3.566 \(\int \cos ^5(c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=160 \[ \frac {a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {2 a^2 \sin ^{n+2}(c+d x)}{d (n+2)}-\frac {a^2 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac {4 a^2 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac {a^2 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac {2 a^2 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac {a^2 \sin ^{n+7}(c+d x)}{d (n+7)} \]

[Out]

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

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Rubi [A]  time = 0.17, antiderivative size = 160, 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} \[ \frac {a^2 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {2 a^2 \sin ^{n+2}(c+d x)}{d (n+2)}-\frac {a^2 \sin ^{n+3}(c+d x)}{d (n+3)}-\frac {4 a^2 \sin ^{n+4}(c+d x)}{d (n+4)}-\frac {a^2 \sin ^{n+5}(c+d x)}{d (n+5)}+\frac {2 a^2 \sin ^{n+6}(c+d x)}{d (n+6)}+\frac {a^2 \sin ^{n+7}(c+d x)}{d (n+7)} \]

Antiderivative was successfully verified.

[In]

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

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Mathematica [A]  time = 0.39, size = 110, normalized size = 0.69 \[ \frac {a^2 \sin ^{n+1}(c+d x) \left (\frac {\sin ^6(c+d x)}{n+7}+\frac {2 \sin ^5(c+d x)}{n+6}-\frac {\sin ^4(c+d x)}{n+5}-\frac {4 \sin ^3(c+d x)}{n+4}-\frac {\sin ^2(c+d x)}{n+3}+\frac {2 \sin (c+d x)}{n+2}+\frac {1}{n+1}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

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fricas [B]  time = 0.83, size = 473, normalized size = 2.96 \[ -\frac {{\left (2 \, {\left (a^{2} n^{6} + 22 \, a^{2} n^{5} + 190 \, a^{2} n^{4} + 820 \, a^{2} n^{3} + 1849 \, a^{2} n^{2} + 2038 \, a^{2} n + 840 \, a^{2}\right )} \cos \left (d x + c\right )^{6} - 16 \, a^{2} n^{4} - 256 \, a^{2} n^{3} - 2 \, {\left (a^{2} n^{6} + 18 \, a^{2} n^{5} + 118 \, a^{2} n^{4} + 348 \, a^{2} n^{3} + 457 \, a^{2} n^{2} + 210 \, a^{2} n\right )} \cos \left (d x + c\right )^{4} - 1376 \, a^{2} n^{2} - 2816 \, a^{2} n - 8 \, {\left (a^{2} n^{5} + 16 \, a^{2} n^{4} + 86 \, a^{2} n^{3} + 176 \, a^{2} n^{2} + 105 \, a^{2} n\right )} \cos \left (d x + c\right )^{2} - 1680 \, a^{2} + {\left ({\left (a^{2} n^{6} + 21 \, a^{2} n^{5} + 175 \, a^{2} n^{4} + 735 \, a^{2} n^{3} + 1624 \, a^{2} n^{2} + 1764 \, a^{2} n + 720 \, a^{2}\right )} \cos \left (d x + c\right )^{6} - 16 \, a^{2} n^{4} - 256 \, a^{2} n^{3} - 2 \, {\left (a^{2} n^{6} + 20 \, a^{2} n^{5} + 159 \, a^{2} n^{4} + 640 \, a^{2} n^{3} + 1364 \, a^{2} n^{2} + 1440 \, a^{2} n + 576 \, a^{2}\right )} \cos \left (d x + c\right )^{4} - 1472 \, a^{2} n^{2} - 3584 \, a^{2} n - 8 \, {\left (a^{2} n^{5} + 17 \, a^{2} n^{4} + 108 \, a^{2} n^{3} + 316 \, a^{2} n^{2} + 416 \, a^{2} n + 192 \, a^{2}\right )} \cos \left (d x + c\right )^{2} - 3072 \, a^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{7} + 28 \, d n^{6} + 322 \, d n^{5} + 1960 \, d n^{4} + 6769 \, d n^{3} + 13132 \, d n^{2} + 13068 \, d n + 5040 \, d} \]

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))^2,x, algorithm="fricas")

[Out]

-(2*(a^2*n^6 + 22*a^2*n^5 + 190*a^2*n^4 + 820*a^2*n^3 + 1849*a^2*n^2 + 2038*a^2*n + 840*a^2)*cos(d*x + c)^6 -
16*a^2*n^4 - 256*a^2*n^3 - 2*(a^2*n^6 + 18*a^2*n^5 + 118*a^2*n^4 + 348*a^2*n^3 + 457*a^2*n^2 + 210*a^2*n)*cos(
d*x + c)^4 - 1376*a^2*n^2 - 2816*a^2*n - 8*(a^2*n^5 + 16*a^2*n^4 + 86*a^2*n^3 + 176*a^2*n^2 + 105*a^2*n)*cos(d
*x + c)^2 - 1680*a^2 + ((a^2*n^6 + 21*a^2*n^5 + 175*a^2*n^4 + 735*a^2*n^3 + 1624*a^2*n^2 + 1764*a^2*n + 720*a^
2)*cos(d*x + c)^6 - 16*a^2*n^4 - 256*a^2*n^3 - 2*(a^2*n^6 + 20*a^2*n^5 + 159*a^2*n^4 + 640*a^2*n^3 + 1364*a^2*
n^2 + 1440*a^2*n + 576*a^2)*cos(d*x + c)^4 - 1472*a^2*n^2 - 3584*a^2*n - 8*(a^2*n^5 + 17*a^2*n^4 + 108*a^2*n^3
+ 316*a^2*n^2 + 416*a^2*n + 192*a^2)*cos(d*x + c)^2 - 3072*a^2)*sin(d*x + c))*sin(d*x + c)^n/(d*n^7 + 28*d*n^
6 + 322*d*n^5 + 1960*d*n^4 + 6769*d*n^3 + 13132*d*n^2 + 13068*d*n + 5040*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))^2,x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 16.81, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{2}\, dx \]

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))^2,x)

[Out]

int(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x)

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maxima [A]  time = 0.97, size = 143, normalized size = 0.89 \[ \frac {\frac {a^{2} \sin \left (d x + c\right )^{n + 7}}{n + 7} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 6}}{n + 6} - \frac {a^{2} \sin \left (d x + c\right )^{n + 5}}{n + 5} - \frac {4 \, a^{2} \sin \left (d x + c\right )^{n + 4}}{n + 4} - \frac {a^{2} \sin \left (d x + c\right )^{n + 3}}{n + 3} + \frac {2 \, a^{2} \sin \left (d x + c\right )^{n + 2}}{n + 2} + \frac {a^{2} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*sin(d*x+c)^n*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

(a^2*sin(d*x + c)^(n + 7)/(n + 7) + 2*a^2*sin(d*x + c)^(n + 6)/(n + 6) - a^2*sin(d*x + c)^(n + 5)/(n + 5) - 4*
a^2*sin(d*x + c)^(n + 4)/(n + 4) - 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 = 14.66, size = 819, normalized size = 5.12 \[ \frac {a^2\,{\sin \left (c+d\,x\right )}^n\,\left (n^6\,1{}\mathrm {i}+n^5\,30{}\mathrm {i}+n^4\,398{}\mathrm {i}+n^3\,2788{}\mathrm {i}+n^2\,10137{}\mathrm {i}+n\,16958{}\mathrm {i}+9240{}\mathrm {i}\right )}{8\,d\,\left (n^7\,1{}\mathrm {i}+n^6\,28{}\mathrm {i}+n^5\,322{}\mathrm {i}+n^4\,1960{}\mathrm {i}+n^3\,6769{}\mathrm {i}+n^2\,13132{}\mathrm {i}+n\,13068{}\mathrm {i}+5040{}\mathrm {i}\right )}-\frac {a^2\,{\sin \left (c+d\,x\right )}^n\,\sin \left (7\,c+7\,d\,x\right )\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )\,1{}\mathrm {i}}{64\,d\,\left (n^7\,1{}\mathrm {i}+n^6\,28{}\mathrm {i}+n^5\,322{}\mathrm {i}+n^4\,1960{}\mathrm {i}+n^3\,6769{}\mathrm {i}+n^2\,13132{}\mathrm {i}+n\,13068{}\mathrm {i}+5040{}\mathrm {i}\right )}+\frac {a^2\,\sin \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^n\,\left (11\,n^6+343\,n^5+4869\,n^4+36773\,n^3+148360\,n^2+296844\,n+226800\right )\,1{}\mathrm {i}}{64\,d\,\left (n^7\,1{}\mathrm {i}+n^6\,28{}\mathrm {i}+n^5\,322{}\mathrm {i}+n^4\,1960{}\mathrm {i}+n^3\,6769{}\mathrm {i}+n^2\,13132{}\mathrm {i}+n\,13068{}\mathrm {i}+5040{}\mathrm {i}\right )}-\frac {a^2\,{\sin \left (c+d\,x\right )}^n\,\cos \left (6\,c+6\,d\,x\right )\,\left (n^6\,1{}\mathrm {i}+n^5\,22{}\mathrm {i}+n^4\,190{}\mathrm {i}+n^3\,820{}\mathrm {i}+n^2\,1849{}\mathrm {i}+n\,2038{}\mathrm {i}+840{}\mathrm {i}\right )}{16\,d\,\left (n^7\,1{}\mathrm {i}+n^6\,28{}\mathrm {i}+n^5\,322{}\mathrm {i}+n^4\,1960{}\mathrm {i}+n^3\,6769{}\mathrm {i}+n^2\,13132{}\mathrm {i}+n\,13068{}\mathrm {i}+5040{}\mathrm {i}\right )}-\frac {a^2\,{\sin \left (c+d\,x\right )}^n\,\cos \left (4\,c+4\,d\,x\right )\,\left (n^6\,1{}\mathrm {i}+n^5\,30{}\mathrm {i}+n^4\,334{}\mathrm {i}+n^3\,1764{}\mathrm {i}+n^2\,4633{}\mathrm {i}+n\,5694{}\mathrm {i}+2520{}\mathrm {i}\right )}{8\,d\,\left (n^7\,1{}\mathrm {i}+n^6\,28{}\mathrm {i}+n^5\,322{}\mathrm {i}+n^4\,1960{}\mathrm {i}+n^3\,6769{}\mathrm {i}+n^2\,13132{}\mathrm {i}+n\,13068{}\mathrm {i}+5040{}\mathrm {i}\right )}-\frac {a^2\,{\sin \left (c+d\,x\right )}^n\,\cos \left (2\,c+2\,d\,x\right )\,\left (-n^6\,1{}\mathrm {i}-n^5\,22{}\mathrm {i}-n^4\,62{}\mathrm {i}+n^3\,1228{}\mathrm {i}+n^2\,9159{}\mathrm {i}+n\,20490{}\mathrm {i}+12600{}\mathrm {i}\right )}{16\,d\,\left (n^7\,1{}\mathrm {i}+n^6\,28{}\mathrm {i}+n^5\,322{}\mathrm {i}+n^4\,1960{}\mathrm {i}+n^3\,6769{}\mathrm {i}+n^2\,13132{}\mathrm {i}+n\,13068{}\mathrm {i}+5040{}\mathrm {i}\right )}+\frac {a^2\,{\sin \left (c+d\,x\right )}^n\,\sin \left (5\,c+5\,d\,x\right )\,\left (3\,n^6+55\,n^5+397\,n^4+1445\,n^3+2792\,n^2+2700\,n+1008\right )\,1{}\mathrm {i}}{64\,d\,\left (n^7\,1{}\mathrm {i}+n^6\,28{}\mathrm {i}+n^5\,322{}\mathrm {i}+n^4\,1960{}\mathrm {i}+n^3\,6769{}\mathrm {i}+n^2\,13132{}\mathrm {i}+n\,13068{}\mathrm {i}+5040{}\mathrm {i}\right )}+\frac {a^2\,{\sin \left (c+d\,x\right )}^n\,\sin \left (3\,c+3\,d\,x\right )\,\left (15\,n^6+419\,n^5+4417\,n^4+22569\,n^3+58568\,n^2+71932\,n+31920\right )\,1{}\mathrm {i}}{64\,d\,\left (n^7\,1{}\mathrm {i}+n^6\,28{}\mathrm {i}+n^5\,322{}\mathrm {i}+n^4\,1960{}\mathrm {i}+n^3\,6769{}\mathrm {i}+n^2\,13132{}\mathrm {i}+n\,13068{}\mathrm {i}+5040{}\mathrm {i}\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))^2,x)

[Out]

(a^2*sin(c + d*x)^n*(n*16958i + n^2*10137i + n^3*2788i + n^4*398i + n^5*30i + n^6*1i + 9240i))/(8*d*(n*13068i
+ n^2*13132i + n^3*6769i + n^4*1960i + n^5*322i + n^6*28i + n^7*1i + 5040i)) - (a^2*sin(c + d*x)^n*sin(7*c + 7
*d*x)*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)*1i)/(64*d*(n*13068i + n^2*13132i + n^3*6769
i + n^4*1960i + n^5*322i + n^6*28i + n^7*1i + 5040i)) + (a^2*sin(c + d*x)*sin(c + d*x)^n*(296844*n + 148360*n^
2 + 36773*n^3 + 4869*n^4 + 343*n^5 + 11*n^6 + 226800)*1i)/(64*d*(n*13068i + n^2*13132i + n^3*6769i + n^4*1960i
+ n^5*322i + n^6*28i + n^7*1i + 5040i)) - (a^2*sin(c + d*x)^n*cos(6*c + 6*d*x)*(n*2038i + n^2*1849i + n^3*820
i + n^4*190i + n^5*22i + n^6*1i + 840i))/(16*d*(n*13068i + n^2*13132i + n^3*6769i + n^4*1960i + n^5*322i + n^6
*28i + n^7*1i + 5040i)) - (a^2*sin(c + d*x)^n*cos(4*c + 4*d*x)*(n*5694i + n^2*4633i + n^3*1764i + n^4*334i + n
^5*30i + n^6*1i + 2520i))/(8*d*(n*13068i + n^2*13132i + n^3*6769i + n^4*1960i + n^5*322i + n^6*28i + n^7*1i +
5040i)) - (a^2*sin(c + d*x)^n*cos(2*c + 2*d*x)*(n*20490i + n^2*9159i + n^3*1228i - n^4*62i - n^5*22i - n^6*1i
+ 12600i))/(16*d*(n*13068i + n^2*13132i + n^3*6769i + n^4*1960i + n^5*322i + n^6*28i + n^7*1i + 5040i)) + (a^2
*sin(c + d*x)^n*sin(5*c + 5*d*x)*(2700*n + 2792*n^2 + 1445*n^3 + 397*n^4 + 55*n^5 + 3*n^6 + 1008)*1i)/(64*d*(n
*13068i + n^2*13132i + n^3*6769i + n^4*1960i + n^5*322i + n^6*28i + n^7*1i + 5040i)) + (a^2*sin(c + d*x)^n*sin
(3*c + 3*d*x)*(71932*n + 58568*n^2 + 22569*n^3 + 4417*n^4 + 419*n^5 + 15*n^6 + 31920)*1i)/(64*d*(n*13068i + n^
2*13132i + n^3*6769i + n^4*1960i + n^5*322i + n^6*28i + n^7*1i + 5040i))

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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))**2,x)

[Out]

Timed out

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