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

Optimal. Leaf size=123 \[ \frac {a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {a \sin ^{n+2}(c+d x)}{d (n+2)}-\frac {2 a \sin ^{n+3}(c+d x)}{d (n+3)}-\frac {2 a \sin ^{n+4}(c+d x)}{d (n+4)}+\frac {a \sin ^{n+5}(c+d x)}{d (n+5)}+\frac {a \sin ^{n+6}(c+d x)}{d (n+6)} \]

[Out]

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

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Rubi [A]  time = 0.12, antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2836, 88} \[ \frac {a \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {a \sin ^{n+2}(c+d x)}{d (n+2)}-\frac {2 a \sin ^{n+3}(c+d x)}{d (n+3)}-\frac {2 a \sin ^{n+4}(c+d x)}{d (n+4)}+\frac {a \sin ^{n+5}(c+d x)}{d (n+5)}+\frac {a \sin ^{n+6}(c+d x)}{d (n+6)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]

[Out]

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

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Mathematica [B]  time = 1.38, size = 345, normalized size = 2.80 \[ \frac {a \sin ^{n+1}(c+d x) \left (2 n^5 \sin (c+d x)+3 n^5 \sin (3 (c+d x))+n^5 \sin (5 (c+d x))+46 n^4 \sin (c+d x)+61 n^4 \sin (3 (c+d x))+15 n^4 \sin (5 (c+d x))+474 n^3 \sin (c+d x)+431 n^3 \sin (3 (c+d x))+85 n^3 \sin (5 (c+d x))+2258 n^2 \sin (c+d x)+1331 n^2 \sin (3 (c+d x))+225 n^2 \sin (5 (c+d x))+8 \left (n^5+20 n^4+147 n^3+484 n^2+692 n+336\right ) \cos (2 (c+d x))+2 \left (n^5+16 n^4+95 n^3+260 n^2+324 n+144\right ) \cos (4 (c+d x))+4468 n \sin (c+d x)+1798 n \sin (3 (c+d x))+274 n \sin (5 (c+d x))+2640 \sin (c+d x)+840 \sin (3 (c+d x))+120 \sin (5 (c+d x))+6 n^5+128 n^4+1114 n^3+4888 n^2+10520 n+8544\right )}{16 d (n+1) (n+2) (n+3) (n+4) (n+5) (n+6)} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*Sin[c + d*x]^n*(a + a*Sin[c + d*x]),x]

[Out]

(a*Sin[c + d*x]^(1 + n)*(8544 + 10520*n + 4888*n^2 + 1114*n^3 + 128*n^4 + 6*n^5 + 8*(336 + 692*n + 484*n^2 + 1
47*n^3 + 20*n^4 + n^5)*Cos[2*(c + d*x)] + 2*(144 + 324*n + 260*n^2 + 95*n^3 + 16*n^4 + n^5)*Cos[4*(c + d*x)] +
2640*Sin[c + d*x] + 4468*n*Sin[c + d*x] + 2258*n^2*Sin[c + d*x] + 474*n^3*Sin[c + d*x] + 46*n^4*Sin[c + d*x]
+ 2*n^5*Sin[c + d*x] + 840*Sin[3*(c + d*x)] + 1798*n*Sin[3*(c + d*x)] + 1331*n^2*Sin[3*(c + d*x)] + 431*n^3*Si
n[3*(c + d*x)] + 61*n^4*Sin[3*(c + d*x)] + 3*n^5*Sin[3*(c + d*x)] + 120*Sin[5*(c + d*x)] + 274*n*Sin[5*(c + d*
x)] + 225*n^2*Sin[5*(c + d*x)] + 85*n^3*Sin[5*(c + d*x)] + 15*n^4*Sin[5*(c + d*x)] + n^5*Sin[5*(c + d*x)]))/(1
6*d*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n))

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fricas [B]  time = 0.73, size = 282, normalized size = 2.29 \[ -\frac {{\left ({\left (a n^{5} + 15 \, a n^{4} + 85 \, a n^{3} + 225 \, a n^{2} + 274 \, a n + 120 \, a\right )} \cos \left (d x + c\right )^{6} - {\left (a n^{5} + 11 \, a n^{4} + 41 \, a n^{3} + 61 \, a n^{2} + 30 \, a n\right )} \cos \left (d x + c\right )^{4} - 8 \, a n^{3} - 72 \, a n^{2} - 4 \, {\left (a n^{4} + 9 \, a n^{3} + 23 \, a n^{2} + 15 \, a n\right )} \cos \left (d x + c\right )^{2} - 184 \, a n - {\left ({\left (a n^{5} + 16 \, a n^{4} + 95 \, a n^{3} + 260 \, a n^{2} + 324 \, a n + 144 \, a\right )} \cos \left (d x + c\right )^{4} + 8 \, a n^{3} + 96 \, a n^{2} + 4 \, {\left (a n^{4} + 13 \, a n^{3} + 56 \, a n^{2} + 92 \, a n + 48 \, a\right )} \cos \left (d x + c\right )^{2} + 352 \, a n + 384 \, a\right )} \sin \left (d x + c\right ) - 120 \, a\right )} \sin \left (d x + c\right )^{n}}{d n^{6} + 21 \, d n^{5} + 175 \, d n^{4} + 735 \, d n^{3} + 1624 \, d n^{2} + 1764 \, d n + 720 \, d} \]

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)),x, algorithm="fricas")

[Out]

-((a*n^5 + 15*a*n^4 + 85*a*n^3 + 225*a*n^2 + 274*a*n + 120*a)*cos(d*x + c)^6 - (a*n^5 + 11*a*n^4 + 41*a*n^3 +
61*a*n^2 + 30*a*n)*cos(d*x + c)^4 - 8*a*n^3 - 72*a*n^2 - 4*(a*n^4 + 9*a*n^3 + 23*a*n^2 + 15*a*n)*cos(d*x + c)^
2 - 184*a*n - ((a*n^5 + 16*a*n^4 + 95*a*n^3 + 260*a*n^2 + 324*a*n + 144*a)*cos(d*x + c)^4 + 8*a*n^3 + 96*a*n^2
+ 4*(a*n^4 + 13*a*n^3 + 56*a*n^2 + 92*a*n + 48*a)*cos(d*x + c)^2 + 352*a*n + 384*a)*sin(d*x + c) - 120*a)*sin
(d*x + c)^n/(d*n^6 + 21*d*n^5 + 175*d*n^4 + 735*d*n^3 + 1624*d*n^2 + 1764*d*n + 720*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)^5*sin(d*x+c)^n*(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 9.86, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )\, dx \]

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)),x)

[Out]

int(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c)),x)

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maxima [A]  time = 0.60, size = 109, normalized size = 0.89 \[ \frac {\frac {a \sin \left (d x + c\right )^{n + 6}}{n + 6} + \frac {a \sin \left (d x + c\right )^{n + 5}}{n + 5} - \frac {2 \, a \sin \left (d x + c\right )^{n + 4}}{n + 4} - \frac {2 \, a \sin \left (d x + c\right )^{n + 3}}{n + 3} + \frac {a \sin \left (d x + c\right )^{n + 2}}{n + 2} + \frac {a \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 13.56, size = 550, normalized size = 4.47 \[ \frac {a\,{\sin \left (c+d\,x\right )}^n\,\left (n^5+23\,n^4+237\,n^3+1129\,n^2+2234\,n+1320\right )}{16\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\cos \left (6\,c+6\,d\,x\right )\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{32\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\cos \left (4\,c+4\,d\,x\right )\,\left (n^5+23\,n^4+173\,n^3+553\,n^2+762\,n+360\right )}{16\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {a\,\sin \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^n\,\left (n^5\,1{}\mathrm {i}+n^4\,24{}\mathrm {i}+n^3\,263{}\mathrm {i}+n^2\,1476{}\mathrm {i}+n\,3876{}\mathrm {i}+3600{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\cos \left (2\,c+2\,d\,x\right )\,\left (-n^5-15\,n^4+43\,n^3+927\,n^2+2670\,n+1800\right )}{32\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\sin \left (5\,c+5\,d\,x\right )\,\left (n^5\,1{}\mathrm {i}+n^4\,16{}\mathrm {i}+n^3\,95{}\mathrm {i}+n^2\,260{}\mathrm {i}+n\,324{}\mathrm {i}+144{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {a\,{\sin \left (c+d\,x\right )}^n\,\sin \left (3\,c+3\,d\,x\right )\,\left (n^5\,3{}\mathrm {i}+n^4\,64{}\mathrm {i}+n^3\,493{}\mathrm {i}+n^2\,1676{}\mathrm {i}+n\,2444{}\mathrm {i}+1200{}\mathrm {i}\right )\,1{}\mathrm {i}}{16\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^5*sin(c + d*x)^n*(a + a*sin(c + d*x)),x)

[Out]

(a*sin(c + d*x)^n*(2234*n + 1129*n^2 + 237*n^3 + 23*n^4 + n^5 + 1320))/(16*d*(1764*n + 1624*n^2 + 735*n^3 + 17
5*n^4 + 21*n^5 + n^6 + 720)) - (a*sin(c + d*x)^n*cos(6*c + 6*d*x)*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 1
20))/(32*d*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) - (a*sin(c + d*x)^n*cos(4*c + 4*d*x)*
(762*n + 553*n^2 + 173*n^3 + 23*n^4 + n^5 + 360))/(16*d*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6
+ 720)) - (a*sin(c + d*x)*sin(c + d*x)^n*(n*3876i + n^2*1476i + n^3*263i + n^4*24i + n^5*1i + 3600i)*1i)/(8*d*
(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) - (a*sin(c + d*x)^n*cos(2*c + 2*d*x)*(2670*n + 9
27*n^2 + 43*n^3 - 15*n^4 - n^5 + 1800))/(32*d*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) -
(a*sin(c + d*x)^n*sin(5*c + 5*d*x)*(n*324i + n^2*260i + n^3*95i + n^4*16i + n^5*1i + 144i)*1i)/(16*d*(1764*n +
1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) - (a*sin(c + d*x)^n*sin(3*c + 3*d*x)*(n*2444i + n^2*1676i
+ n^3*493i + n^4*64i + n^5*3i + 1200i)*1i)/(16*d*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)
)

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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)**5*sin(d*x+c)**n*(a+a*sin(d*x+c)),x)

[Out]

Timed out

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