*x + c)^n*sin(d*x + c)^5 - 261*n*sin(d*x + c)^n*sin(d*x + c)^7 + 57*n^2*sin(d*x + c)^n*sin(d*x + c
)^5 - 405*sin(d*x + c)^n*sin(d*x + c)^7 - n^3*sin(d*x + c)^n*sin(d*x + c)^3 + 333*n*sin(d*x + c)^n*sin(d*x + c
)^5 - 21*n^2*sin(d*x + c)^n*sin(d*x + c)^3 + 567*sin(d*x + c)^n*sin(d*x + c)^5 - 143*n*sin(d*x + c)^n*sin(d*x
+ c)^3 - 315*sin(d*x + c)^n*sin(d*x + c)^3)*a^3/(n^4 + 24*n^3 + 206*n^2 + 744*n + 945) + 3*(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*si
n(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*sin(d*x + c)^2 - 192*si
n(d*x + c)^n*sin(d*x + c)^2)*a^3/(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^3/(n^4 + 16*n^3 + 86*n^2 + 176*n + 105))/d