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(d*x+c)^(1+n)/d/(1+n)+2*a^2*sin(d*x+c)^(2+n)/d/(2+n)-2*a^2*sin(d*x+c)^(3+n)/d/(3+n)-6*a^2*sin(d*x+c)^(4
+n)/d/(4+n)+6*a^2*sin(d*x+c)^(6+n)/d/(6+n)+2*a^2*sin(d*x+c)^(7+n)/d/(7+n)-2*a^2*sin(d*x+c)^(8+n)/d/(8+n)-a^2*s
in(d*x+c)^(9+n)/d/(9+n)

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Rubi [A]  time = 0.18, antiderivative size = 184, normalized size of antiderivative = 1.00, 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 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]))

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]

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*}

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Mathematica [A]  time = 0.78, 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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fricas [B]  time = 0.57, size = 628, normalized size = 3.41 \[ -\frac {{\left (2 \, {\left (a^{2} n^{7} + 32 \, a^{2} n^{6} + 414 \, a^{2} n^{5} + 2788 \, a^{2} n^{4} + 10469 \, a^{2} n^{3} + 21708 \, a^{2} n^{2} + 22716 \, a^{2} n + 9072 \, a^{2}\right )} \cos \left (d x + c\right )^{8} - 2 \, {\left (a^{2} n^{7} + 26 \, a^{2} n^{6} + 258 \, a^{2} n^{5} + 1240 \, a^{2} n^{4} + 3029 \, a^{2} n^{3} + 3534 \, a^{2} n^{2} + 1512 \, a^{2} n\right )} \cos \left (d x + c\right )^{6} - 96 \, a^{2} n^{4} - 1920 \, a^{2} n^{3} - 12 \, {\left (a^{2} n^{6} + 22 \, a^{2} n^{5} + 170 \, a^{2} n^{4} + 560 \, a^{2} n^{3} + 789 \, a^{2} n^{2} + 378 \, a^{2} n\right )} \cos \left (d x + c\right )^{4} - 12480 \, a^{2} n^{2} - 28800 \, a^{2} n - 48 \, {\left (a^{2} n^{5} + 20 \, a^{2} n^{4} + 130 \, a^{2} n^{3} + 300 \, a^{2} n^{2} + 189 \, a^{2} n\right )} \cos \left (d x + c\right )^{2} - 18144 \, a^{2} + {\left ({\left (a^{2} n^{7} + 31 \, a^{2} n^{6} + 391 \, a^{2} n^{5} + 2581 \, a^{2} n^{4} + 9544 \, a^{2} n^{3} + 19564 \, a^{2} n^{2} + 20304 \, a^{2} n + 8064 \, a^{2}\right )} \cos \left (d x + c\right )^{8} - 2 \, {\left (a^{2} n^{7} + 29 \, a^{2} n^{6} + 343 \, a^{2} n^{5} + 2135 \, a^{2} n^{4} + 7504 \, a^{2} n^{3} + 14756 \, a^{2} n^{2} + 14832 \, a^{2} n + 5760 \, a^{2}\right )} \cos \left (d x + c\right )^{6} - 96 \, a^{2} n^{4} - 1920 \, a^{2} n^{3} - 12 \, {\left (a^{2} n^{6} + 24 \, a^{2} n^{5} + 223 \, a^{2} n^{4} + 1020 \, a^{2} n^{3} + 2404 \, a^{2} n^{2} + 2736 \, a^{2} n + 1152 \, a^{2}\right )} \cos \left (d x + c\right )^{4} - 13440 \, a^{2} n^{2} - 38400 \, a^{2} n - 48 \, {\left (a^{2} n^{5} + 21 \, a^{2} n^{4} + 160 \, a^{2} n^{3} + 540 \, a^{2} n^{2} + 784 \, a^{2} n + 384 \, a^{2}\right )} \cos \left (d x + c\right )^{2} - 36864 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{8} + 40 \, d n^{7} + 670 \, d n^{6} + 6100 \, d n^{5} + 32773 \, d n^{4} + 105460 \, d n^{3} + 196380 \, d n^{2} + 190800 \, d n + 72576 \, d} \]

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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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]

Timed out

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maple [F]  time = 28.83, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{2}\, dx \]

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 [A]  time = 0.32, size = 165, normalized size = 0.90 \[ -\frac {\frac {a^{2} \sin \left (d x + c\right )^{n + 9}}{n + 9} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 8}}{n + 8} - \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 7}}{n + 7} - \frac {6 \, a^{2} \sin \left (d x + c\right )^{n + 6}}{n + 6} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{n + 4}}{n + 4} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 3}}{n + 3} - \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 2}}{n + 2} - \frac {a^{2} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \]

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]

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

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mupad [B]  time = 16.45, size = 1142, normalized size = 6.21 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)^n*(n*1507788i + n^2*868332i + n^3*238585i + n^4*37844i + n^5*3702i + n^6*208i + n^7*5i + 843
696i))/(64*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i + 7
2576i)) - (a^2*sin(c + d*x)^n*sin(9*c + 9*d*x)*(20304*n + 19564*n^2 + 9544*n^3 + 2581*n^4 + 391*n^5 + 31*n^6 +
n^7 + 8064)*1i)/(256*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i +
n^8*1i + 72576i)) + (a^2*sin(c + d*x)*sin(c + d*x)^n*(6799248*n + 3169500*n^2 + 770632*n^3 + 111993*n^4 + 102
67*n^5 + 555*n^6 + 13*n^7 + 5588352)*1i)/(128*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100
i + n^6*670i + n^7*40i + n^8*1i + 72576i)) - (a^2*sin(c + d*x)^n*cos(8*c + 8*d*x)*(n*22716i + n^2*21708i + n^3
*10469i + n^4*2788i + n^5*414i + n^6*32i + n^7*1i + 9072i))/(64*d*(n*190800i + n^2*196380i + n^3*105460i + n^4
*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i + 72576i)) - (a^2*sin(c + d*x)^n*cos(6*c + 6*d*x)*(n*43920i
+ n^2*39882i + n^3*17909i + n^4*4336i + n^5*570i + n^6*38i + n^7*1i + 18144i))/(16*d*(n*190800i + n^2*196380i
+ n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i + 72576i)) - (a^2*sin(c + d*x)^n*cos(4*c
+ 4*d*x)*(n*140868i + n^2*111816i + n^3*41669i + n^4*7996i + n^5*822i + n^6*44i + n^7*1i + 63504i))/(16*d*(n*1
90800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i + 72576i)) + (a^2*si
n(c + d*x)^n*cos(2*c + 2*d*x)*(n^3*2549i - n^2*59958i - n*186480i + n^4*3568i + n^5*570i + n^6*38i + n^7*1i -
127008i))/(16*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i
+ 72576i)) - (a^2*sin(c + d*x)^n*sin(7*c + 7*d*x)*(23472*n + 18900*n^2 + 6776*n^3 + 987*n^4 - 7*n^5 - 15*n^6 -
n^7 + 10368)*1i)/(256*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i + n^6*670i + n^7*40i
+ n^8*1i + 72576i)) + (a^2*sin(c + d*x)^n*sin(5*c + 5*d*x)*(178128*n + 165132*n^2 + 76280*n^3 + 19149*n^4 + 26
27*n^5 + 183*n^6 + 5*n^7 + 72576)*1i)/(64*d*(n*190800i + n^2*196380i + n^3*105460i + n^4*32773i + n^5*6100i +
n^6*670i + n^7*40i + n^8*1i + 72576i)) + (a^2*sin(c + d*x)^n*sin(3*c + 3*d*x)*(1120944*n + 889556*n^2 + 338024
*n^3 + 68603*n^4 + 7661*n^5 + 449*n^6 + 11*n^7 + 508032)*1i)/(64*d*(n*190800i + n^2*196380i + n^3*105460i + n^
4*32773i + n^5*6100i + n^6*670i + n^7*40i + n^8*1i + 72576i))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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