3.700 \(\int \frac {\cos ^7(c+d x) \sin ^n(c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=137 \[ \frac {\sin ^{n+1}(c+d x)}{a d (n+1)}-\frac {\sin ^{n+2}(c+d x)}{a d (n+2)}-\frac {2 \sin ^{n+3}(c+d x)}{a d (n+3)}+\frac {2 \sin ^{n+4}(c+d x)}{a d (n+4)}+\frac {\sin ^{n+5}(c+d x)}{a d (n+5)}-\frac {\sin ^{n+6}(c+d x)}{a d (n+6)} \]

[Out]

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

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Rubi [A]  time = 0.16, antiderivative size = 137, 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} \[ \frac {\sin ^{n+1}(c+d x)}{a d (n+1)}-\frac {\sin ^{n+2}(c+d x)}{a d (n+2)}-\frac {2 \sin ^{n+3}(c+d x)}{a d (n+3)}+\frac {2 \sin ^{n+4}(c+d x)}{a d (n+4)}+\frac {\sin ^{n+5}(c+d x)}{a d (n+5)}-\frac {\sin ^{n+6}(c+d x)}{a d (n+6)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.33, size = 95, normalized size = 0.69 \[ \frac {\sin ^{n+1}(c+d x) \left (-\frac {\sin ^5(c+d x)}{n+6}+\frac {\sin ^4(c+d x)}{n+5}+\frac {2 \sin ^3(c+d x)}{n+4}-\frac {2 \sin ^2(c+d x)}{n+3}-\frac {\sin (c+d x)}{n+2}+\frac {1}{n+1}\right )}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 243, normalized size = 1.77 \[ \frac {{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} \cos \left (d x + c\right )^{6} - {\left (n^{5} + 11 \, n^{4} + 41 \, n^{3} + 61 \, n^{2} + 30 \, n\right )} \cos \left (d x + c\right )^{4} - 8 \, n^{3} - 4 \, {\left (n^{4} + 9 \, n^{3} + 23 \, n^{2} + 15 \, n\right )} \cos \left (d x + c\right )^{2} - 72 \, n^{2} + {\left ({\left (n^{5} + 16 \, n^{4} + 95 \, n^{3} + 260 \, n^{2} + 324 \, n + 144\right )} \cos \left (d x + c\right )^{4} + 8 \, n^{3} + 4 \, {\left (n^{4} + 13 \, n^{3} + 56 \, n^{2} + 92 \, n + 48\right )} \cos \left (d x + c\right )^{2} + 96 \, n^{2} + 352 \, n + 384\right )} \sin \left (d x + c\right ) - 184 \, n - 120\right )} \sin \left (d x + c\right )^{n}}{a d n^{6} + 21 \, a d n^{5} + 175 \, a d n^{4} + 735 \, a d n^{3} + 1624 \, a d n^{2} + 1764 \, a d n + 720 \, a d} \]

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)),x, algorithm="fricas")

[Out]

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

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

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)),x, algorithm="giac")

[Out]

integrate(sin(d*x + c)^n*cos(d*x + c)^7/(a*sin(d*x + c) + a), x)

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maple [F]  time = 9.34, size = 0, normalized size = 0.00 \[ \int \frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right )}{a +a \sin \left (d x +c \right )}\, dx \]

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)),x)

[Out]

int(cos(d*x+c)^7*sin(d*x+c)^n/(a+a*sin(d*x+c)),x)

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maxima [A]  time = 0.37, size = 241, normalized size = 1.76 \[ -\frac {{\left ({\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} \sin \left (d x + c\right )^{6} - {\left (n^{5} + 16 \, n^{4} + 95 \, n^{3} + 260 \, n^{2} + 324 \, n + 144\right )} \sin \left (d x + c\right )^{5} - 2 \, {\left (n^{5} + 17 \, n^{4} + 107 \, n^{3} + 307 \, n^{2} + 396 \, n + 180\right )} \sin \left (d x + c\right )^{4} + 2 \, {\left (n^{5} + 18 \, n^{4} + 121 \, n^{3} + 372 \, n^{2} + 508 \, n + 240\right )} \sin \left (d x + c\right )^{3} + {\left (n^{5} + 19 \, n^{4} + 137 \, n^{3} + 461 \, n^{2} + 702 \, n + 360\right )} \sin \left (d x + c\right )^{2} - {\left (n^{5} + 20 \, n^{4} + 155 \, n^{3} + 580 \, n^{2} + 1044 \, n + 720\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{{\left (n^{6} + 21 \, n^{5} + 175 \, n^{4} + 735 \, n^{3} + 1624 \, n^{2} + 1764 \, n + 720\right )} a d} \]

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)),x, algorithm="maxima")

[Out]

-((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*sin(d*x + c)^6 - (n^5 + 16*n^4 + 95*n^3 + 260*n^2 + 324*n +
144)*sin(d*x + c)^5 - 2*(n^5 + 17*n^4 + 107*n^3 + 307*n^2 + 396*n + 180)*sin(d*x + c)^4 + 2*(n^5 + 18*n^4 + 12
1*n^3 + 372*n^2 + 508*n + 240)*sin(d*x + c)^3 + (n^5 + 19*n^4 + 137*n^3 + 461*n^2 + 702*n + 360)*sin(d*x + c)^
2 - (n^5 + 20*n^4 + 155*n^3 + 580*n^2 + 1044*n + 720)*sin(d*x + c))*sin(d*x + c)^n/((n^6 + 21*n^5 + 175*n^4 +
735*n^3 + 1624*n^2 + 1764*n + 720)*a*d)

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mupad [B]  time = 14.29, size = 568, normalized size = 4.15 \[ \frac {{\sin \left (c+d\,x\right )}^n\,\cos \left (6\,c+6\,d\,x\right )\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )}{32\,a\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {{\sin \left (c+d\,x\right )}^n\,\left (4\,n^5+92\,n^4+948\,n^3+4516\,n^2+8936\,n+5280\right )}{64\,a\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {\sin \left (c+d\,x\right )\,{\sin \left (c+d\,x\right )}^n\,\left (n^5\,4{}\mathrm {i}+n^4\,96{}\mathrm {i}+n^3\,1052{}\mathrm {i}+n^2\,5904{}\mathrm {i}+n\,15504{}\mathrm {i}+14400{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,a\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {{\sin \left (c+d\,x\right )}^n\,\cos \left (4\,c+4\,d\,x\right )\,\left (2\,n^5+46\,n^4+346\,n^3+1106\,n^2+1524\,n+720\right )}{32\,a\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}+\frac {{\sin \left (c+d\,x\right )}^n\,\cos \left (2\,c+2\,d\,x\right )\,\left (-n^5-15\,n^4+43\,n^3+927\,n^2+2670\,n+1800\right )}{32\,a\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {{\sin \left (c+d\,x\right )}^n\,\sin \left (5\,c+5\,d\,x\right )\,\left (n^5\,2{}\mathrm {i}+n^4\,32{}\mathrm {i}+n^3\,190{}\mathrm {i}+n^2\,520{}\mathrm {i}+n\,648{}\mathrm {i}+288{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,a\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )}-\frac {{\sin \left (c+d\,x\right )}^n\,\sin \left (3\,c+3\,d\,x\right )\,\left (n^5\,6{}\mathrm {i}+n^4\,128{}\mathrm {i}+n^3\,986{}\mathrm {i}+n^2\,3352{}\mathrm {i}+n\,4888{}\mathrm {i}+2400{}\mathrm {i}\right )\,1{}\mathrm {i}}{32\,a\,d\,\left (n^6+21\,n^5+175\,n^4+735\,n^3+1624\,n^2+1764\,n+720\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cos(c + d*x)^7*sin(c + d*x)^n)/(a + a*sin(c + d*x)),x)

[Out]

(sin(c + d*x)^n*cos(6*c + 6*d*x)*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))/(32*a*d*(1764*n + 1624*n^2 +
735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) - (sin(c + d*x)^n*(8936*n + 4516*n^2 + 948*n^3 + 92*n^4 + 4*n^5 + 52
80))/(64*a*d*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) - (sin(c + d*x)*sin(c + d*x)^n*(n*1
5504i + n^2*5904i + n^3*1052i + n^4*96i + n^5*4i + 14400i)*1i)/(32*a*d*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4
+ 21*n^5 + n^6 + 720)) + (sin(c + d*x)^n*cos(4*c + 4*d*x)*(1524*n + 1106*n^2 + 346*n^3 + 46*n^4 + 2*n^5 + 720)
)/(32*a*d*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) + (sin(c + d*x)^n*cos(2*c + 2*d*x)*(26
70*n + 927*n^2 + 43*n^3 - 15*n^4 - n^5 + 1800))/(32*a*d*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6
+ 720)) - (sin(c + d*x)^n*sin(5*c + 5*d*x)*(n*648i + n^2*520i + n^3*190i + n^4*32i + n^5*2i + 288i)*1i)/(32*a*
d*(1764*n + 1624*n^2 + 735*n^3 + 175*n^4 + 21*n^5 + n^6 + 720)) - (sin(c + d*x)^n*sin(3*c + 3*d*x)*(n*4888i +
n^2*3352i + n^3*986i + n^4*128i + n^5*6i + 2400i)*1i)/(32*a*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)**7*sin(d*x+c)**n/(a+a*sin(d*x+c)),x)

[Out]

Timed out

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