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[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))

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Rubi [A]  time = 0.119184, antiderivative size = 123, normalized size of antiderivative = 1., 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 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]

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]))

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*}

Mathematica [B]  time = 1.35988, size = 345, normalized size = 2.8 \[ \frac{a \sin ^{n+1}(c+d x) \left (2 n^5 \sin (c+d x)+3 n^5 \sin (3 (c+d x))+n^5 \sin (5 (c+d x))+46 n^4 \sin (c+d x)+61 n^4 \sin (3 (c+d x))+15 n^4 \sin (5 (c+d x))+474 n^3 \sin (c+d x)+431 n^3 \sin (3 (c+d x))+85 n^3 \sin (5 (c+d x))+2258 n^2 \sin (c+d x)+1331 n^2 \sin (3 (c+d x))+225 n^2 \sin (5 (c+d x))+8 \left (n^5+20 n^4+147 n^3+484 n^2+692 n+336\right ) \cos (2 (c+d x))+2 \left (n^5+16 n^4+95 n^3+260 n^2+324 n+144\right ) \cos (4 (c+d x))+4468 n \sin (c+d x)+1798 n \sin (3 (c+d x))+274 n \sin (5 (c+d x))+2640 \sin (c+d x)+840 \sin (3 (c+d x))+120 \sin (5 (c+d x))+6 n^5+128 n^4+1114 n^3+4888 n^2+10520 n+8544\right )}{16 d (n+1) (n+2) (n+3) (n+4) (n+5) (n+6)} \]

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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Maple [F]  time = 4.082, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{5} \left ( \sin \left ( dx+c \right ) \right ) ^{n} \left ( a+a\sin \left ( dx+c \right ) \right ) \, dx \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [B]  time = 1.27655, size = 713, normalized size = 5.8 \begin{align*} -\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} \end{align*}

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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Sympy [A]  time = 173.843, size = 8534, normalized size = 69.38 \begin{align*} \text{result too large to display} \end{align*}

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]

Piecewise((x*(a*sin(c) + a)*sin(c)**n*cos(c)**5, Eq(d, 0)), (a*log(sin(c + d*x))/d - 8*a/(15*d*sin(c + d*x)) +
a*cos(c + d*x)**2/(2*d*sin(c + d*x)**2) + 4*a*cos(c + d*x)**2/(15*d*sin(c + d*x)**3) - a*cos(c + d*x)**4/(4*d
*sin(c + d*x)**4) - a*cos(c + d*x)**4/(5*d*sin(c + d*x)**5), Eq(n, -6)), (a*log(sin(c + d*x))/d + 8*a*sin(c +
d*x)/(3*d) + 4*a*cos(c + d*x)**2/(3*d*sin(c + d*x)) + a*cos(c + d*x)**2/(2*d*sin(c + d*x)**2) - a*cos(c + d*x)
**4/(3*d*sin(c + d*x)**3) - a*cos(c + d*x)**4/(4*d*sin(c + d*x)**4), Eq(n, -5)), (48*a*log(tan(c/2 + d*x/2)**2
+ 1)*tan(c/2 + d*x/2)**7/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 9
6*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**5/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24
*d*tan(c/2 + d*x/2)**3) + 48*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**3/(24*d*tan(c/2 + d*x/2)**7 + 48
*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 48*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**7/(24*d*tan(
c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 96*a*log(tan(c/2 + d*x/2))*tan(c/2 +
d*x/2)**5/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 48*a*log(tan(c/2
+ d*x/2))*tan(c/2 + d*x/2)**3/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3)
- a*tan(c/2 + d*x/2)**10/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 3
*a*tan(c/2 + d*x/2)**9/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 19*a
*tan(c/2 + d*x/2)**8/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 27*a*t
an(c/2 + d*x/2)**7/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 110*a*ta
n(c/2 + d*x/2)**6/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 110*a*tan
(c/2 + d*x/2)**4/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 27*a*tan(c
/2 + d*x/2)**3/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) + 19*a*tan(c/2
+ d*x/2)**2/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - 3*a*tan(c/2 +
d*x/2)/(24*d*tan(c/2 + d*x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3) - a/(24*d*tan(c/2 + d*
x/2)**7 + 48*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3), Eq(n, -4)), (48*a*log(tan(c/2 + d*x/2)**2 + 1)
*tan(c/2 + d*x/2)**8/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*ta
n(c/2 + d*x/2)**2) + 144*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**6/(24*d*tan(c/2 + d*x/2)**8 + 72*d*t
an(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) + 144*a*log(tan(c/2 + d*x/2)**2 + 1)
*tan(c/2 + d*x/2)**4/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*ta
n(c/2 + d*x/2)**2) + 48*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**2/(24*d*tan(c/2 + d*x/2)**8 + 72*d*ta
n(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 48*a*log(tan(c/2 + d*x/2))*tan(c/2
+ d*x/2)**8/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d
*x/2)**2) - 144*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**6/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)*
*6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 144*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**4/(2
4*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 48
*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c
/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 3*a*tan(c/2 + d*x/2)**10/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2
+ d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 12*a*tan(c/2 + d*x/2)**9/(24*d*tan(c/2 +
d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 144*a*tan(c/2 +
d*x/2)**7/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*
x/2)**2) + 63*a*tan(c/2 + d*x/2)**6/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/
2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 200*a*tan(c/2 + d*x/2)**5/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2
)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) + 63*a*tan(c/2 + d*x/2)**4/(24*d*tan(c/2 + d*x/2)*
*8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2) - 144*a*tan(c/2 + d*x/2)*
*3/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*d*tan(c/2 + d*x/2)**2)
- 12*a*tan(c/2 + d*x/2)/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 + 24*
d*tan(c/2 + d*x/2)**2) - 3*a/(24*d*tan(c/2 + d*x/2)**8 + 72*d*tan(c/2 + d*x/2)**6 + 72*d*tan(c/2 + d*x/2)**4 +
24*d*tan(c/2 + d*x/2)**2), Eq(n, -3)), (-6*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**9/(6*d*tan(c/2 +
d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x
/2)) - 24*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**7/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)*
*7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 36*a*log(tan(c/2 + d*x/2)**
2 + 1)*tan(c/2 + d*x/2)**5/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24
*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 24*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**3/(6*d*ta
n(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c
/2 + d*x/2)) - 6*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x
/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) + 6*a*log(tan(c/2 + d*x/2
))*tan(c/2 + d*x/2)**9/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*t
an(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) + 24*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**7/(6*d*tan(c/2 + d*x
/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)
) + 36*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**5/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*
tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) + 24*a*log(tan(c/2 + d*x/2))*tan(c/2 +
d*x/2)**3/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/
2)**3 + 6*d*tan(c/2 + d*x/2)) + 6*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan
(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 3*a*tan(c/2 +
d*x/2)**10/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*
x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 39*a*tan(c/2 + d*x/2)**8/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7
+ 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 24*a*tan(c/2 + d*x/2)**7/(6*d
*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*ta
n(c/2 + d*x/2)) - 86*a*tan(c/2 + d*x/2)**6/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2
+ d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 24*a*tan(c/2 + d*x/2)**5/(6*d*tan(c/2 + d*x/2
)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2))
- 86*a*tan(c/2 + d*x/2)**4/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24
*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 24*a*tan(c/2 + d*x/2)**3/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(
c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)) - 39*a*tan(c/2 +
d*x/2)**2/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/2)**5 + 24*d*tan(c/2 + d*x
/2)**3 + 6*d*tan(c/2 + d*x/2)) - 3*a/(6*d*tan(c/2 + d*x/2)**9 + 24*d*tan(c/2 + d*x/2)**7 + 36*d*tan(c/2 + d*x/
2)**5 + 24*d*tan(c/2 + d*x/2)**3 + 6*d*tan(c/2 + d*x/2)), Eq(n, -2)), (-15*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(
c/2 + d*x/2)**10/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan
(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) - 75*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**8/(1
5*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 +
75*d*tan(c/2 + d*x/2)**2 + 15*d) - 150*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**6/(15*d*tan(c/2 + d*x/
2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x
/2)**2 + 15*d) - 150*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**4/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(
c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) - 7
5*a*log(tan(c/2 + d*x/2)**2 + 1)*tan(c/2 + d*x/2)**2/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 1
50*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) - 15*a*log(tan(c/2 + d
*x/2)**2 + 1)/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/
2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) + 15*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**10/(15*d*tan(c
/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(
c/2 + d*x/2)**2 + 15*d) + 75*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**8/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan
(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) +
150*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**6/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*
tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) + 150*a*log(tan(c/2 + d*x/2
))*tan(c/2 + d*x/2)**4/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150
*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) + 75*a*log(tan(c/2 + d*x/2))*tan(c/2 + d*x/2)**2/(15
*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 7
5*d*tan(c/2 + d*x/2)**2 + 15*d) + 15*a*log(tan(c/2 + d*x/2))/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2
)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) + 12*a*tan(c/2
+ d*x/2)**10/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/
2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) + 30*a*tan(c/2 + d*x/2)**9/(15*d*tan(c/2 + d*x/2)**10 + 75*d*
tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d)
+ 40*a*tan(c/2 + d*x/2)**7/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6
+ 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) + 116*a*tan(c/2 + d*x/2)**5/(15*d*tan(c/2 + d*x
/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*
x/2)**2 + 15*d) + 40*a*tan(c/2 + d*x/2)**3/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c
/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d) + 30*a*tan(c/2 + d*x/2)/(15*d*ta
n(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*t
an(c/2 + d*x/2)**2 + 15*d) + 12*a/(15*d*tan(c/2 + d*x/2)**10 + 75*d*tan(c/2 + d*x/2)**8 + 150*d*tan(c/2 + d*x/
2)**6 + 150*d*tan(c/2 + d*x/2)**4 + 75*d*tan(c/2 + d*x/2)**2 + 15*d), Eq(n, -1)), (a*n**5*sin(c + d*x)**2*sin(
c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) +
a*n**5*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**
2 + 1764*d*n + 720*d) + 4*a*n**4*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**6 + 21*d*n**5 + 175*d*n
**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 4*a*n**4*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**2/
(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 19*a*n**4*sin(c + d*x)**2*si
n(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d)
+ 20*a*n**4*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*
d*n**2 + 1764*d*n + 720*d) + 8*a*n**3*sin(c + d*x)**6*sin(c + d*x)**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) + 8*a*n**3*sin(c + d*x)**5*sin(c + d*x)**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) + 60*a*n**3*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)*
*2/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 68*a*n**3*sin(c + d*x)**3
*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*
d) + 137*a*n**3*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3
+ 1624*d*n**2 + 1764*d*n + 720*d) + 155*a*n**3*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**
5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 72*a*n**2*sin(c + d*x)**6*sin(c + d*x)**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) + 96*a*n**2*sin(c + d*x)**5*sin(c
+ d*x)**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) + 308*a*n**2*sin(c +
d*x)**4*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*
n + 720*d) + 416*a*n**2*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735
*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 461*a*n**2*sin(c + d*x)**2*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6
+ 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 580*a*n**2*sin(c + d*x)*sin(c + d*x
)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 184*a*n
*sin(c + d*x)**6*sin(c + d*x)**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) + 352*a*n*sin(c + d*x)**5*sin(c + d*x)**n/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 176
4*d*n + 720*d) + 612*a*n*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 73
5*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 1072*a*n*sin(c + d*x)**3*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**6
+ 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 702*a*n*sin(c + d*x)**2*sin(c + d*x)
**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 1044*a*n
*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 17
64*d*n + 720*d) + 120*a*sin(c + d*x)**6*sin(c + d*x)**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) + 384*a*sin(c + d*x)**5*sin(c + d*x)**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) + 360*a*sin(c + d*x)**4*sin(c + d*x)**n*cos(c + d*x)**2/(d*n**6 + 21*d*n**
5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 960*a*sin(c + d*x)**3*sin(c + d*x)**n*cos(c +
d*x)**2/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720*d) + 360*a*sin(c + d*x)**
2*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*d*n**2 + 1764*d*n + 720
*d) + 720*a*sin(c + d*x)*sin(c + d*x)**n*cos(c + d*x)**4/(d*n**6 + 21*d*n**5 + 175*d*n**4 + 735*d*n**3 + 1624*
d*n**2 + 1764*d*n + 720*d), True))

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Giac [B]  time = 1.14342, size = 512, normalized size = 4.16 \begin{align*} \frac{\frac{{\left (n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} + 6 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} - 2 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 8 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{6} - 16 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} - 24 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 10 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 24 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2}\right )} a}{n^{3} + 12 \, n^{2} + 44 \, n + 48} + \frac{{\left (n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 4 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} - 2 \, n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 3 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} - 12 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) - 10 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 8 \, n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 15 \, \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )\right )} a}{n^{3} + 9 \, n^{2} + 23 \, n + 15}}{d} \end{align*}

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]

((n^2*sin(d*x + c)^n*sin(d*x + c)^6 + 6*n*sin(d*x + c)^n*sin(d*x + c)^6 - 2*n^2*sin(d*x + c)^n*sin(d*x + c)^4
+ 8*sin(d*x + c)^n*sin(d*x + c)^6 - 16*n*sin(d*x + c)^n*sin(d*x + c)^4 + n^2*sin(d*x + c)^n*sin(d*x + c)^2 - 2
4*sin(d*x + c)^n*sin(d*x + c)^4 + 10*n*sin(d*x + c)^n*sin(d*x + c)^2 + 24*sin(d*x + c)^n*sin(d*x + c)^2)*a/(n^
3 + 12*n^2 + 44*n + 48) + (n^2*sin(d*x + c)^n*sin(d*x + c)^5 + 4*n*sin(d*x + c)^n*sin(d*x + c)^5 - 2*n^2*sin(d
*x + c)^n*sin(d*x + c)^3 + 3*sin(d*x + c)^n*sin(d*x + c)^5 - 12*n*sin(d*x + c)^n*sin(d*x + c)^3 + n^2*sin(d*x
+ c)^n*sin(d*x + c) - 10*sin(d*x + c)^n*sin(d*x + c)^3 + 8*n*sin(d*x + c)^n*sin(d*x + c) + 15*sin(d*x + c)^n*s
in(d*x + c))*a/(n^3 + 9*n^2 + 23*n + 15))/d