3.565 \(\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=181 \[ \frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{a^3 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{5 a^3 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac{5 a^3 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{a^3 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{3 a^3 \sin ^{n+7}(c+d x)}{d (n+7)}+\frac{a^3 \sin ^{n+8}(c+d x)}{d (n+8)} \]

[Out]

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

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Rubi [A]  time = 0.180944, antiderivative size = 181, 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} \[ \frac{a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac{a^3 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{5 a^3 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac{5 a^3 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac{a^3 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{3 a^3 \sin ^{n+7}(c+d x)}{d (n+7)}+\frac{a^3 \sin ^{n+8}(c+d x)}{d (n+8)} \]

Antiderivative was successfully verified.

[In]

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

Mathematica [A]  time = 0.564413, size = 123, normalized size = 0.68 \[ \frac{a^3 \sin ^{n+1}(c+d x) \left (\frac{\sin ^7(c+d x)}{n+8}+\frac{3 \sin ^6(c+d x)}{n+7}+\frac{\sin ^5(c+d x)}{n+6}-\frac{5 \sin ^4(c+d x)}{n+5}-\frac{5 \sin ^3(c+d x)}{n+4}+\frac{\sin ^2(c+d x)}{n+3}+\frac{3 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

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Maple [F]  time = 11.083, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{5} \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)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x)

[Out]

int(cos(d*x+c)^5*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)^5*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 = 1.48964, size = 1554, normalized size = 8.59 \begin{align*} \frac{{\left ({\left (a^{3} n^{7} + 28 \, a^{3} n^{6} + 322 \, a^{3} n^{5} + 1960 \, a^{3} n^{4} + 6769 \, a^{3} n^{3} + 13132 \, a^{3} n^{2} + 13068 \, a^{3} n + 5040 \, a^{3}\right )} \cos \left (d x + c\right )^{8} + 32 \, a^{3} n^{5} + 720 \, a^{3} n^{4} -{\left (5 \, a^{3} n^{7} + 142 \, a^{3} n^{6} + 1654 \, a^{3} n^{5} + 10180 \, a^{3} n^{4} + 35485 \, a^{3} n^{3} + 69358 \, a^{3} n^{2} + 69416 \, a^{3} n + 26880 \, a^{3}\right )} \cos \left (d x + c\right )^{6} + 6080 \, a^{3} n^{3} + 23520 \, a^{3} n^{2} + 2 \,{\left (2 \, a^{3} n^{7} + 49 \, a^{3} n^{6} + 470 \, a^{3} n^{5} + 2230 \, a^{3} n^{4} + 5438 \, a^{3} n^{3} + 6361 \, a^{3} n^{2} + 2730 \, a^{3} n\right )} \cos \left (d x + c\right )^{4} + 39968 \, a^{3} n + 21840 \, a^{3} + 8 \,{\left (2 \, a^{3} n^{6} + 45 \, a^{3} n^{5} + 380 \, a^{3} n^{4} + 1470 \, a^{3} n^{3} + 2498 \, a^{3} n^{2} + 1365 \, a^{3} n\right )} \cos \left (d x + c\right )^{2} +{\left (32 \, a^{3} n^{5} + 720 \, a^{3} n^{4} - 3 \,{\left (a^{3} n^{7} + 29 \, a^{3} n^{6} + 343 \, a^{3} n^{5} + 2135 \, a^{3} n^{4} + 7504 \, a^{3} n^{3} + 14756 \, a^{3} n^{2} + 14832 \, a^{3} n + 5760 \, a^{3}\right )} \cos \left (d x + c\right )^{6} + 6080 \, a^{3} n^{3} + 24000 \, a^{3} n^{2} + 2 \,{\left (2 \, a^{3} n^{7} + 53 \, a^{3} n^{6} + 566 \, a^{3} n^{5} + 3155 \, a^{3} n^{4} + 9908 \, a^{3} n^{3} + 17492 \, a^{3} n^{2} + 15984 \, a^{3} n + 5760 \, a^{3}\right )} \cos \left (d x + c\right )^{4} + 44288 \, a^{3} n + 30720 \, a^{3} + 8 \,{\left (2 \, a^{3} n^{6} + 47 \, a^{3} n^{5} + 425 \, a^{3} n^{4} + 1880 \, a^{3} n^{3} + 4268 \, a^{3} n^{2} + 4688 \, a^{3} n + 1920 \, a^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\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)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

((a^3*n^7 + 28*a^3*n^6 + 322*a^3*n^5 + 1960*a^3*n^4 + 6769*a^3*n^3 + 13132*a^3*n^2 + 13068*a^3*n + 5040*a^3)*c
os(d*x + c)^8 + 32*a^3*n^5 + 720*a^3*n^4 - (5*a^3*n^7 + 142*a^3*n^6 + 1654*a^3*n^5 + 10180*a^3*n^4 + 35485*a^3
*n^3 + 69358*a^3*n^2 + 69416*a^3*n + 26880*a^3)*cos(d*x + c)^6 + 6080*a^3*n^3 + 23520*a^3*n^2 + 2*(2*a^3*n^7 +
49*a^3*n^6 + 470*a^3*n^5 + 2230*a^3*n^4 + 5438*a^3*n^3 + 6361*a^3*n^2 + 2730*a^3*n)*cos(d*x + c)^4 + 39968*a^
3*n + 21840*a^3 + 8*(2*a^3*n^6 + 45*a^3*n^5 + 380*a^3*n^4 + 1470*a^3*n^3 + 2498*a^3*n^2 + 1365*a^3*n)*cos(d*x
+ c)^2 + (32*a^3*n^5 + 720*a^3*n^4 - 3*(a^3*n^7 + 29*a^3*n^6 + 343*a^3*n^5 + 2135*a^3*n^4 + 7504*a^3*n^3 + 147
56*a^3*n^2 + 14832*a^3*n + 5760*a^3)*cos(d*x + c)^6 + 6080*a^3*n^3 + 24000*a^3*n^2 + 2*(2*a^3*n^7 + 53*a^3*n^6
+ 566*a^3*n^5 + 3155*a^3*n^4 + 9908*a^3*n^3 + 17492*a^3*n^2 + 15984*a^3*n + 5760*a^3)*cos(d*x + c)^4 + 44288*
a^3*n + 30720*a^3 + 8*(2*a^3*n^6 + 47*a^3*n^5 + 425*a^3*n^4 + 1880*a^3*n^3 + 4268*a^3*n^2 + 4688*a^3*n + 1920*
a^3)*cos(d*x + c)^2)*sin(d*x + c))*sin(d*x + c)^n/(d*n^8 + 36*d*n^7 + 546*d*n^6 + 4536*d*n^5 + 22449*d*n^4 + 6
7284*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)**5*sin(d*x+c)**n*(a+a*sin(d*x+c))**3,x)

[Out]

Timed out

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Giac [B]  time = 1.23205, size = 1038, normalized size = 5.73 \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)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

((n^2*sin(d*x + c)^n*sin(d*x + c)^8 + 10*n*sin(d*x + c)^n*sin(d*x + c)^8 - 2*n^2*sin(d*x + c)^n*sin(d*x + c)^6
+ 24*sin(d*x + c)^n*sin(d*x + c)^8 - 24*n*sin(d*x + c)^n*sin(d*x + c)^6 + n^2*sin(d*x + c)^n*sin(d*x + c)^4 -
64*sin(d*x + c)^n*sin(d*x + c)^6 + 14*n*sin(d*x + c)^n*sin(d*x + c)^4 + 48*sin(d*x + c)^n*sin(d*x + c)^4)*a^3
/(n^3 + 18*n^2 + 104*n + 192) + 3*(n^2*sin(d*x + c)^n*sin(d*x + c)^7 + 8*n*sin(d*x + c)^n*sin(d*x + c)^7 - 2*n
^2*sin(d*x + c)^n*sin(d*x + c)^5 + 15*sin(d*x + c)^n*sin(d*x + c)^7 - 20*n*sin(d*x + c)^n*sin(d*x + c)^5 + n^2
*sin(d*x + c)^n*sin(d*x + c)^3 - 42*sin(d*x + c)^n*sin(d*x + c)^5 + 12*n*sin(d*x + c)^n*sin(d*x + c)^3 + 35*si
n(d*x + c)^n*sin(d*x + c)^3)*a^3/(n^3 + 15*n^2 + 71*n + 105) + 3*(n^2*sin(d*x + c)^n*sin(d*x + c)^6 + 6*n*sin(
d*x + c)^n*sin(d*x + c)^6 - 2*n^2*sin(d*x + c)^n*sin(d*x + c)^4 + 8*sin(d*x + c)^n*sin(d*x + c)^6 - 16*n*sin(d
*x + c)^n*sin(d*x + c)^4 + n^2*sin(d*x + c)^n*sin(d*x + c)^2 - 24*sin(d*x + c)^n*sin(d*x + c)^4 + 10*n*sin(d*x
+ c)^n*sin(d*x + c)^2 + 24*sin(d*x + c)^n*sin(d*x + c)^2)*a^3/(n^3 + 12*n^2 + 44*n + 48) + (n^2*sin(d*x + c)^
n*sin(d*x + c)^5 + 4*n*sin(d*x + c)^n*sin(d*x + c)^5 - 2*n^2*sin(d*x + c)^n*sin(d*x + c)^3 + 3*sin(d*x + c)^n*
sin(d*x + c)^5 - 12*n*sin(d*x + c)^n*sin(d*x + c)^3 + n^2*sin(d*x + c)^n*sin(d*x + c) - 10*sin(d*x + c)^n*sin(
d*x + c)^3 + 8*n*sin(d*x + c)^n*sin(d*x + c) + 15*sin(d*x + c)^n*sin(d*x + c))*a^3/(n^3 + 9*n^2 + 23*n + 15))/
d