3.698 \(\int \cos ^7(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.175899, 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} \[ \frac{a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac{2 a^2 \sin ^{n+2}(c+d x)}{d (n+2)}-\frac{2 a^2 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac{6 a^2 \sin ^{n+4}(c+d x)}{d (n+4)}+\frac{6 a^2 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac{2 a^2 \sin ^{n+7}(c+d x)}{d (n+7)}-\frac{2 a^2 \sin ^{n+8}(c+d x)}{d (n+8)}-\frac{a^2 \sin ^{n+9}(c+d x)}{d (n+9)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]

[Out]

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

Mathematica [A]  time = 0.73897, size = 126, normalized size = 0.68 \[ \frac{a^2 \sin ^{n+1}(c+d x) \left (-\frac{\sin ^8(c+d x)}{n+9}-\frac{2 \sin ^7(c+d x)}{n+8}+\frac{2 \sin ^6(c+d x)}{n+7}+\frac{6 \sin ^5(c+d x)}{n+6}-\frac{6 \sin ^3(c+d x)}{n+4}-\frac{2 \sin ^2(c+d x)}{n+3}+\frac{2 \sin (c+d x)}{n+2}+\frac{1}{n+1}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^2,x]

[Out]

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

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Maple [F]  time = 13.977, 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 ) ^{2}\, 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))^2,x)

[Out]

int(cos(d*x+c)^7*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,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))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.64247, size = 1567, normalized size = 8.52 \begin{align*} -\frac{{\left (2 \,{\left (a^{2} n^{7} + 32 \, a^{2} n^{6} + 414 \, a^{2} n^{5} + 2788 \, a^{2} n^{4} + 10469 \, a^{2} n^{3} + 21708 \, a^{2} n^{2} + 22716 \, a^{2} n + 9072 \, a^{2}\right )} \cos \left (d x + c\right )^{8} - 2 \,{\left (a^{2} n^{7} + 26 \, a^{2} n^{6} + 258 \, a^{2} n^{5} + 1240 \, a^{2} n^{4} + 3029 \, a^{2} n^{3} + 3534 \, a^{2} n^{2} + 1512 \, a^{2} n\right )} \cos \left (d x + c\right )^{6} - 96 \, a^{2} n^{4} - 1920 \, a^{2} n^{3} - 12 \,{\left (a^{2} n^{6} + 22 \, a^{2} n^{5} + 170 \, a^{2} n^{4} + 560 \, a^{2} n^{3} + 789 \, a^{2} n^{2} + 378 \, a^{2} n\right )} \cos \left (d x + c\right )^{4} - 12480 \, a^{2} n^{2} - 28800 \, a^{2} n - 48 \,{\left (a^{2} n^{5} + 20 \, a^{2} n^{4} + 130 \, a^{2} n^{3} + 300 \, a^{2} n^{2} + 189 \, a^{2} n\right )} \cos \left (d x + c\right )^{2} - 18144 \, a^{2} +{\left ({\left (a^{2} n^{7} + 31 \, a^{2} n^{6} + 391 \, a^{2} n^{5} + 2581 \, a^{2} n^{4} + 9544 \, a^{2} n^{3} + 19564 \, a^{2} n^{2} + 20304 \, a^{2} n + 8064 \, a^{2}\right )} \cos \left (d x + c\right )^{8} - 2 \,{\left (a^{2} n^{7} + 29 \, a^{2} n^{6} + 343 \, a^{2} n^{5} + 2135 \, a^{2} n^{4} + 7504 \, a^{2} n^{3} + 14756 \, a^{2} n^{2} + 14832 \, a^{2} n + 5760 \, a^{2}\right )} \cos \left (d x + c\right )^{6} - 96 \, a^{2} n^{4} - 1920 \, a^{2} n^{3} - 12 \,{\left (a^{2} n^{6} + 24 \, a^{2} n^{5} + 223 \, a^{2} n^{4} + 1020 \, a^{2} n^{3} + 2404 \, a^{2} n^{2} + 2736 \, a^{2} n + 1152 \, a^{2}\right )} \cos \left (d x + c\right )^{4} - 13440 \, a^{2} n^{2} - 38400 \, a^{2} n - 48 \,{\left (a^{2} n^{5} + 21 \, a^{2} n^{4} + 160 \, a^{2} n^{3} + 540 \, a^{2} n^{2} + 784 \, a^{2} n + 384 \, a^{2}\right )} \cos \left (d x + c\right )^{2} - 36864 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{8} + 40 \, d n^{7} + 670 \, d n^{6} + 6100 \, d n^{5} + 32773 \, d n^{4} + 105460 \, d n^{3} + 196380 \, d n^{2} + 190800 \, d n + 72576 \, 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))^2,x, algorithm="fricas")

[Out]

-(2*(a^2*n^7 + 32*a^2*n^6 + 414*a^2*n^5 + 2788*a^2*n^4 + 10469*a^2*n^3 + 21708*a^2*n^2 + 22716*a^2*n + 9072*a^
2)*cos(d*x + c)^8 - 2*(a^2*n^7 + 26*a^2*n^6 + 258*a^2*n^5 + 1240*a^2*n^4 + 3029*a^2*n^3 + 3534*a^2*n^2 + 1512*
a^2*n)*cos(d*x + c)^6 - 96*a^2*n^4 - 1920*a^2*n^3 - 12*(a^2*n^6 + 22*a^2*n^5 + 170*a^2*n^4 + 560*a^2*n^3 + 789
*a^2*n^2 + 378*a^2*n)*cos(d*x + c)^4 - 12480*a^2*n^2 - 28800*a^2*n - 48*(a^2*n^5 + 20*a^2*n^4 + 130*a^2*n^3 +
300*a^2*n^2 + 189*a^2*n)*cos(d*x + c)^2 - 18144*a^2 + ((a^2*n^7 + 31*a^2*n^6 + 391*a^2*n^5 + 2581*a^2*n^4 + 95
44*a^2*n^3 + 19564*a^2*n^2 + 20304*a^2*n + 8064*a^2)*cos(d*x + c)^8 - 2*(a^2*n^7 + 29*a^2*n^6 + 343*a^2*n^5 +
2135*a^2*n^4 + 7504*a^2*n^3 + 14756*a^2*n^2 + 14832*a^2*n + 5760*a^2)*cos(d*x + c)^6 - 96*a^2*n^4 - 1920*a^2*n
^3 - 12*(a^2*n^6 + 24*a^2*n^5 + 223*a^2*n^4 + 1020*a^2*n^3 + 2404*a^2*n^2 + 2736*a^2*n + 1152*a^2)*cos(d*x + c
)^4 - 13440*a^2*n^2 - 38400*a^2*n - 48*(a^2*n^5 + 21*a^2*n^4 + 160*a^2*n^3 + 540*a^2*n^2 + 784*a^2*n + 384*a^2
)*cos(d*x + c)^2 - 36864*a^2)*sin(d*x + c))*sin(d*x + c)^n/(d*n^8 + 40*d*n^7 + 670*d*n^6 + 6100*d*n^5 + 32773*
d*n^4 + 105460*d*n^3 + 196380*d*n^2 + 190800*d*n + 72576*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))**2,x)

[Out]

Timed out

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Giac [B]  time = 1.32205, size = 1376, normalized size = 7.48 \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))^2,x, algorithm="giac")

[Out]

-((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 + 71*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*s
in(d*x + c)^3 - 315*sin(d*x + c)^n*sin(d*x + c)^3)*a^2/(n^4 + 24*n^3 + 206*n^2 + 744*n + 945) + 2*(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*s
in(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 - 19
2*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*sin(d*x + c)^2 -
192*sin(d*x + c)^n*sin(d*x + c)^2)*a^2/(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*si
n(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^2/(n^4 + 16*n^3 + 86*n^2 + 176*n + 105))/d