3.913 \(\int \cos (e+f x) (a+a \sin (e+f x))^4 (c+d \sin (e+f x))^n \, dx\)
Optimal. Leaf size=175 \[ \frac {a^4 (c-d)^4 (c+d \sin (e+f x))^{n+1}}{d^5 f (n+1)}-\frac {4 a^4 (c-d)^3 (c+d \sin (e+f x))^{n+2}}{d^5 f (n+2)}+\frac {6 a^4 (c-d)^2 (c+d \sin (e+f x))^{n+3}}{d^5 f (n+3)}-\frac {4 a^4 (c-d) (c+d \sin (e+f x))^{n+4}}{d^5 f (n+4)}+\frac {a^4 (c+d \sin (e+f x))^{n+5}}{d^5 f (n+5)} \]
[Out]
a^4*(c-d)^4*(c+d*sin(f*x+e))^(1+n)/d^5/f/(1+n)-4*a^4*(c-d)^3*(c+d*sin(f*x+e))^(2+n)/d^5/f/(2+n)+6*a^4*(c-d)^2*
(c+d*sin(f*x+e))^(3+n)/d^5/f/(3+n)-4*a^4*(c-d)*(c+d*sin(f*x+e))^(4+n)/d^5/f/(4+n)+a^4*(c+d*sin(f*x+e))^(5+n)/d
^5/f/(5+n)
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Rubi [A] time = 0.21, antiderivative size = 175, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used =
{2833, 43} \[ \frac {a^4 (c-d)^4 (c+d \sin (e+f x))^{n+1}}{d^5 f (n+1)}-\frac {4 a^4 (c-d)^3 (c+d \sin (e+f x))^{n+2}}{d^5 f (n+2)}+\frac {6 a^4 (c-d)^2 (c+d \sin (e+f x))^{n+3}}{d^5 f (n+3)}-\frac {4 a^4 (c-d) (c+d \sin (e+f x))^{n+4}}{d^5 f (n+4)}+\frac {a^4 (c+d \sin (e+f x))^{n+5}}{d^5 f (n+5)} \]
Antiderivative was successfully verified.
[In]
Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^4*(c + d*Sin[e + f*x])^n,x]
[Out]
(a^4*(c - d)^4*(c + d*Sin[e + f*x])^(1 + n))/(d^5*f*(1 + n)) - (4*a^4*(c - d)^3*(c + d*Sin[e + f*x])^(2 + n))/
(d^5*f*(2 + n)) + (6*a^4*(c - d)^2*(c + d*Sin[e + f*x])^(3 + n))/(d^5*f*(3 + n)) - (4*a^4*(c - d)*(c + d*Sin[e
+ f*x])^(4 + n))/(d^5*f*(4 + n)) + (a^4*(c + d*Sin[e + f*x])^(5 + n))/(d^5*f*(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 (e+f x) (a+a \sin (e+f x))^4 (c+d \sin (e+f x))^n \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^4 \left (c+\frac {d x}{a}\right )^n \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^4 (c-d)^4 \left (c+\frac {d x}{a}\right )^n}{d^4}-\frac {4 a^4 (c-d)^3 \left (c+\frac {d x}{a}\right )^{1+n}}{d^4}+\frac {6 a^4 (c-d)^2 \left (c+\frac {d x}{a}\right )^{2+n}}{d^4}-\frac {4 a^4 (c-d) \left (c+\frac {d x}{a}\right )^{3+n}}{d^4}+\frac {a^4 \left (c+\frac {d x}{a}\right )^{4+n}}{d^4}\right ) \, dx,x,a \sin (e+f x)\right )}{a f}\\ &=\frac {a^4 (c-d)^4 (c+d \sin (e+f x))^{1+n}}{d^5 f (1+n)}-\frac {4 a^4 (c-d)^3 (c+d \sin (e+f x))^{2+n}}{d^5 f (2+n)}+\frac {6 a^4 (c-d)^2 (c+d \sin (e+f x))^{3+n}}{d^5 f (3+n)}-\frac {4 a^4 (c-d) (c+d \sin (e+f x))^{4+n}}{d^5 f (4+n)}+\frac {a^4 (c+d \sin (e+f x))^{5+n}}{d^5 f (5+n)}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 130, normalized size = 0.74 \[ \frac {a^4 (c+d \sin (e+f x))^{n+1} \left (-\frac {4 (c-d)^3 (c+d \sin (e+f x))}{n+2}+\frac {6 (c-d)^2 (c+d \sin (e+f x))^2}{n+3}-\frac {4 (c-d) (c+d \sin (e+f x))^3}{n+4}+\frac {(c+d \sin (e+f x))^4}{n+5}+\frac {(c-d)^4}{n+1}\right )}{d^5 f} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^4*(c + d*Sin[e + f*x])^n,x]
[Out]
(a^4*(c + d*Sin[e + f*x])^(1 + n)*((c - d)^4/(1 + n) - (4*(c - d)^3*(c + d*Sin[e + f*x]))/(2 + n) + (6*(c - d)
^2*(c + d*Sin[e + f*x])^2)/(3 + n) - (4*(c - d)*(c + d*Sin[e + f*x])^3)/(4 + n) + (c + d*Sin[e + f*x])^4/(5 +
n)))/(d^5*f)
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fricas [B] time = 0.59, size = 902, normalized size = 5.15 \[ \frac {{\left (24 \, a^{4} c^{5} - 120 \, a^{4} c^{4} d + 240 \, a^{4} c^{3} d^{2} - 240 \, a^{4} c^{2} d^{3} + 120 \, a^{4} c d^{4} + 360 \, a^{4} d^{5} + 8 \, {\left (a^{4} c d^{4} + a^{4} d^{5}\right )} n^{4} + {\left (120 \, a^{4} d^{5} + {\left (a^{4} c d^{4} + 4 \, a^{4} d^{5}\right )} n^{4} + 2 \, {\left (3 \, a^{4} c d^{4} + 22 \, a^{4} d^{5}\right )} n^{3} + {\left (11 \, a^{4} c d^{4} + 164 \, a^{4} d^{5}\right )} n^{2} + 2 \, {\left (3 \, a^{4} c d^{4} + 122 \, a^{4} d^{5}\right )} n\right )} \cos \left (f x + e\right )^{4} - 16 \, {\left (a^{4} c^{2} d^{3} - 5 \, a^{4} c d^{4} - 6 \, a^{4} d^{5}\right )} n^{3} + 8 \, {\left (3 \, a^{4} c^{3} d^{2} - 15 \, a^{4} c^{2} d^{3} + 32 \, a^{4} c d^{4} + 50 \, a^{4} d^{5}\right )} n^{2} - 4 \, {\left (120 \, a^{4} d^{5} + {\left (2 \, a^{4} c d^{4} + 3 \, a^{4} d^{5}\right )} n^{4} - {\left (3 \, a^{4} c^{2} d^{3} - 18 \, a^{4} c d^{4} - 35 \, a^{4} d^{5}\right )} n^{3} + {\left (3 \, a^{4} c^{3} d^{2} - 18 \, a^{4} c^{2} d^{3} + 49 \, a^{4} c d^{4} + 141 \, a^{4} d^{5}\right )} n^{2} + {\left (3 \, a^{4} c^{3} d^{2} - 15 \, a^{4} c^{2} d^{3} + 33 \, a^{4} c d^{4} + 229 \, a^{4} d^{5}\right )} n\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left (3 \, a^{4} c^{4} d - 15 \, a^{4} c^{3} d^{2} + 31 \, a^{4} c^{2} d^{3} - 35 \, a^{4} c d^{4} - 84 \, a^{4} d^{5}\right )} n + {\left (384 \, a^{4} d^{5} + 8 \, {\left (a^{4} c d^{4} + a^{4} d^{5}\right )} n^{4} + {\left (a^{4} d^{5} n^{4} + 10 \, a^{4} d^{5} n^{3} + 35 \, a^{4} d^{5} n^{2} + 50 \, a^{4} d^{5} n + 24 \, a^{4} d^{5}\right )} \cos \left (f x + e\right )^{4} - 16 \, {\left (a^{4} c^{2} d^{3} - 5 \, a^{4} c d^{4} - 6 \, a^{4} d^{5}\right )} n^{3} + 8 \, {\left (3 \, a^{4} c^{3} d^{2} - 15 \, a^{4} c^{2} d^{3} + 32 \, a^{4} c d^{4} + 50 \, a^{4} d^{5}\right )} n^{2} - 4 \, {\left (72 \, a^{4} d^{5} + {\left (a^{4} c d^{4} + 2 \, a^{4} d^{5}\right )} n^{4} - {\left (a^{4} c^{2} d^{3} - 8 \, a^{4} c d^{4} - 23 \, a^{4} d^{5}\right )} n^{3} - {\left (3 \, a^{4} c^{2} d^{3} - 17 \, a^{4} c d^{4} - 91 \, a^{4} d^{5}\right )} n^{2} - 2 \, {\left (a^{4} c^{2} d^{3} - 5 \, a^{4} c d^{4} - 71 \, a^{4} d^{5}\right )} n\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left (3 \, a^{4} c^{4} d - 15 \, a^{4} c^{3} d^{2} + 31 \, a^{4} c^{2} d^{3} - 35 \, a^{4} c d^{4} - 84 \, a^{4} d^{5}\right )} n\right )} \sin \left (f x + e\right )\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{d^{5} f n^{5} + 15 \, d^{5} f n^{4} + 85 \, d^{5} f n^{3} + 225 \, d^{5} f n^{2} + 274 \, d^{5} f n + 120 \, d^{5} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(f*x+e)*(a+a*sin(f*x+e))^4*(c+d*sin(f*x+e))^n,x, algorithm="fricas")
[Out]
(24*a^4*c^5 - 120*a^4*c^4*d + 240*a^4*c^3*d^2 - 240*a^4*c^2*d^3 + 120*a^4*c*d^4 + 360*a^4*d^5 + 8*(a^4*c*d^4 +
a^4*d^5)*n^4 + (120*a^4*d^5 + (a^4*c*d^4 + 4*a^4*d^5)*n^4 + 2*(3*a^4*c*d^4 + 22*a^4*d^5)*n^3 + (11*a^4*c*d^4
+ 164*a^4*d^5)*n^2 + 2*(3*a^4*c*d^4 + 122*a^4*d^5)*n)*cos(f*x + e)^4 - 16*(a^4*c^2*d^3 - 5*a^4*c*d^4 - 6*a^4*d
^5)*n^3 + 8*(3*a^4*c^3*d^2 - 15*a^4*c^2*d^3 + 32*a^4*c*d^4 + 50*a^4*d^5)*n^2 - 4*(120*a^4*d^5 + (2*a^4*c*d^4 +
3*a^4*d^5)*n^4 - (3*a^4*c^2*d^3 - 18*a^4*c*d^4 - 35*a^4*d^5)*n^3 + (3*a^4*c^3*d^2 - 18*a^4*c^2*d^3 + 49*a^4*c
*d^4 + 141*a^4*d^5)*n^2 + (3*a^4*c^3*d^2 - 15*a^4*c^2*d^3 + 33*a^4*c*d^4 + 229*a^4*d^5)*n)*cos(f*x + e)^2 - 8*
(3*a^4*c^4*d - 15*a^4*c^3*d^2 + 31*a^4*c^2*d^3 - 35*a^4*c*d^4 - 84*a^4*d^5)*n + (384*a^4*d^5 + 8*(a^4*c*d^4 +
a^4*d^5)*n^4 + (a^4*d^5*n^4 + 10*a^4*d^5*n^3 + 35*a^4*d^5*n^2 + 50*a^4*d^5*n + 24*a^4*d^5)*cos(f*x + e)^4 - 16
*(a^4*c^2*d^3 - 5*a^4*c*d^4 - 6*a^4*d^5)*n^3 + 8*(3*a^4*c^3*d^2 - 15*a^4*c^2*d^3 + 32*a^4*c*d^4 + 50*a^4*d^5)*
n^2 - 4*(72*a^4*d^5 + (a^4*c*d^4 + 2*a^4*d^5)*n^4 - (a^4*c^2*d^3 - 8*a^4*c*d^4 - 23*a^4*d^5)*n^3 - (3*a^4*c^2*
d^3 - 17*a^4*c*d^4 - 91*a^4*d^5)*n^2 - 2*(a^4*c^2*d^3 - 5*a^4*c*d^4 - 71*a^4*d^5)*n)*cos(f*x + e)^2 - 8*(3*a^4
*c^4*d - 15*a^4*c^3*d^2 + 31*a^4*c^2*d^3 - 35*a^4*c*d^4 - 84*a^4*d^5)*n)*sin(f*x + e))*(d*sin(f*x + e) + c)^n/
(d^5*f*n^5 + 15*d^5*f*n^4 + 85*d^5*f*n^3 + 225*d^5*f*n^2 + 274*d^5*f*n + 120*d^5*f)
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giac [B] time = 0.24, size = 1840, normalized size = 10.51 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(f*x+e)*(a+a*sin(f*x+e))^4*(c+d*sin(f*x+e))^n,x, algorithm="giac")
[Out]
(((d*sin(f*x + e) + c)^5*(d*sin(f*x + e) + c)^n*n^4 - 4*(d*sin(f*x + e) + c)^4*(d*sin(f*x + e) + c)^n*c*n^4 +
6*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*c^2*n^4 - 4*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c^3*
n^4 + (d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^4*n^4 + 10*(d*sin(f*x + e) + c)^5*(d*sin(f*x + e) + c)^n*n
^3 - 44*(d*sin(f*x + e) + c)^4*(d*sin(f*x + e) + c)^n*c*n^3 + 72*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n
*c^2*n^3 - 52*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c^3*n^3 + 14*(d*sin(f*x + e) + c)*(d*sin(f*x + e)
+ c)^n*c^4*n^3 + 35*(d*sin(f*x + e) + c)^5*(d*sin(f*x + e) + c)^n*n^2 - 164*(d*sin(f*x + e) + c)^4*(d*sin(f*x
+ e) + c)^n*c*n^2 + 294*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*c^2*n^2 - 236*(d*sin(f*x + e) + c)^2*(d*
sin(f*x + e) + c)^n*c^3*n^2 + 71*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^4*n^2 + 50*(d*sin(f*x + e) + c)
^5*(d*sin(f*x + e) + c)^n*n - 244*(d*sin(f*x + e) + c)^4*(d*sin(f*x + e) + c)^n*c*n + 468*(d*sin(f*x + e) + c)
^3*(d*sin(f*x + e) + c)^n*c^2*n - 428*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c^3*n + 154*(d*sin(f*x + e
) + c)*(d*sin(f*x + e) + c)^n*c^4*n + 24*(d*sin(f*x + e) + c)^5*(d*sin(f*x + e) + c)^n - 120*(d*sin(f*x + e) +
c)^4*(d*sin(f*x + e) + c)^n*c + 240*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*c^2 - 240*(d*sin(f*x + e) +
c)^2*(d*sin(f*x + e) + c)^n*c^3 + 120*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^4)*a^4/(d^4*n^5 + 15*d^4*
n^4 + 85*d^4*n^3 + 225*d^4*n^2 + 274*d^4*n + 120*d^4) + 4*((d*sin(f*x + e) + c)^4*(d*sin(f*x + e) + c)^n*n^3 -
3*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*c*n^3 + 3*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c^2*n
^3 - (d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^3*n^3 + 6*(d*sin(f*x + e) + c)^4*(d*sin(f*x + e) + c)^n*n^2
- 21*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*c*n^2 + 24*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c
^2*n^2 - 9*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^3*n^2 + 11*(d*sin(f*x + e) + c)^4*(d*sin(f*x + e) + c
)^n*n - 42*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*c*n + 57*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^
n*c^2*n - 26*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^3*n + 6*(d*sin(f*x + e) + c)^4*(d*sin(f*x + e) + c)
^n - 24*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*c + 36*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c^2
- 24*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^3)*a^4/(d^3*n^4 + 10*d^3*n^3 + 35*d^3*n^2 + 50*d^3*n + 24*
d^3) + 6*((d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*n^2 - 2*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*
c*n^2 + (d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^2*n^2 + 3*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n*
n - 8*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c*n + 5*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^2*n
+ 2*(d*sin(f*x + e) + c)^3*(d*sin(f*x + e) + c)^n - 6*(d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*c + 6*(d*s
in(f*x + e) + c)*(d*sin(f*x + e) + c)^n*c^2)*a^4/(d^2*n^3 + 6*d^2*n^2 + 11*d^2*n + 6*d^2) + (d*sin(f*x + e) +
c)^(n + 1)*a^4/(n + 1) + 4*((d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n*n - (d*sin(f*x + e) + c)*(d*sin(f*x
+ e) + c)^n*c*n + (d*sin(f*x + e) + c)^2*(d*sin(f*x + e) + c)^n - 2*(d*sin(f*x + e) + c)*(d*sin(f*x + e) + c)^
n*c)*a^4/((n^2 + 3*n + 2)*d))/(d*f)
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maple [F] time = 4.38, size = 0, normalized size = 0.00 \[ \int \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{4} \left (c +d \sin \left (f x +e \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(f*x+e)*(a+a*sin(f*x+e))^4*(c+d*sin(f*x+e))^n,x)
[Out]
int(cos(f*x+e)*(a+a*sin(f*x+e))^4*(c+d*sin(f*x+e))^n,x)
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maxima [B] time = 1.30, size = 486, normalized size = 2.78 \[ \frac {\frac {4 \, {\left (d^{2} {\left (n + 1\right )} \sin \left (f x + e\right )^{2} + c d n \sin \left (f x + e\right ) - c^{2}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} a^{4}}{{\left (n^{2} + 3 \, n + 2\right )} d^{2}} + \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n + 1} a^{4}}{d {\left (n + 1\right )}} + \frac {6 \, {\left ({\left (n^{2} + 3 \, n + 2\right )} d^{3} \sin \left (f x + e\right )^{3} + {\left (n^{2} + n\right )} c d^{2} \sin \left (f x + e\right )^{2} - 2 \, c^{2} d n \sin \left (f x + e\right ) + 2 \, c^{3}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} a^{4}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{3}} + \frac {4 \, {\left ({\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{4} \sin \left (f x + e\right )^{4} + {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c d^{3} \sin \left (f x + e\right )^{3} - 3 \, {\left (n^{2} + n\right )} c^{2} d^{2} \sin \left (f x + e\right )^{2} + 6 \, c^{3} d n \sin \left (f x + e\right ) - 6 \, c^{4}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} a^{4}}{{\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d^{4}} + \frac {{\left ({\left (n^{4} + 10 \, n^{3} + 35 \, n^{2} + 50 \, n + 24\right )} d^{5} \sin \left (f x + e\right )^{5} + {\left (n^{4} + 6 \, n^{3} + 11 \, n^{2} + 6 \, n\right )} c d^{4} \sin \left (f x + e\right )^{4} - 4 \, {\left (n^{3} + 3 \, n^{2} + 2 \, n\right )} c^{2} d^{3} \sin \left (f x + e\right )^{3} + 12 \, {\left (n^{2} + n\right )} c^{3} d^{2} \sin \left (f x + e\right )^{2} - 24 \, c^{4} d n \sin \left (f x + e\right ) + 24 \, c^{5}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} a^{4}}{{\left (n^{5} + 15 \, n^{4} + 85 \, n^{3} + 225 \, n^{2} + 274 \, n + 120\right )} d^{5}}}{f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(f*x+e)*(a+a*sin(f*x+e))^4*(c+d*sin(f*x+e))^n,x, algorithm="maxima")
[Out]
(4*(d^2*(n + 1)*sin(f*x + e)^2 + c*d*n*sin(f*x + e) - c^2)*(d*sin(f*x + e) + c)^n*a^4/((n^2 + 3*n + 2)*d^2) +
(d*sin(f*x + e) + c)^(n + 1)*a^4/(d*(n + 1)) + 6*((n^2 + 3*n + 2)*d^3*sin(f*x + e)^3 + (n^2 + n)*c*d^2*sin(f*x
+ e)^2 - 2*c^2*d*n*sin(f*x + e) + 2*c^3)*(d*sin(f*x + e) + c)^n*a^4/((n^3 + 6*n^2 + 11*n + 6)*d^3) + 4*((n^3
+ 6*n^2 + 11*n + 6)*d^4*sin(f*x + e)^4 + (n^3 + 3*n^2 + 2*n)*c*d^3*sin(f*x + e)^3 - 3*(n^2 + n)*c^2*d^2*sin(f*
x + e)^2 + 6*c^3*d*n*sin(f*x + e) - 6*c^4)*(d*sin(f*x + e) + c)^n*a^4/((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^4
) + ((n^4 + 10*n^3 + 35*n^2 + 50*n + 24)*d^5*sin(f*x + e)^5 + (n^4 + 6*n^3 + 11*n^2 + 6*n)*c*d^4*sin(f*x + e)^
4 - 4*(n^3 + 3*n^2 + 2*n)*c^2*d^3*sin(f*x + e)^3 + 12*(n^2 + n)*c^3*d^2*sin(f*x + e)^2 - 24*c^4*d*n*sin(f*x +
e) + 24*c^5)*(d*sin(f*x + e) + c)^n*a^4/((n^5 + 15*n^4 + 85*n^3 + 225*n^2 + 274*n + 120)*d^5))/f
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mupad [B] time = 18.14, size = 863, normalized size = 4.93 \[ \frac {a^4\,\sin \left (5\,e+5\,f\,x\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n\,\left (n^4+10\,n^3+35\,n^2+50\,n+24\right )\,1{}\mathrm {i}}{16\,f\,\left (n^5\,1{}\mathrm {i}+n^4\,15{}\mathrm {i}+n^3\,85{}\mathrm {i}+n^2\,225{}\mathrm {i}+n\,274{}\mathrm {i}+120{}\mathrm {i}\right )}+\frac {a^4\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n\,\left (c^5\,192{}\mathrm {i}-c^4\,d\,n\,192{}\mathrm {i}-c^4\,d\,960{}\mathrm {i}+c^3\,d^2\,n^2\,144{}\mathrm {i}+c^3\,d^2\,n\,912{}\mathrm {i}+c^3\,d^2\,1920{}\mathrm {i}-c^2\,d^3\,n^3\,80{}\mathrm {i}-c^2\,d^3\,n^2\,672{}\mathrm {i}-c^2\,d^3\,n\,1744{}\mathrm {i}-c^2\,d^3\,1920{}\mathrm {i}+c\,d^4\,n^4\,35{}\mathrm {i}+c\,d^4\,n^3\,370{}\mathrm {i}+c\,d^4\,n^2\,1297{}\mathrm {i}+c\,d^4\,n\,1730{}\mathrm {i}+c\,d^4\,960{}\mathrm {i}+d^5\,n^4\,28{}\mathrm {i}+d^5\,n^3\,340{}\mathrm {i}+d^5\,n^2\,1436{}\mathrm {i}+d^5\,n\,2444{}\mathrm {i}+d^5\,1320{}\mathrm {i}\right )}{8\,d^5\,f\,\left (n^5\,1{}\mathrm {i}+n^4\,15{}\mathrm {i}+n^3\,85{}\mathrm {i}+n^2\,225{}\mathrm {i}+n\,274{}\mathrm {i}+120{}\mathrm {i}\right )}+\frac {a^4\,\sin \left (e+f\,x\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n\,\left (-192\,c^4\,n+192\,c^3\,d\,n^2+960\,c^3\,d\,n-120\,c^2\,d^2\,n^3-936\,c^2\,d^2\,n^2-1968\,c^2\,d^2\,n+56\,c\,d^3\,n^4+576\,c\,d^3\,n^3+1912\,c\,d^3\,n^2+2160\,c\,d^3\,n+49\,d^4\,n^4+594\,d^4\,n^3+2507\,d^4\,n^2+4290\,d^4\,n+2520\,d^4\right )\,1{}\mathrm {i}}{8\,d^4\,f\,\left (n^5\,1{}\mathrm {i}+n^4\,15{}\mathrm {i}+n^3\,85{}\mathrm {i}+n^2\,225{}\mathrm {i}+n\,274{}\mathrm {i}+120{}\mathrm {i}\right )}+\frac {a^4\,\cos \left (4\,e+4\,f\,x\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n\,\left (d\,20{}\mathrm {i}+c\,n\,1{}\mathrm {i}+d\,n\,4{}\mathrm {i}\right )\,\left (n^3+6\,n^2+11\,n+6\right )}{8\,d\,f\,\left (n^5\,1{}\mathrm {i}+n^4\,15{}\mathrm {i}+n^3\,85{}\mathrm {i}+n^2\,225{}\mathrm {i}+n\,274{}\mathrm {i}+120{}\mathrm {i}\right )}-\frac {a^4\,\sin \left (3\,e+3\,f\,x\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n\,\left (n^2+3\,n+2\right )\,\left (-16\,c^2\,n+16\,c\,d\,n^2+80\,c\,d\,n+29\,d^2\,n^2+251\,d^2\,n+540\,d^2\right )\,1{}\mathrm {i}}{16\,d^2\,f\,\left (n^5\,1{}\mathrm {i}+n^4\,15{}\mathrm {i}+n^3\,85{}\mathrm {i}+n^2\,225{}\mathrm {i}+n\,274{}\mathrm {i}+120{}\mathrm {i}\right )}-\frac {a^4\,\cos \left (2\,e+2\,f\,x\right )\,\left (n+1\right )\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n\,\left (c^3\,n\,12{}\mathrm {i}-c^2\,d\,n^2\,12{}\mathrm {i}-c^2\,d\,n\,60{}\mathrm {i}+c\,d^2\,n^3\,7{}\mathrm {i}+c\,d^2\,n^2\,59{}\mathrm {i}+c\,d^2\,n\,126{}\mathrm {i}+d^3\,n^3\,8{}\mathrm {i}+d^3\,n^2\,88{}\mathrm {i}+d^3\,n\,312{}\mathrm {i}+d^3\,360{}\mathrm {i}\right )}{2\,d^3\,f\,\left (n^5\,1{}\mathrm {i}+n^4\,15{}\mathrm {i}+n^3\,85{}\mathrm {i}+n^2\,225{}\mathrm {i}+n\,274{}\mathrm {i}+120{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(e + f*x)*(a + a*sin(e + f*x))^4*(c + d*sin(e + f*x))^n,x)
[Out]
(a^4*sin(5*e + 5*f*x)*(c + d*sin(e + f*x))^n*(50*n + 35*n^2 + 10*n^3 + n^4 + 24)*1i)/(16*f*(n*274i + n^2*225i
+ n^3*85i + n^4*15i + n^5*1i + 120i)) + (a^4*(c + d*sin(e + f*x))^n*(c*d^4*960i - c^4*d*960i + d^5*n*2444i + c
^5*192i + d^5*1320i - c^2*d^3*1920i + c^3*d^2*1920i + d^5*n^2*1436i + d^5*n^3*340i + d^5*n^4*28i - c^2*d^3*n*1
744i + c^3*d^2*n*912i + c*d^4*n^2*1297i + c*d^4*n^3*370i + c*d^4*n^4*35i - c^2*d^3*n^2*672i + c^3*d^2*n^2*144i
- c^2*d^3*n^3*80i + c*d^4*n*1730i - c^4*d*n*192i))/(8*d^5*f*(n*274i + n^2*225i + n^3*85i + n^4*15i + n^5*1i +
120i)) + (a^4*sin(e + f*x)*(c + d*sin(e + f*x))^n*(4290*d^4*n - 192*c^4*n + 2520*d^4 + 2507*d^4*n^2 + 594*d^4
*n^3 + 49*d^4*n^4 - 1968*c^2*d^2*n + 1912*c*d^3*n^2 + 192*c^3*d*n^2 + 576*c*d^3*n^3 + 56*c*d^3*n^4 - 936*c^2*d
^2*n^2 - 120*c^2*d^2*n^3 + 2160*c*d^3*n + 960*c^3*d*n)*1i)/(8*d^4*f*(n*274i + n^2*225i + n^3*85i + n^4*15i + n
^5*1i + 120i)) + (a^4*cos(4*e + 4*f*x)*(c + d*sin(e + f*x))^n*(d*20i + c*n*1i + d*n*4i)*(11*n + 6*n^2 + n^3 +
6))/(8*d*f*(n*274i + n^2*225i + n^3*85i + n^4*15i + n^5*1i + 120i)) - (a^4*sin(3*e + 3*f*x)*(c + d*sin(e + f*x
))^n*(3*n + n^2 + 2)*(251*d^2*n - 16*c^2*n + 540*d^2 + 29*d^2*n^2 + 80*c*d*n + 16*c*d*n^2)*1i)/(16*d^2*f*(n*27
4i + n^2*225i + n^3*85i + n^4*15i + n^5*1i + 120i)) - (a^4*cos(2*e + 2*f*x)*(n + 1)*(c + d*sin(e + f*x))^n*(c^
3*n*12i + d^3*n*312i + d^3*360i + d^3*n^2*88i + d^3*n^3*8i + c*d^2*n^2*59i - c^2*d*n^2*12i + c*d^2*n^3*7i + c*
d^2*n*126i - c^2*d*n*60i))/(2*d^3*f*(n*274i + n^2*225i + n^3*85i + n^4*15i + n^5*1i + 120i))
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sympy [A] time = 138.04, size = 11900, normalized size = 68.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(f*x+e)*(a+a*sin(f*x+e))**4*(c+d*sin(f*x+e))**n,x)
[Out]
Piecewise((c**n*(a**4*sin(e + f*x)**5/(5*f) + a**4*sin(e + f*x)**4/f + 2*a**4*sin(e + f*x)**3/f + 2*a**4*sin(e
+ f*x)**2/f + a**4*sin(e + f*x)/f), Eq(d, 0)), (x*(c + d*sin(e))**n*(a*sin(e) + a)**4*cos(e), Eq(f, 0)), (12*
a**4*c**4*log(c/d + sin(e + f*x))/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin(e + f*x)*
*2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4) + 25*a**4*c**4/(12*c**4*d**5*f + 48*c**3*d**6*f*
sin(e + f*x) + 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4) + 48*
a**4*c**3*d*log(c/d + sin(e + f*x))*sin(e + f*x)/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*
f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4) + 88*a**4*c**3*d*sin(e + f*x)/(12
*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12
*d**9*f*sin(e + f*x)**4) - 12*a**4*c**3*d/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin(e
+ f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4) + 72*a**4*c**2*d**2*log(c/d + sin(e + f*
x))*sin(e + f*x)**2/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8
*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4) + 108*a**4*c**2*d**2*sin(e + f*x)**2/(12*c**4*d**5*f + 48*c**3
*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**
4) - 48*a**4*c**2*d**2*sin(e + f*x)/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin(e + f*x
)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4) - 6*a**4*c**2*d**2/(12*c**4*d**5*f + 48*c**3*d
**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4)
+ 48*a**4*c*d**3*log(c/d + sin(e + f*x))*sin(e + f*x)**3/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c
**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4) + 48*a**4*c*d**3*sin(e +
f*x)**3/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e +
f*x)**3 + 12*d**9*f*sin(e + f*x)**4) - 72*a**4*c*d**3*sin(e + f*x)**2/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e +
f*x) + 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4) - 24*a**4*c*
d**3*sin(e + f*x)/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f
*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4) - 4*a**4*c*d**3/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) +
72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4) + 12*a**4*d**4*log(c
/d + sin(e + f*x))*sin(e + f*x)**4/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin(e + f*x)
**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4) - 48*a**4*d**4*sin(e + f*x)**3/(12*c**4*d**5*f
+ 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e
+ f*x)**4) - 36*a**4*d**4*sin(e + f*x)**2/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin(
e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4) - 16*a**4*d**4*sin(e + f*x)/(12*c**4*d*
*5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin(e + f*x)**2 + 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*
sin(e + f*x)**4) - 3*a**4*d**4/(12*c**4*d**5*f + 48*c**3*d**6*f*sin(e + f*x) + 72*c**2*d**7*f*sin(e + f*x)**2
+ 48*c*d**8*f*sin(e + f*x)**3 + 12*d**9*f*sin(e + f*x)**4), Eq(n, -5)), (-12*a**4*c**4*log(c/d + sin(e + f*x))
/(3*c**3*d**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) - 22*a**
4*c**4/(3*c**3*d**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) -
36*a**4*c**3*d*log(c/d + sin(e + f*x))*sin(e + f*x)/(3*c**3*d**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*s
in(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) + 12*a**4*c**3*d*log(c/d + sin(e + f*x))/(3*c**3*d**5*f + 9*c**2*d*
*6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) - 54*a**4*c**3*d*sin(e + f*x)/(3*c*
*3*d**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) + 22*a**4*c**3
*d/(3*c**3*d**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) - 36*a
**4*c**2*d**2*log(c/d + sin(e + f*x))*sin(e + f*x)**2/(3*c**3*d**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f
*sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) + 36*a**4*c**2*d**2*log(c/d + sin(e + f*x))*sin(e + f*x)/(3*c**3*
d**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) - 36*a**4*c**2*d*
*2*sin(e + f*x)**2/(3*c**3*d**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f*sin(e +
f*x)**3) + 54*a**4*c**2*d**2*sin(e + f*x)/(3*c**3*d**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*
x)**2 + 3*d**8*f*sin(e + f*x)**3) - 6*a**4*c**2*d**2/(3*c**3*d**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*
sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) - 12*a**4*c*d**3*log(c/d + sin(e + f*x))*sin(e + f*x)**3/(3*c**3*d
**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) + 36*a**4*c*d**3*l
og(c/d + sin(e + f*x))*sin(e + f*x)**2/(3*c**3*d**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*x)**
2 + 3*d**8*f*sin(e + f*x)**3) + 36*a**4*c*d**3*sin(e + f*x)**2/(3*c**3*d**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9
*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) - 18*a**4*c*d**3*sin(e + f*x)/(3*c**3*d**5*f + 9*c**2*d*
*6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) - 2*a**4*c*d**3/(3*c**3*d**5*f + 9*
c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) + 12*a**4*d**4*log(c/d + sin
(e + f*x))*sin(e + f*x)**3/(3*c**3*d**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f
*sin(e + f*x)**3) + 3*a**4*d**4*sin(e + f*x)**4/(3*c**3*d**5*f + 9*c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*sin(e
+ f*x)**2 + 3*d**8*f*sin(e + f*x)**3) - 18*a**4*d**4*sin(e + f*x)**2/(3*c**3*d**5*f + 9*c**2*d**6*f*sin(e + f
*x) + 9*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) - 6*a**4*d**4*sin(e + f*x)/(3*c**3*d**5*f + 9*c**
2*d**6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3) - a**4*d**4/(3*c**3*d**5*f + 9*
c**2*d**6*f*sin(e + f*x) + 9*c*d**7*f*sin(e + f*x)**2 + 3*d**8*f*sin(e + f*x)**3), Eq(n, -4)), (12*a**4*c**4*l
og(c/d + sin(e + f*x))/(2*c**2*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) + 18*a**4*c**4/(2*
c**2*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) + 24*a**4*c**3*d*log(c/d + sin(e + f*x))*sin
(e + f*x)/(2*c**2*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) - 24*a**4*c**3*d*log(c/d + sin(
e + f*x))/(2*c**2*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) + 24*a**4*c**3*d*sin(e + f*x)/(
2*c**2*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) - 36*a**4*c**3*d/(2*c**2*d**5*f + 4*c*d**6
*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) + 12*a**4*c**2*d**2*log(c/d + sin(e + f*x))*sin(e + f*x)**2/(2*c**
2*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) - 48*a**4*c**2*d**2*log(c/d + sin(e + f*x))*sin
(e + f*x)/(2*c**2*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) + 12*a**4*c**2*d**2*log(c/d + s
in(e + f*x))/(2*c**2*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) - 48*a**4*c**2*d**2*sin(e +
f*x)/(2*c**2*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) + 18*a**4*c**2*d**2/(2*c**2*d**5*f +
4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) - 24*a**4*c*d**3*log(c/d + sin(e + f*x))*sin(e + f*x)**2/
(2*c**2*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) + 24*a**4*c*d**3*log(c/d + sin(e + f*x))*
sin(e + f*x)/(2*c**2*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) - 4*a**4*c*d**3*sin(e + f*x)
**3/(2*c**2*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) + 24*a**4*c*d**3*sin(e + f*x)/(2*c**2
*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) - 4*a**4*c*d**3/(2*c**2*d**5*f + 4*c*d**6*f*sin(
e + f*x) + 2*d**7*f*sin(e + f*x)**2) + 12*a**4*d**4*log(c/d + sin(e + f*x))*sin(e + f*x)**2/(2*c**2*d**5*f + 4
*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2) + a**4*d**4*sin(e + f*x)**4/(2*c**2*d**5*f + 4*c*d**6*f*sin
(e + f*x) + 2*d**7*f*sin(e + f*x)**2) + 8*a**4*d**4*sin(e + f*x)**3/(2*c**2*d**5*f + 4*c*d**6*f*sin(e + f*x) +
2*d**7*f*sin(e + f*x)**2) - 8*a**4*d**4*sin(e + f*x)/(2*c**2*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(
e + f*x)**2) - a**4*d**4/(2*c**2*d**5*f + 4*c*d**6*f*sin(e + f*x) + 2*d**7*f*sin(e + f*x)**2), Eq(n, -3)), (-1
2*a**4*c**4*log(c/d + sin(e + f*x))/(3*c*d**5*f + 3*d**6*f*sin(e + f*x)) - 12*a**4*c**4/(3*c*d**5*f + 3*d**6*f
*sin(e + f*x)) - 12*a**4*c**3*d*log(c/d + sin(e + f*x))*sin(e + f*x)/(3*c*d**5*f + 3*d**6*f*sin(e + f*x)) + 36
*a**4*c**3*d*log(c/d + sin(e + f*x))/(3*c*d**5*f + 3*d**6*f*sin(e + f*x)) + 36*a**4*c**3*d/(3*c*d**5*f + 3*d**
6*f*sin(e + f*x)) + 36*a**4*c**2*d**2*log(c/d + sin(e + f*x))*sin(e + f*x)/(3*c*d**5*f + 3*d**6*f*sin(e + f*x)
) - 36*a**4*c**2*d**2*log(c/d + sin(e + f*x))/(3*c*d**5*f + 3*d**6*f*sin(e + f*x)) + 6*a**4*c**2*d**2*sin(e +
f*x)**2/(3*c*d**5*f + 3*d**6*f*sin(e + f*x)) - 36*a**4*c**2*d**2/(3*c*d**5*f + 3*d**6*f*sin(e + f*x)) - 36*a**
4*c*d**3*log(c/d + sin(e + f*x))*sin(e + f*x)/(3*c*d**5*f + 3*d**6*f*sin(e + f*x)) + 12*a**4*c*d**3*log(c/d +
sin(e + f*x))/(3*c*d**5*f + 3*d**6*f*sin(e + f*x)) - 2*a**4*c*d**3*sin(e + f*x)**3/(3*c*d**5*f + 3*d**6*f*sin(
e + f*x)) - 18*a**4*c*d**3*sin(e + f*x)**2/(3*c*d**5*f + 3*d**6*f*sin(e + f*x)) + 12*a**4*c*d**3/(3*c*d**5*f +
3*d**6*f*sin(e + f*x)) + 12*a**4*d**4*log(c/d + sin(e + f*x))*sin(e + f*x)/(3*c*d**5*f + 3*d**6*f*sin(e + f*x
)) + a**4*d**4*sin(e + f*x)**4/(3*c*d**5*f + 3*d**6*f*sin(e + f*x)) + 6*a**4*d**4*sin(e + f*x)**3/(3*c*d**5*f
+ 3*d**6*f*sin(e + f*x)) + 18*a**4*d**4*sin(e + f*x)**2/(3*c*d**5*f + 3*d**6*f*sin(e + f*x)) - 3*a**4*d**4/(3*
c*d**5*f + 3*d**6*f*sin(e + f*x)), Eq(n, -2)), (a**4*c**4*log(c/d + sin(e + f*x))/(d**5*f) - 4*a**4*c**3*log(c
/d + sin(e + f*x))/(d**4*f) - a**4*c**3*sin(e + f*x)/(d**4*f) + 6*a**4*c**2*log(c/d + sin(e + f*x))/(d**3*f) +
a**4*c**2*sin(e + f*x)**2/(2*d**3*f) + 4*a**4*c**2*sin(e + f*x)/(d**3*f) - 4*a**4*c*log(c/d + sin(e + f*x))/(
d**2*f) - a**4*c*sin(e + f*x)**3/(3*d**2*f) - 2*a**4*c*sin(e + f*x)**2/(d**2*f) - 6*a**4*c*sin(e + f*x)/(d**2*
f) + a**4*log(c/d + sin(e + f*x))/(d*f) + a**4*sin(e + f*x)**4/(4*d*f) + 4*a**4*sin(e + f*x)**3/(3*d*f) + 3*a*
*4*sin(e + f*x)**2/(d*f) + 4*a**4*sin(e + f*x)/(d*f), Eq(n, -1)), (24*a**4*c**5*(c + d*sin(e + f*x))**n/(d**5*
f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) - 24*a**4*c**4*d*n*(c
+ d*sin(e + f*x))**n*sin(e + f*x)/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*
f*n + 120*d**5*f) - 24*a**4*c**4*d*n*(c + d*sin(e + f*x))**n/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 +
225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) - 120*a**4*c**4*d*(c + d*sin(e + f*x))**n/(d**5*f*n**5 + 15*d**5*
f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 12*a**4*c**3*d**2*n**2*(c + d*sin(e +
f*x))**n*sin(e + f*x)**2/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 12
0*d**5*f) + 24*a**4*c**3*d**2*n**2*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**
5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 12*a**4*c**3*d**2*n**2*(c + d*sin(e + f*x))**n/(d**5
*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 12*a**4*c**3*d**2*n
*(c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 27
4*d**5*f*n + 120*d**5*f) + 120*a**4*c**3*d**2*n*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**5*f*n**5 + 15*d**5*f*
n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 108*a**4*c**3*d**2*n*(c + d*sin(e + f*x
))**n/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 240*a**4
*c**3*d**2*(c + d*sin(e + f*x))**n/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5
*f*n + 120*d**5*f) - 4*a**4*c**2*d**3*n**3*(c + d*sin(e + f*x))**n*sin(e + f*x)**3/(d**5*f*n**5 + 15*d**5*f*n*
*4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) - 12*a**4*c**2*d**3*n**3*(c + d*sin(e + f*x
))**n*sin(e + f*x)**2/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d*
*5*f) - 12*a**4*c**2*d**3*n**3*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*
n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) - 4*a**4*c**2*d**3*n**3*(c + d*sin(e + f*x))**n/(d**5*f*n*
*5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) - 12*a**4*c**2*d**3*n**2*(
c + d*sin(e + f*x))**n*sin(e + f*x)**3/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*
d**5*f*n + 120*d**5*f) - 72*a**4*c**2*d**3*n**2*(c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**5*f*n**5 + 15*d**5
*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) - 108*a**4*c**2*d**3*n**2*(c + d*sin(e
+ f*x))**n*sin(e + f*x)/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120
*d**5*f) - 48*a**4*c**2*d**3*n**2*(c + d*sin(e + f*x))**n/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225
*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) - 8*a**4*c**2*d**3*n*(c + d*sin(e + f*x))**n*sin(e + f*x)**3/(d**5*f
*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) - 60*a**4*c**2*d**3*n*(
c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*
d**5*f*n + 120*d**5*f) - 240*a**4*c**2*d**3*n*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**5*f*n**5 + 15*d**5*f*n*
*4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) - 188*a**4*c**2*d**3*n*(c + d*sin(e + f*x))
**n/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) - 240*a**4*c
**2*d**3*(c + d*sin(e + f*x))**n/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f
*n + 120*d**5*f) + a**4*c*d**4*n**4*(c + d*sin(e + f*x))**n*sin(e + f*x)**4/(d**5*f*n**5 + 15*d**5*f*n**4 + 85
*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 4*a**4*c*d**4*n**4*(c + d*sin(e + f*x))**n*sin(e
+ f*x)**3/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 6*a
**4*c*d**4*n**4*(c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d
**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 4*a**4*c*d**4*n**4*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**5*f*n**5
+ 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + a**4*c*d**4*n**4*(c + d*si
n(e + f*x))**n/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) +
6*a**4*c*d**4*n**3*(c + d*sin(e + f*x))**n*sin(e + f*x)**4/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 2
25*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 32*a**4*c*d**4*n**3*(c + d*sin(e + f*x))**n*sin(e + f*x)**3/(d**
5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 60*a**4*c*d**4*n**
3*(c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 2
74*d**5*f*n + 120*d**5*f) + 48*a**4*c*d**4*n**3*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**5*f*n**5 + 15*d**5*f*
n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 14*a**4*c*d**4*n**3*(c + d*sin(e + f*x)
)**n/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 11*a**4*c
*d**4*n**2*(c + d*sin(e + f*x))**n*sin(e + f*x)**4/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f
*n**2 + 274*d**5*f*n + 120*d**5*f) + 68*a**4*c*d**4*n**2*(c + d*sin(e + f*x))**n*sin(e + f*x)**3/(d**5*f*n**5
+ 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 174*a**4*c*d**4*n**2*(c + d
*sin(e + f*x))**n*sin(e + f*x)**2/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*
f*n + 120*d**5*f) + 188*a**4*c*d**4*n**2*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**5*f*n**5 + 15*d**5*f*n**4 +
85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 71*a**4*c*d**4*n**2*(c + d*sin(e + f*x))**n/(d
**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 6*a**4*c*d**4*n*
(c + d*sin(e + f*x))**n*sin(e + f*x)**4/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274
*d**5*f*n + 120*d**5*f) + 40*a**4*c*d**4*n*(c + d*sin(e + f*x))**n*sin(e + f*x)**3/(d**5*f*n**5 + 15*d**5*f*n*
*4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 120*a**4*c*d**4*n*(c + d*sin(e + f*x))**n
*sin(e + f*x)**2/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f)
+ 240*a**4*c*d**4*n*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225
*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 154*a**4*c*d**4*n*(c + d*sin(e + f*x))**n/(d**5*f*n**5 + 15*d**5*f
*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 120*a**4*c*d**4*(c + d*sin(e + f*x))**
n/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + a**4*d**5*n*
*4*(c + d*sin(e + f*x))**n*sin(e + f*x)**5/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 +
274*d**5*f*n + 120*d**5*f) + 4*a**4*d**5*n**4*(c + d*sin(e + f*x))**n*sin(e + f*x)**4/(d**5*f*n**5 + 15*d**5*f
*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 6*a**4*d**5*n**4*(c + d*sin(e + f*x))*
*n*sin(e + f*x)**3/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*
f) + 4*a**4*d**5*n**4*(c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 +
225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + a**4*d**5*n**4*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**5*f*n*
*5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 10*a**4*d**5*n**3*(c + d
*sin(e + f*x))**n*sin(e + f*x)**5/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*
f*n + 120*d**5*f) + 44*a**4*d**5*n**3*(c + d*sin(e + f*x))**n*sin(e + f*x)**4/(d**5*f*n**5 + 15*d**5*f*n**4 +
85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 72*a**4*d**5*n**3*(c + d*sin(e + f*x))**n*sin(
e + f*x)**3/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 52
*a**4*d**5*n**3*(c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d
**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 14*a**4*d**5*n**3*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**5*f*n**5
+ 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 35*a**4*d**5*n**2*(c + d*si
n(e + f*x))**n*sin(e + f*x)**5/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n
+ 120*d**5*f) + 164*a**4*d**5*n**2*(c + d*sin(e + f*x))**n*sin(e + f*x)**4/(d**5*f*n**5 + 15*d**5*f*n**4 + 85
*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 294*a**4*d**5*n**2*(c + d*sin(e + f*x))**n*sin(e
+ f*x)**3/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 236
*a**4*d**5*n**2*(c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d
**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 71*a**4*d**5*n**2*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**5*f*n**5
+ 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 50*a**4*d**5*n*(c + d*sin(e
+ f*x))**n*sin(e + f*x)**5/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n +
120*d**5*f) + 244*a**4*d**5*n*(c + d*sin(e + f*x))**n*sin(e + f*x)**4/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*
f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 468*a**4*d**5*n*(c + d*sin(e + f*x))**n*sin(e + f*x)**
3/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 428*a**4*d**
5*n*(c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 +
274*d**5*f*n + 120*d**5*f) + 154*a**4*d**5*n*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**5*f*n**5 + 15*d**5*f*n*
*4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 24*a**4*d**5*(c + d*sin(e + f*x))**n*sin(
e + f*x)**5/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 12
0*a**4*d**5*(c + d*sin(e + f*x))**n*sin(e + f*x)**4/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*
f*n**2 + 274*d**5*f*n + 120*d**5*f) + 240*a**4*d**5*(c + d*sin(e + f*x))**n*sin(e + f*x)**3/(d**5*f*n**5 + 15*
d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f) + 240*a**4*d**5*(c + d*sin(e + f*x
))**n*sin(e + f*x)**2/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 225*d**5*f*n**2 + 274*d**5*f*n + 120*d*
*5*f) + 120*a**4*d**5*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**5*f*n**5 + 15*d**5*f*n**4 + 85*d**5*f*n**3 + 22
5*d**5*f*n**2 + 274*d**5*f*n + 120*d**5*f), True))
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