\(\int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^4 \, dx\) [258]

Optimal result

Integrand size = 27, antiderivative size = 114 \[ \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {4 a^4 \sin ^{2+n}(c+d x)}{d (2+n)}+\frac {6 a^4 \sin ^{3+n}(c+d x)}{d (3+n)}+\frac {4 a^4 \sin ^{4+n}(c+d x)}{d (4+n)}+\frac {a^4 \sin ^{5+n}(c+d x)}{d (5+n)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2912, 45} \[ \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {4 a^4 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac {6 a^4 \sin ^{n+3}(c+d x)}{d (n+3)}+\frac {4 a^4 \sin ^{n+4}(c+d x)}{d (n+4)}+\frac {a^4 \sin ^{n+5}(c+d x)}{d (n+5)} \]

[In]

Int[Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^4,x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2912

Int[cos[(e_.) + (f_.)*(x_)]*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)
])^(n_.), x_Symbol] :> Dist[1/(b*f), Subst[Int[(a + x)^m*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (\frac {x}{a}\right )^n (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {\text {Subst}\left (\int \left (a^4 \left (\frac {x}{a}\right )^n+4 a^4 \left (\frac {x}{a}\right )^{1+n}+6 a^4 \left (\frac {x}{a}\right )^{2+n}+4 a^4 \left (\frac {x}{a}\right )^{3+n}+a^4 \left (\frac {x}{a}\right )^{4+n}\right ) \, dx,x,a \sin (c+d x)\right )}{a d} \\ & = \frac {a^4 \sin ^{1+n}(c+d x)}{d (1+n)}+\frac {4 a^4 \sin ^{2+n}(c+d x)}{d (2+n)}+\frac {6 a^4 \sin ^{3+n}(c+d x)}{d (3+n)}+\frac {4 a^4 \sin ^{4+n}(c+d x)}{d (4+n)}+\frac {a^4 \sin ^{5+n}(c+d x)}{d (5+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.70 \[ \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4 \sin ^{1+n}(c+d x) \left (\frac {1}{1+n}+\frac {4 \sin (c+d x)}{2+n}+\frac {6 \sin ^2(c+d x)}{3+n}+\frac {4 \sin ^3(c+d x)}{4+n}+\frac {\sin ^4(c+d x)}{5+n}\right )}{d} \]

[In]

Integrate[Cos[c + d*x]*Sin[c + d*x]^n*(a + a*Sin[c + d*x])^4,x]

[Out]

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

Maple [A] (verified)

Time = 3.78 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.34

method	result	size
derivativedivides	\(\frac {a^{4} \sin \left (d x +c \right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (1+n \right )}+\frac {a^{4} \left (\sin ^{5}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (5+n \right )}+\frac {4 a^{4} \left (\sin ^{2}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (2+n \right )}+\frac {6 a^{4} \left (\sin ^{3}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (3+n \right )}+\frac {4 a^{4} \left (\sin ^{4}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (4+n \right )}\)	\(153\)
default	\(\frac {a^{4} \sin \left (d x +c \right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (1+n \right )}+\frac {a^{4} \left (\sin ^{5}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (5+n \right )}+\frac {4 a^{4} \left (\sin ^{2}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (2+n \right )}+\frac {6 a^{4} \left (\sin ^{3}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (3+n \right )}+\frac {4 a^{4} \left (\sin ^{4}\left (d x +c \right )\right ) {\mathrm e}^{n \ln \left (\sin \left (d x +c \right )\right )}}{d \left (4+n \right )}\)	\(153\)
parallelrisch	\(\frac {\left (2640-64 \left (5+n \right ) \left (1+n \right ) \left (3+n \right )^{2} \cos \left (2 d x +2 c \right )+8 \left (n^{4}+11 n^{3}+41 n^{2}+61 n +30\right ) \cos \left (4 d x +4 c \right )+\left (-29 n^{4}-338 n^{3}-1351 n^{2}-2122 n -1080\right ) \sin \left (3 d x +3 c \right )+\left (n^{4}+10 n^{3}+35 n^{2}+50 n +24\right ) \sin \left (5 d x +5 c \right )+2 \left (49 n^{4}+594 n^{3}+2507 n^{2}+4290 n +2520\right ) \sin \left (d x +c \right )+56 n^{4}+680 n^{3}+2872 n^{2}+4888 n \right ) \left (\sin ^{n}\left (d x +c \right )\right ) a^{4}}{16 \left (5+n \right ) \left (3+n \right ) \left (1+n \right ) \left (2+n \right ) d \left (4+n \right )}\)	\(199\)

[In]

int(cos(d*x+c)*sin(d*x+c)^n*(a+a*sin(d*x+c))^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (114) = 228\).

Time = 0.29 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.65 \[ \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {{\left (8 \, a^{4} n^{4} + 96 \, a^{4} n^{3} + 400 \, a^{4} n^{2} + 672 \, a^{4} n + 4 \, {\left (a^{4} n^{4} + 11 \, a^{4} n^{3} + 41 \, a^{4} n^{2} + 61 \, a^{4} n + 30 \, a^{4}\right )} \cos \left (d x + c\right )^{4} + 360 \, a^{4} - 4 \, {\left (3 \, a^{4} n^{4} + 35 \, a^{4} n^{3} + 141 \, a^{4} n^{2} + 229 \, a^{4} n + 120 \, a^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (8 \, a^{4} n^{4} + 96 \, a^{4} n^{3} + 400 \, a^{4} n^{2} + 672 \, a^{4} n + {\left (a^{4} n^{4} + 10 \, a^{4} n^{3} + 35 \, a^{4} n^{2} + 50 \, a^{4} n + 24 \, a^{4}\right )} \cos \left (d x + c\right )^{4} + 384 \, a^{4} - 4 \, {\left (2 \, a^{4} n^{4} + 23 \, a^{4} n^{3} + 91 \, a^{4} n^{2} + 142 \, a^{4} n + 72 \, a^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{5} + 15 \, d n^{4} + 85 \, d n^{3} + 225 \, d n^{2} + 274 \, d n + 120 \, d} \]

[In]

integrate(cos(d*x+c)*sin(d*x+c)^n*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

(8*a^4*n^4 + 96*a^4*n^3 + 400*a^4*n^2 + 672*a^4*n + 4*(a^4*n^4 + 11*a^4*n^3 + 41*a^4*n^2 + 61*a^4*n + 30*a^4)*
cos(d*x + c)^4 + 360*a^4 - 4*(3*a^4*n^4 + 35*a^4*n^3 + 141*a^4*n^2 + 229*a^4*n + 120*a^4)*cos(d*x + c)^2 + (8*
a^4*n^4 + 96*a^4*n^3 + 400*a^4*n^2 + 672*a^4*n + (a^4*n^4 + 10*a^4*n^3 + 35*a^4*n^2 + 50*a^4*n + 24*a^4)*cos(d
*x + c)^4 + 384*a^4 - 4*(2*a^4*n^4 + 23*a^4*n^3 + 91*a^4*n^2 + 142*a^4*n + 72*a^4)*cos(d*x + c)^2)*sin(d*x + c
))*sin(d*x + c)^n/(d*n^5 + 15*d*n^4 + 85*d*n^3 + 225*d*n^2 + 274*d*n + 120*d)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1833 vs. \(2 (97) = 194\).

Time = 3.61 (sec) , antiderivative size = 1833, normalized size of antiderivative = 16.08 \[ \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^4 \, dx=\text {Too large to display} \]

[In]

integrate(cos(d*x+c)*sin(d*x+c)**n*(a+a*sin(d*x+c))**4,x)

[Out]

Piecewise((x*(a*sin(c) + a)**4*sin(c)**n*cos(c), Eq(d, 0)), (a**4*log(sin(c + d*x))/d - 4*a**4/(d*sin(c + d*x)
) - 3*a**4/(d*sin(c + d*x)**2) - 4*a**4/(3*d*sin(c + d*x)**3) - a**4/(4*d*sin(c + d*x)**4), Eq(n, -5)), (4*a**
4*log(sin(c + d*x))/d + a**4*sin(c + d*x)/d - 6*a**4/(d*sin(c + d*x)) - 2*a**4/(d*sin(c + d*x)**2) - a**4/(3*d
*sin(c + d*x)**3), Eq(n, -4)), (6*a**4*log(sin(c + d*x))/d + a**4*sin(c + d*x)**2/(2*d) + 4*a**4*sin(c + d*x)/
d - 4*a**4/(d*sin(c + d*x)) - a**4/(2*d*sin(c + d*x)**2), Eq(n, -3)), (4*a**4*log(sin(c + d*x))/d + a**4*sin(c
+ d*x)**3/(3*d) + 2*a**4*sin(c + d*x)**2/d + 6*a**4*sin(c + d*x)/d - a**4/(d*sin(c + d*x)), Eq(n, -2)), (a**4
*log(sin(c + d*x))/d + a**4*sin(c + d*x)**4/(4*d) + 4*a**4*sin(c + d*x)**3/(3*d) + 3*a**4*sin(c + d*x)**2/d +
4*a**4*sin(c + d*x)/d, Eq(n, -1)), (a**4*n**4*sin(c + d*x)**5*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3
+ 225*d*n**2 + 274*d*n + 120*d) + 4*a**4*n**4*sin(c + d*x)**4*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3
+ 225*d*n**2 + 274*d*n + 120*d) + 6*a**4*n**4*sin(c + d*x)**3*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3
+ 225*d*n**2 + 274*d*n + 120*d) + 4*a**4*n**4*sin(c + d*x)**2*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3
+ 225*d*n**2 + 274*d*n + 120*d) + a**4*n**4*sin(c + d*x)*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225
*d*n**2 + 274*d*n + 120*d) + 10*a**4*n**3*sin(c + d*x)**5*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 22
5*d*n**2 + 274*d*n + 120*d) + 44*a**4*n**3*sin(c + d*x)**4*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 2
25*d*n**2 + 274*d*n + 120*d) + 72*a**4*n**3*sin(c + d*x)**3*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 +
225*d*n**2 + 274*d*n + 120*d) + 52*a**4*n**3*sin(c + d*x)**2*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 +
225*d*n**2 + 274*d*n + 120*d) + 14*a**4*n**3*sin(c + d*x)*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 2
25*d*n**2 + 274*d*n + 120*d) + 35*a**4*n**2*sin(c + d*x)**5*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 +
225*d*n**2 + 274*d*n + 120*d) + 164*a**4*n**2*sin(c + d*x)**4*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3
+ 225*d*n**2 + 274*d*n + 120*d) + 294*a**4*n**2*sin(c + d*x)**3*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**
3 + 225*d*n**2 + 274*d*n + 120*d) + 236*a**4*n**2*sin(c + d*x)**2*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n
**3 + 225*d*n**2 + 274*d*n + 120*d) + 71*a**4*n**2*sin(c + d*x)*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**
3 + 225*d*n**2 + 274*d*n + 120*d) + 50*a**4*n*sin(c + d*x)**5*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3
+ 225*d*n**2 + 274*d*n + 120*d) + 244*a**4*n*sin(c + d*x)**4*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 +
225*d*n**2 + 274*d*n + 120*d) + 468*a**4*n*sin(c + d*x)**3*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 +
225*d*n**2 + 274*d*n + 120*d) + 428*a**4*n*sin(c + d*x)**2*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 2
25*d*n**2 + 274*d*n + 120*d) + 154*a**4*n*sin(c + d*x)*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d
*n**2 + 274*d*n + 120*d) + 24*a**4*sin(c + d*x)**5*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**
2 + 274*d*n + 120*d) + 120*a**4*sin(c + d*x)**4*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 +
274*d*n + 120*d) + 240*a**4*sin(c + d*x)**3*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 27
4*d*n + 120*d) + 240*a**4*sin(c + d*x)**2*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d
*n + 120*d) + 120*a**4*sin(c + d*x)*sin(c + d*x)**n/(d*n**5 + 15*d*n**4 + 85*d*n**3 + 225*d*n**2 + 274*d*n + 1
20*d), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {\frac {a^{4} \sin \left (d x + c\right )^{n + 5}}{n + 5} + \frac {4 \, a^{4} \sin \left (d x + c\right )^{n + 4}}{n + 4} + \frac {6 \, a^{4} \sin \left (d x + c\right )^{n + 3}}{n + 3} + \frac {4 \, a^{4} \sin \left (d x + c\right )^{n + 2}}{n + 2} + \frac {a^{4} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \]

[In]

integrate(cos(d*x+c)*sin(d*x+c)^n*(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.50 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.11 \[ \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {\frac {a^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5}}{n + 5} + \frac {4 \, a^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4}}{n + 4} + \frac {6 \, a^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3}}{n + 3} + \frac {4 \, a^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2}}{n + 2} + \frac {a^{4} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \]

[In]

integrate(cos(d*x+c)*sin(d*x+c)^n*(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 13.88 (sec) , antiderivative size = 370, normalized size of antiderivative = 3.25 \[ \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^4 \, dx=\frac {a^4\,{\sin \left (c+d\,x\right )}^n\,\left (4888\,n+5040\,\sin \left (c+d\,x\right )-2880\,\cos \left (2\,c+2\,d\,x\right )+240\,\cos \left (4\,c+4\,d\,x\right )-1080\,\sin \left (3\,c+3\,d\,x\right )+24\,\sin \left (5\,c+5\,d\,x\right )+8580\,n\,\sin \left (c+d\,x\right )-5376\,n\,\cos \left (2\,c+2\,d\,x\right )+488\,n\,\cos \left (4\,c+4\,d\,x\right )-2122\,n\,\sin \left (3\,c+3\,d\,x\right )+50\,n\,\sin \left (5\,c+5\,d\,x\right )+5014\,n^2\,\sin \left (c+d\,x\right )+1188\,n^3\,\sin \left (c+d\,x\right )+98\,n^4\,\sin \left (c+d\,x\right )+2872\,n^2+680\,n^3+56\,n^4-3200\,n^2\,\cos \left (2\,c+2\,d\,x\right )-768\,n^3\,\cos \left (2\,c+2\,d\,x\right )-64\,n^4\,\cos \left (2\,c+2\,d\,x\right )+328\,n^2\,\cos \left (4\,c+4\,d\,x\right )+88\,n^3\,\cos \left (4\,c+4\,d\,x\right )+8\,n^4\,\cos \left (4\,c+4\,d\,x\right )-1351\,n^2\,\sin \left (3\,c+3\,d\,x\right )-338\,n^3\,\sin \left (3\,c+3\,d\,x\right )-29\,n^4\,\sin \left (3\,c+3\,d\,x\right )+35\,n^2\,\sin \left (5\,c+5\,d\,x\right )+10\,n^3\,\sin \left (5\,c+5\,d\,x\right )+n^4\,\sin \left (5\,c+5\,d\,x\right )+2640\right )}{16\,d\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )} \]

[In]

int(cos(c + d*x)*sin(c + d*x)^n*(a + a*sin(c + d*x))^4,x)

[Out]

(a^4*sin(c + d*x)^n*(4888*n + 5040*sin(c + d*x) - 2880*cos(2*c + 2*d*x) + 240*cos(4*c + 4*d*x) - 1080*sin(3*c
+ 3*d*x) + 24*sin(5*c + 5*d*x) + 8580*n*sin(c + d*x) - 5376*n*cos(2*c + 2*d*x) + 488*n*cos(4*c + 4*d*x) - 2122
*n*sin(3*c + 3*d*x) + 50*n*sin(5*c + 5*d*x) + 5014*n^2*sin(c + d*x) + 1188*n^3*sin(c + d*x) + 98*n^4*sin(c + d
*x) + 2872*n^2 + 680*n^3 + 56*n^4 - 3200*n^2*cos(2*c + 2*d*x) - 768*n^3*cos(2*c + 2*d*x) - 64*n^4*cos(2*c + 2*
d*x) + 328*n^2*cos(4*c + 4*d*x) + 88*n^3*cos(4*c + 4*d*x) + 8*n^4*cos(4*c + 4*d*x) - 1351*n^2*sin(3*c + 3*d*x)
- 338*n^3*sin(3*c + 3*d*x) - 29*n^4*sin(3*c + 3*d*x) + 35*n^2*sin(5*c + 5*d*x) + 10*n^3*sin(5*c + 5*d*x) + n^
4*sin(5*c + 5*d*x) + 2640))/(16*d*(274*n + 225*n^2 + 85*n^3 + 15*n^4 + n^5 + 120))