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

Optimal. Leaf size=114 \[ \frac {a^4 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {4 a^4 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac {6 a^4 \sin ^{n+3}(c+d x)}{d (n+3)}+\frac {4 a^4 \sin ^{n+4}(c+d x)}{d (n+4)}+\frac {a^4 \sin ^{n+5}(c+d x)}{d (n+5)} \]

[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)

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Rubi [A]  time = 0.12, antiderivative size = 114, 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 = {2833, 43} \[ \frac {a^4 \sin ^{n+1}(c+d x)}{d (n+1)}+\frac {4 a^4 \sin ^{n+2}(c+d x)}{d (n+2)}+\frac {6 a^4 \sin ^{n+3}(c+d x)}{d (n+3)}+\frac {4 a^4 \sin ^{n+4}(c+d x)}{d (n+4)}+\frac {a^4 \sin ^{n+5}(c+d x)}{d (n+5)} \]

Antiderivative was successfully verified.

[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 43

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 2833

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*x)/b)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[
{a, b, c, d, e, f, m, n}, x]

Rubi steps

\begin {align*} \int \cos (c+d x) \sin ^n(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac {\operatorname {Subst}\left (\int \left (\frac {x}{a}\right )^n (a+x)^4 \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {\operatorname {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*}

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Mathematica [A]  time = 0.27, size = 80, normalized size = 0.70 \[ \frac {a^4 \sin ^{n+1}(c+d x) \left (\frac {\sin ^4(c+d x)}{n+5}+\frac {4 \sin ^3(c+d x)}{n+4}+\frac {6 \sin ^2(c+d x)}{n+3}+\frac {4 \sin (c+d x)}{n+2}+\frac {1}{n+1}\right )}{d} \]

Antiderivative was successfully verified.

[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

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fricas [B]  time = 0.56, size = 302, normalized size = 2.65 \[ \frac {{\left (8 \, a^{4} n^{4} + 96 \, a^{4} n^{3} + 400 \, a^{4} n^{2} + 672 \, a^{4} n + 4 \, {\left (a^{4} n^{4} + 11 \, a^{4} n^{3} + 41 \, a^{4} n^{2} + 61 \, a^{4} n + 30 \, a^{4}\right )} \cos \left (d x + c\right )^{4} + 360 \, a^{4} - 4 \, {\left (3 \, a^{4} n^{4} + 35 \, a^{4} n^{3} + 141 \, a^{4} n^{2} + 229 \, a^{4} n + 120 \, a^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (8 \, a^{4} n^{4} + 96 \, a^{4} n^{3} + 400 \, a^{4} n^{2} + 672 \, a^{4} n + {\left (a^{4} n^{4} + 10 \, a^{4} n^{3} + 35 \, a^{4} n^{2} + 50 \, a^{4} n + 24 \, a^{4}\right )} \cos \left (d x + c\right )^{4} + 384 \, a^{4} - 4 \, {\left (2 \, a^{4} n^{4} + 23 \, a^{4} n^{3} + 91 \, a^{4} n^{2} + 142 \, a^{4} n + 72 \, a^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sin \left (d x + c\right )^{n}}{d n^{5} + 15 \, d n^{4} + 85 \, d n^{3} + 225 \, d n^{2} + 274 \, d n + 120 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.35, size = 593, normalized size = 5.20 \[ \frac {a^{4} n^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 4 \, a^{4} n^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 10 \, a^{4} n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 6 \, a^{4} n^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 44 \, a^{4} n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 35 \, a^{4} n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 4 \, a^{4} n^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 72 \, a^{4} n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 164 \, a^{4} n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 50 \, a^{4} n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + a^{4} n^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 52 \, a^{4} n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 294 \, a^{4} n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 244 \, a^{4} n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 24 \, a^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{5} + 14 \, a^{4} n^{3} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 236 \, a^{4} n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 468 \, a^{4} n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 120 \, a^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{4} + 71 \, a^{4} n^{2} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 428 \, a^{4} n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 240 \, a^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{3} + 154 \, a^{4} n \sin \left (d x + c\right )^{n} \sin \left (d x + c\right ) + 240 \, a^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )^{2} + 120 \, a^{4} \sin \left (d x + c\right )^{n} \sin \left (d x + c\right )}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^4*n^4*sin(d*x + c)^n*sin(d*x + c)^5 + 4*a^4*n^4*sin(d*x + c)^n*sin(d*x + c)^4 + 10*a^4*n^3*sin(d*x + c)^n*s
in(d*x + c)^5 + 6*a^4*n^4*sin(d*x + c)^n*sin(d*x + c)^3 + 44*a^4*n^3*sin(d*x + c)^n*sin(d*x + c)^4 + 35*a^4*n^
2*sin(d*x + c)^n*sin(d*x + c)^5 + 4*a^4*n^4*sin(d*x + c)^n*sin(d*x + c)^2 + 72*a^4*n^3*sin(d*x + c)^n*sin(d*x
+ c)^3 + 164*a^4*n^2*sin(d*x + c)^n*sin(d*x + c)^4 + 50*a^4*n*sin(d*x + c)^n*sin(d*x + c)^5 + a^4*n^4*sin(d*x
+ c)^n*sin(d*x + c) + 52*a^4*n^3*sin(d*x + c)^n*sin(d*x + c)^2 + 294*a^4*n^2*sin(d*x + c)^n*sin(d*x + c)^3 + 2
44*a^4*n*sin(d*x + c)^n*sin(d*x + c)^4 + 24*a^4*sin(d*x + c)^n*sin(d*x + c)^5 + 14*a^4*n^3*sin(d*x + c)^n*sin(
d*x + c) + 236*a^4*n^2*sin(d*x + c)^n*sin(d*x + c)^2 + 468*a^4*n*sin(d*x + c)^n*sin(d*x + c)^3 + 120*a^4*sin(d
*x + c)^n*sin(d*x + c)^4 + 71*a^4*n^2*sin(d*x + c)^n*sin(d*x + c) + 428*a^4*n*sin(d*x + c)^n*sin(d*x + c)^2 +
240*a^4*sin(d*x + c)^n*sin(d*x + c)^3 + 154*a^4*n*sin(d*x + c)^n*sin(d*x + c) + 240*a^4*sin(d*x + c)^n*sin(d*x
+ c)^2 + 120*a^4*sin(d*x + c)^n*sin(d*x + c))/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*d)

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maple [F]  time = 8.36, size = 0, normalized size = 0.00 \[ \int \cos \left (d x +c \right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +a \sin \left (d x +c \right )\right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.31, size = 103, normalized size = 0.90 \[ \frac {\frac {a^{4} \sin \left (d x + c\right )^{n + 5}}{n + 5} + \frac {4 \, a^{4} \sin \left (d x + c\right )^{n + 4}}{n + 4} + \frac {6 \, a^{4} \sin \left (d x + c\right )^{n + 3}}{n + 3} + \frac {4 \, a^{4} \sin \left (d x + c\right )^{n + 2}}{n + 2} + \frac {a^{4} \sin \left (d x + c\right )^{n + 1}}{n + 1}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 11.76, size = 370, normalized size = 3.25 \[ \frac {a^4\,{\sin \left (c+d\,x\right )}^n\,\left (4888\,n+5040\,\sin \left (c+d\,x\right )-2880\,\cos \left (2\,c+2\,d\,x\right )+240\,\cos \left (4\,c+4\,d\,x\right )-1080\,\sin \left (3\,c+3\,d\,x\right )+24\,\sin \left (5\,c+5\,d\,x\right )+8580\,n\,\sin \left (c+d\,x\right )-5376\,n\,\cos \left (2\,c+2\,d\,x\right )+488\,n\,\cos \left (4\,c+4\,d\,x\right )-2122\,n\,\sin \left (3\,c+3\,d\,x\right )+50\,n\,\sin \left (5\,c+5\,d\,x\right )+5014\,n^2\,\sin \left (c+d\,x\right )+1188\,n^3\,\sin \left (c+d\,x\right )+98\,n^4\,\sin \left (c+d\,x\right )+2872\,n^2+680\,n^3+56\,n^4-3200\,n^2\,\cos \left (2\,c+2\,d\,x\right )-768\,n^3\,\cos \left (2\,c+2\,d\,x\right )-64\,n^4\,\cos \left (2\,c+2\,d\,x\right )+328\,n^2\,\cos \left (4\,c+4\,d\,x\right )+88\,n^3\,\cos \left (4\,c+4\,d\,x\right )+8\,n^4\,\cos \left (4\,c+4\,d\,x\right )-1351\,n^2\,\sin \left (3\,c+3\,d\,x\right )-338\,n^3\,\sin \left (3\,c+3\,d\,x\right )-29\,n^4\,\sin \left (3\,c+3\,d\,x\right )+35\,n^2\,\sin \left (5\,c+5\,d\,x\right )+10\,n^3\,\sin \left (5\,c+5\,d\,x\right )+n^4\,\sin \left (5\,c+5\,d\,x\right )+2640\right )}{16\,d\,\left (n^5+15\,n^4+85\,n^3+225\,n^2+274\,n+120\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 61.69, size = 1833, normalized size = 16.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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