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(d*x+c)^(1+n)/d/(1+n)+3*a^3*sin(d*x+c)^(2+n)/d/(2+n)-8*a^3*sin(d*x+c)^(4+n)/d/(4+n)-6*a^3*sin(d*x+c)^(5
+n)/d/(5+n)+6*a^3*sin(d*x+c)^(6+n)/d/(6+n)+8*a^3*sin(d*x+c)^(7+n)/d/(7+n)-3*a^3*sin(d*x+c)^(9+n)/d/(9+n)-a^3*s
in(d*x+c)^(10+n)/d/(10+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} \[ \frac {a^3 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {3 a^3 \sin ^{n+2}(c+d x)}{d (n+2)}-\frac {8 a^3 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac {6 a^3 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac {6 a^3 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac {8 a^3 \sin ^{n+7}(c+d x)}{d (n+7)}-\frac {3 a^3 \sin ^{n+9}(c+d x)}{d (n+9)}-\frac {a^3 \sin ^{n+10}(c+d x)}{d (n+10)} \]

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 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))^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*}

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Mathematica [A]  time = 0.96, 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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fricas [B]  time = 0.58, size = 697, normalized size = 3.79 \[ \frac {{\left ({\left (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}\right )} \cos \left (d x + c\right )^{10} - 5 \, {\left (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}\right )} \cos \left (d x + c\right )^{8} + 192 \, a^{3} n^{4} + 4 \, {\left (a^{3} n^{7} + 28 \, a^{3} n^{6} + 304 \, a^{3} n^{5} + 1618 \, a^{3} n^{4} + 4375 \, a^{3} n^{3} + 5554 \, a^{3} n^{2} + 2520 \, a^{3} n\right )} \cos \left (d x + c\right )^{6} + 4224 \, a^{3} n^{3} + 31488 \, a^{3} n^{2} + 24 \, {\left (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\right )} \cos \left (d x + c\right )^{4} + 87936 \, a^{3} n + 60480 \, a^{3} + 96 \, {\left (a^{3} n^{5} + 22 \, a^{3} n^{4} + 164 \, a^{3} n^{3} + 458 \, a^{3} n^{2} + 315 \, a^{3} n\right )} \cos \left (d x + c\right )^{2} - {\left (3 \, {\left (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}\right )} \cos \left (d x + c\right )^{8} - 192 \, a^{3} n^{4} - 4 \, {\left (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}\right )} \cos \left (d x + c\right )^{6} - 4224 \, a^{3} n^{3} - 31488 \, a^{3} n^{2} - 24 \, {\left (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}\right )} \cos \left (d x + c\right )^{4} - 93696 \, a^{3} n - 92160 \, a^{3} - 96 \, {\left (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}\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} + 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} \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Eval
uation time: 39.12Not invertible Error: Bad Argument Value

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maple [F]  time = 41.96, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{3}\, dx \]

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))^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 [A]  time = 0.35, size = 165, normalized size = 0.90 \[ -\frac {\frac {a^{3} \sin \left (d x + c\right )^{n + 10}}{n + 10} + \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 9}}{n + 9} - \frac {8 \, a^{3} \sin \left (d x + c\right )^{n + 7}}{n + 7} - \frac {6 \, a^{3} \sin \left (d x + c\right )^{n + 6}}{n + 6} + \frac {6 \, a^{3} \sin \left (d x + c\right )^{n + 5}}{n + 5} + \frac {8 \, a^{3} \sin \left (d x + c\right )^{n + 4}}{n + 4} - \frac {3 \, a^{3} \sin \left (d x + c\right )^{n + 2}}{n + 2} - \frac {a^{3} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \]

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))^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 17.09, size = 1130, normalized size = 6.14 \[ \text {result too large to display} \]

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))^3,x)

[Out]

(3*a^3*sin(c + d*x)^n*(6117676*n + 3058196*n^2 + 755233*n^3 + 109542*n^4 + 9800*n^5 + 502*n^6 + 11*n^7 + 37144
80))/(256*d*(373560*n + 353884*n^2 + 172634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) - (
5*a^3*sin(c + d*x)^n*cos(8*c + 8*d*x)*(35844*n + 31804*n^2 + 14083*n^3 + 3442*n^4 + 472*n^5 + 34*n^6 + n^7 + 1
5120))/(256*d*(373560*n + 353884*n^2 + 172634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) +
(a^3*sin(c + d*x)^n*cos(10*c + 10*d*x)*(35844*n + 31804*n^2 + 14083*n^3 + 3442*n^4 + 472*n^5 + 34*n^6 + n^7 +
15120))/(512*d*(373560*n + 353884*n^2 + 172634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200))
- (a^3*sin(c + d*x)*sin(c + d*x)^n*(n*16168200i + n^2*7143148i + n^3*1614322i + n^4*215083i + n^5*18019i + n^
6*889i + n^7*19i + 13759200i)*1i)/(128*d*(373560*n + 353884*n^2 + 172634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6
+ 44*n^7 + n^8 + 151200)) - (3*a^3*sin(c + d*x)^n*cos(6*c + 6*d*x)*(1320260*n + 1100668*n^2 + 446515*n^3 + 974
26*n^4 + 11608*n^5 + 706*n^6 + 17*n^7 + 579600))/(512*d*(373560*n + 353884*n^2 + 172634*n^3 + 48503*n^4 + 8162
*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) - (a^3*sin(c + d*x)^n*cos(4*c + 4*d*x)*(1729500*n + 1246276*n^2 + 413
653*n^3 + 71710*n^4 + 6760*n^5 + 334*n^6 + 7*n^7 + 831600))/(64*d*(373560*n + 353884*n^2 + 172634*n^3 + 48503*
n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) + (a^3*sin(c + d*x)^n*cos(2*c + 2*d*x)*(122059*n^3 - 239536
4*n^2 - 9293340*n + 119842*n^4 + 17176*n^5 + 1042*n^6 + 25*n^7 - 6879600))/(256*d*(373560*n + 353884*n^2 + 172
634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) + (a^3*sin(c + d*x)^n*sin(9*c + 9*d*x)*(n*3
9640i + n^2*34916i + n^3*15302i + n^4*3689i + n^5*497i + n^6*35i + n^7*1i + 16800i)*3i)/(256*d*(373560*n + 353
884*n^2 + 172634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) - (a^3*sin(c + d*x)^n*sin(5*c
+ 5*d*x)*(n*97464i + n^2*117044i + n^3*66110i + n^4*18845i + n^5*2741i + n^6*191i + n^7*5i + 30240i)*1i)/(64*d
*(373560*n + 353884*n^2 + 172634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151200)) + (a^3*sin(c +
d*x)^n*sin(7*c + 7*d*x)*(n*538680i + n^2*445172i + n^3*177758i + n^4*37709i + n^5*4277i + n^6*239i + n^7*5i +
237600i)*1i)/(256*d*(373560*n + 353884*n^2 + 172634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 + 44*n^7 + n^8 + 151
200)) - (a^3*sin(c + d*x)^n*sin(3*c + 3*d*x)*(n*763320i + n^2*586164i + n^3*211966i + n^4*40253i + n^5*4149i +
n^6*223i + n^7*5i + 352800i)*3i)/(64*d*(373560*n + 353884*n^2 + 172634*n^3 + 48503*n^4 + 8162*n^5 + 812*n^6 +
44*n^7 + n^8 + 151200))

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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))**3,x)

[Out]

Timed out

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