3.915 \(\int \cos (e+f x) (a+a \sin (e+f x))^2 (c+d \sin (e+f x))^n \, dx\)

Optimal. Leaf size=101 \[ \frac {a^2 (c-d)^2 (c+d \sin (e+f x))^{n+1}}{d^3 f (n+1)}-\frac {2 a^2 (c-d) (c+d \sin (e+f x))^{n+2}}{d^3 f (n+2)}+\frac {a^2 (c+d \sin (e+f x))^{n+3}}{d^3 f (n+3)} \]

[Out]

a^2*(c-d)^2*(c+d*sin(f*x+e))^(1+n)/d^3/f/(1+n)-2*a^2*(c-d)*(c+d*sin(f*x+e))^(2+n)/d^3/f/(2+n)+a^2*(c+d*sin(f*x
+e))^(3+n)/d^3/f/(3+n)

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Rubi [A]  time = 0.15, antiderivative size = 101, 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} \[ \frac {a^2 (c-d)^2 (c+d \sin (e+f x))^{n+1}}{d^3 f (n+1)}-\frac {2 a^2 (c-d) (c+d \sin (e+f x))^{n+2}}{d^3 f (n+2)}+\frac {a^2 (c+d \sin (e+f x))^{n+3}}{d^3 f (n+3)} \]

Antiderivative was successfully verified.

[In]

Int[Cos[e + f*x]*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]

[Out]

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

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Mathematica [A]  time = 0.36, size = 78, normalized size = 0.77 \[ \frac {a^2 (c+d \sin (e+f x))^{n+1} \left (-\frac {2 (c-d) (c+d \sin (e+f x))}{n+2}+\frac {(c+d \sin (e+f x))^2}{n+3}+\frac {(c-d)^2}{n+1}\right )}{d^3 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[e + f*x]*(a + a*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^n,x]

[Out]

(a^2*(c + d*Sin[e + f*x])^(1 + n)*((c - d)^2/(1 + n) - (2*(c - d)*(c + d*Sin[e + f*x]))/(2 + n) + (c + d*Sin[e
+ f*x])^2/(3 + n)))/(d^3*f)

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fricas [B]  time = 0.51, size = 294, normalized size = 2.91 \[ \frac {{\left (2 \, a^{2} c^{3} - 6 \, a^{2} c^{2} d + 6 \, a^{2} c d^{2} + 6 \, a^{2} d^{3} + 2 \, {\left (a^{2} c d^{2} + a^{2} d^{3}\right )} n^{2} - {\left (6 \, a^{2} d^{3} + {\left (a^{2} c d^{2} + 2 \, a^{2} d^{3}\right )} n^{2} + {\left (a^{2} c d^{2} + 8 \, a^{2} d^{3}\right )} n\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (a^{2} c^{2} d - 3 \, a^{2} c d^{2} - 4 \, a^{2} d^{3}\right )} n + {\left (8 \, a^{2} d^{3} + 2 \, {\left (a^{2} c d^{2} + a^{2} d^{3}\right )} n^{2} - {\left (a^{2} d^{3} n^{2} + 3 \, a^{2} d^{3} n + 2 \, a^{2} d^{3}\right )} \cos \left (f x + e\right )^{2} - 2 \, {\left (a^{2} c^{2} d - 3 \, a^{2} c d^{2} - 4 \, a^{2} d^{3}\right )} n\right )} \sin \left (f x + e\right )\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{d^{3} f n^{3} + 6 \, d^{3} f n^{2} + 11 \, d^{3} f n + 6 \, d^{3} f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^n,x, algorithm="fricas")

[Out]

(2*a^2*c^3 - 6*a^2*c^2*d + 6*a^2*c*d^2 + 6*a^2*d^3 + 2*(a^2*c*d^2 + a^2*d^3)*n^2 - (6*a^2*d^3 + (a^2*c*d^2 + 2
*a^2*d^3)*n^2 + (a^2*c*d^2 + 8*a^2*d^3)*n)*cos(f*x + e)^2 - 2*(a^2*c^2*d - 3*a^2*c*d^2 - 4*a^2*d^3)*n + (8*a^2
*d^3 + 2*(a^2*c*d^2 + a^2*d^3)*n^2 - (a^2*d^3*n^2 + 3*a^2*d^3*n + 2*a^2*d^3)*cos(f*x + e)^2 - 2*(a^2*c^2*d - 3
*a^2*c*d^2 - 4*a^2*d^3)*n)*sin(f*x + e))*(d*sin(f*x + e) + c)^n/(d^3*f*n^3 + 6*d^3*f*n^2 + 11*d^3*f*n + 6*d^3*
f)

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giac [B]  time = 0.19, size = 463, normalized size = 4.58 \[ \frac {\frac {{\left ({\left (d \sin \left (f x + e\right ) + c\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} n^{2} - 2 \, {\left (d \sin \left (f x + e\right ) + c\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c n^{2} + {\left (d \sin \left (f x + e\right ) + c\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c^{2} n^{2} + 3 \, {\left (d \sin \left (f x + e\right ) + c\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} n - 8 \, {\left (d \sin \left (f x + e\right ) + c\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c n + 5 \, {\left (d \sin \left (f x + e\right ) + c\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c^{2} n + 2 \, {\left (d \sin \left (f x + e\right ) + c\right )}^{3} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} - 6 \, {\left (d \sin \left (f x + e\right ) + c\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c + 6 \, {\left (d \sin \left (f x + e\right ) + c\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c^{2}\right )} a^{2}}{d^{2} n^{3} + 6 \, d^{2} n^{2} + 11 \, d^{2} n + 6 \, d^{2}} + \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n + 1} a^{2}}{n + 1} + \frac {2 \, {\left ({\left (d \sin \left (f x + e\right ) + c\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} n - {\left (d \sin \left (f x + e\right ) + c\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c n + {\left (d \sin \left (f x + e\right ) + c\right )}^{2} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} - 2 \, {\left (d \sin \left (f x + e\right ) + c\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} c\right )} a^{2}}{{\left (n^{2} + 3 \, n + 2\right )} d}}{d f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^n,x, algorithm="giac")

[Out]

(((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*s
in(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*sin(f*x +
e) + c)*(d*sin(f*x + e) + c)^n*c^2)*a^2/(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^2/(n + 1) + 2*((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^2
/((n^2 + 3*n + 2)*d))/(d*f)

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maple [F]  time = 3.25, size = 0, normalized size = 0.00 \[ \int \cos \left (f x +e \right ) \left (a +a \sin \left (f x +e \right )\right )^{2} \left (c +d \sin \left (f x +e \right )\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(f*x+e)*(a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^n,x)

[Out]

int(cos(f*x+e)*(a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^n,x)

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maxima [A]  time = 0.71, size = 183, normalized size = 1.81 \[ \frac {\frac {2 \, {\left (d^{2} {\left (n + 1\right )} \sin \left (f x + e\right )^{2} + c d n \sin \left (f x + e\right ) - c^{2}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} a^{2}}{{\left (n^{2} + 3 \, n + 2\right )} d^{2}} + \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n + 1} a^{2}}{d {\left (n + 1\right )}} + \frac {{\left ({\left (n^{2} + 3 \, n + 2\right )} d^{3} \sin \left (f x + e\right )^{3} + {\left (n^{2} + n\right )} c d^{2} \sin \left (f x + e\right )^{2} - 2 \, c^{2} d n \sin \left (f x + e\right ) + 2 \, c^{3}\right )} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} a^{2}}{{\left (n^{3} + 6 \, n^{2} + 11 \, n + 6\right )} d^{3}}}{f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))^2*(c+d*sin(f*x+e))^n,x, algorithm="maxima")

[Out]

(2*(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^2/((n^2 + 3*n + 2)*d^2) +
(d*sin(f*x + e) + c)^(n + 1)*a^2/(d*(n + 1)) + ((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^2/((n^3 + 6*n^2 + 11*n + 6)*d^3))/f

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mupad [B]  time = 11.26, size = 302, normalized size = 2.99 \[ \frac {a^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n\,\left (24\,c\,d^2-24\,c^2\,d+16\,d^3\,n+30\,d^3\,\sin \left (e+f\,x\right )+8\,c^3+12\,d^3-12\,d^3\,\cos \left (2\,e+2\,f\,x\right )+4\,d^3\,n^2-2\,d^3\,\sin \left (3\,e+3\,f\,x\right )+29\,d^3\,n\,\sin \left (e+f\,x\right )+6\,c\,d^2\,n^2-16\,d^3\,n\,\cos \left (2\,e+2\,f\,x\right )-3\,d^3\,n\,\sin \left (3\,e+3\,f\,x\right )+7\,d^3\,n^2\,\sin \left (e+f\,x\right )-4\,d^3\,n^2\,\cos \left (2\,e+2\,f\,x\right )-d^3\,n^2\,\sin \left (3\,e+3\,f\,x\right )+22\,c\,d^2\,n-8\,c^2\,d\,n-2\,c\,d^2\,n^2\,\cos \left (2\,e+2\,f\,x\right )+24\,c\,d^2\,n\,\sin \left (e+f\,x\right )-8\,c^2\,d\,n\,\sin \left (e+f\,x\right )-2\,c\,d^2\,n\,\cos \left (2\,e+2\,f\,x\right )+8\,c\,d^2\,n^2\,\sin \left (e+f\,x\right )\right )}{4\,d^3\,f\,\left (n^3+6\,n^2+11\,n+6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(e + f*x)*(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x))^n,x)

[Out]

(a^2*(c + d*sin(e + f*x))^n*(24*c*d^2 - 24*c^2*d + 16*d^3*n + 30*d^3*sin(e + f*x) + 8*c^3 + 12*d^3 - 12*d^3*co
s(2*e + 2*f*x) + 4*d^3*n^2 - 2*d^3*sin(3*e + 3*f*x) + 29*d^3*n*sin(e + f*x) + 6*c*d^2*n^2 - 16*d^3*n*cos(2*e +
2*f*x) - 3*d^3*n*sin(3*e + 3*f*x) + 7*d^3*n^2*sin(e + f*x) - 4*d^3*n^2*cos(2*e + 2*f*x) - d^3*n^2*sin(3*e + 3
*f*x) + 22*c*d^2*n - 8*c^2*d*n - 2*c*d^2*n^2*cos(2*e + 2*f*x) + 24*c*d^2*n*sin(e + f*x) - 8*c^2*d*n*sin(e + f*
x) - 2*c*d^2*n*cos(2*e + 2*f*x) + 8*c*d^2*n^2*sin(e + f*x)))/(4*d^3*f*(11*n + 6*n^2 + n^3 + 6))

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sympy [A]  time = 20.59, size = 2159, normalized size = 21.38 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(f*x+e)*(a+a*sin(f*x+e))**2*(c+d*sin(f*x+e))**n,x)

[Out]

Piecewise((c**n*(a**2*sin(e + f*x)**3/(3*f) + a**2*sin(e + f*x)**2/f + a**2*sin(e + f*x)/f), Eq(d, 0)), (x*(c
+ d*sin(e))**n*(a*sin(e) + a)**2*cos(e), Eq(f, 0)), (2*a**2*c**2*log(c/d + sin(e + f*x))/(2*c**2*d**3*f + 4*c*
d**4*f*sin(e + f*x) + 2*d**5*f*sin(e + f*x)**2) + 3*a**2*c**2/(2*c**2*d**3*f + 4*c*d**4*f*sin(e + f*x) + 2*d**
5*f*sin(e + f*x)**2) + 4*a**2*c*d*log(c/d + sin(e + f*x))*sin(e + f*x)/(2*c**2*d**3*f + 4*c*d**4*f*sin(e + f*x
) + 2*d**5*f*sin(e + f*x)**2) + 4*a**2*c*d*sin(e + f*x)/(2*c**2*d**3*f + 4*c*d**4*f*sin(e + f*x) + 2*d**5*f*si
n(e + f*x)**2) - 2*a**2*c*d/(2*c**2*d**3*f + 4*c*d**4*f*sin(e + f*x) + 2*d**5*f*sin(e + f*x)**2) + 2*a**2*d**2
*log(c/d + sin(e + f*x))*sin(e + f*x)**2/(2*c**2*d**3*f + 4*c*d**4*f*sin(e + f*x) + 2*d**5*f*sin(e + f*x)**2)
- 4*a**2*d**2*sin(e + f*x)/(2*c**2*d**3*f + 4*c*d**4*f*sin(e + f*x) + 2*d**5*f*sin(e + f*x)**2) - a**2*d**2/(2
*c**2*d**3*f + 4*c*d**4*f*sin(e + f*x) + 2*d**5*f*sin(e + f*x)**2), Eq(n, -3)), (-2*a**2*c**2*log(c/d + sin(e
+ f*x))/(c*d**3*f + d**4*f*sin(e + f*x)) - 2*a**2*c**2/(c*d**3*f + d**4*f*sin(e + f*x)) - 2*a**2*c*d*log(c/d +
sin(e + f*x))*sin(e + f*x)/(c*d**3*f + d**4*f*sin(e + f*x)) + 2*a**2*c*d*log(c/d + sin(e + f*x))/(c*d**3*f +
d**4*f*sin(e + f*x)) + 2*a**2*c*d/(c*d**3*f + d**4*f*sin(e + f*x)) + 2*a**2*d**2*log(c/d + sin(e + f*x))*sin(e
+ f*x)/(c*d**3*f + d**4*f*sin(e + f*x)) + a**2*d**2*sin(e + f*x)**2/(c*d**3*f + d**4*f*sin(e + f*x)) - a**2*d
**2/(c*d**3*f + d**4*f*sin(e + f*x)), Eq(n, -2)), (a**2*c**2*log(c/d + sin(e + f*x))/(d**3*f) - 2*a**2*c*log(c
/d + sin(e + f*x))/(d**2*f) - a**2*c*sin(e + f*x)/(d**2*f) + a**2*log(c/d + sin(e + f*x))/(d*f) + a**2*sin(e +
f*x)**2/(2*d*f) + 2*a**2*sin(e + f*x)/(d*f), Eq(n, -1)), (2*a**2*c**3*(c + d*sin(e + f*x))**n/(d**3*f*n**3 +
6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) - 2*a**2*c**2*d*n*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**3*f*n**3 +
6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) - 2*a**2*c**2*d*n*(c + d*sin(e + f*x))**n/(d**3*f*n**3 + 6*d**3*f*n**2
+ 11*d**3*f*n + 6*d**3*f) - 6*a**2*c**2*d*(c + d*sin(e + f*x))**n/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n
+ 6*d**3*f) + a**2*c*d**2*n**2*(c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*
f*n + 6*d**3*f) + 2*a**2*c*d**2*n**2*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d*
*3*f*n + 6*d**3*f) + a**2*c*d**2*n**2*(c + d*sin(e + f*x))**n/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d
**3*f) + a**2*c*d**2*n*(c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*
d**3*f) + 6*a**2*c*d**2*n*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*
d**3*f) + 5*a**2*c*d**2*n*(c + d*sin(e + f*x))**n/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 6*a
**2*c*d**2*(c + d*sin(e + f*x))**n/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + a**2*d**3*n**2*(c
+ d*sin(e + f*x))**n*sin(e + f*x)**3/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 2*a**2*d**3*n**2
*(c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + a**2*d**3*n*
*2*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 3*a**2*d**3*n
*(c + d*sin(e + f*x))**n*sin(e + f*x)**3/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 8*a**2*d**3*
n*(c + d*sin(e + f*x))**n*sin(e + f*x)**2/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 5*a**2*d**3
*n*(c + d*sin(e + f*x))**n*sin(e + f*x)/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 2*a**2*d**3*(
c + d*sin(e + f*x))**n*sin(e + f*x)**3/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 6*a**2*d**3*(c
+ d*sin(e + f*x))**n*sin(e + f*x)**2/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f) + 6*a**2*d**3*(c
+ d*sin(e + f*x))**n*sin(e + f*x)/(d**3*f*n**3 + 6*d**3*f*n**2 + 11*d**3*f*n + 6*d**3*f), True))

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