\(\int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx\) [256]

Optimal result

Integrand size = 26, antiderivative size = 34 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7} \]

[Out]

1/7*a^3*c^3*cos(f*x+e)^7/f/(c-c*sin(f*x+e))^7

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2815, 2750} \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7} \]

[In]

Int[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^4,x]

[Out]

(a^3*c^3*Cos[e + f*x]^7)/(7*f*(c - c*Sin[e + f*x])^7)

Rule 2750

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*m)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^
2, 0] && EqQ[Simplify[m + p + 1], 0] &&  !ILtQ[p, 0]

Rule 2815

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Di
st[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&
EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0,
n, m] || LtQ[m, n, 0]))

Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^7} \, dx \\ & = \frac {a^3 c^3 \cos ^7(e+f x)}{7 f (c-c \sin (e+f x))^7} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(34)=68\).

Time = 2.36 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.74 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {a^3 \left (35 \cos \left (\frac {1}{2} (e+f x)\right )-21 \cos \left (\frac {3}{2} (e+f x)\right )-7 \cos \left (\frac {5}{2} (e+f x)\right )+\cos \left (\frac {7}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{28 c^4 f (-1+\sin (e+f x))^4} \]

[In]

Integrate[(a + a*Sin[e + f*x])^3/(c - c*Sin[e + f*x])^4,x]

[Out]

(a^3*(35*Cos[(e + f*x)/2] - 21*Cos[(3*(e + f*x))/2] - 7*Cos[(5*(e + f*x))/2] + Cos[(7*(e + f*x))/2])*(Cos[(e +
f*x)/2] - Sin[(e + f*x)/2]))/(28*c^4*f*(-1 + Sin[e + f*x])^4)

Maple [A] (verified)

Time = 0.97 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.88

method	result	size
parallelrisch	\(-\frac {2 a^{3} \left (\frac {1}{7}+\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )+5 \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+3 \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{f \,c^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\)	\(64\)
risch	\(-\frac {2 \left (7 a^{3} {\mathrm e}^{6 i \left (f x +e \right )}-35 a^{3} {\mathrm e}^{4 i \left (f x +e \right )}+21 a^{3} {\mathrm e}^{2 i \left (f x +e \right )}-a^{3}\right )}{7 \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )^{7} f \,c^{4}}\)	\(71\)
derivativedivides	\(\frac {2 a^{3} \left (-\frac {64}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {6}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {48}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {40}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {20}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{4}}\)	\(118\)
default	\(\frac {2 a^{3} \left (-\frac {64}{7 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}-\frac {6}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}-\frac {32}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{6}}-\frac {48}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{5}}-\frac {1}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}-\frac {40}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{4}}-\frac {20}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}\right )}{f \,c^{4}}\)	\(118\)
norman	\(\frac {-\frac {2 a^{3}}{7 c f}-\frac {48 a^{3} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {202 a^{3} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {352 a^{3} \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{7 c f}-\frac {42 a^{3} \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {16 a^{3} \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {2 a^{3} \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} c^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{7}}\)	\(177\)

[In]

int((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x,method=_RETURNVERBOSE)

[Out]

-2*a^3*(1/7+tan(1/2*f*x+1/2*e)^6+5*tan(1/2*f*x+1/2*e)^4+3*tan(1/2*f*x+1/2*e)^2)/f/c^4/(tan(1/2*f*x+1/2*e)-1)^7

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 222 vs. \(2 (33) = 66\).

Time = 0.25 (sec) , antiderivative size = 222, normalized size of antiderivative = 6.53 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {a^{3} \cos \left (f x + e\right )^{4} - 3 \, a^{3} \cos \left (f x + e\right )^{3} - 8 \, a^{3} \cos \left (f x + e\right )^{2} + 4 \, a^{3} \cos \left (f x + e\right ) + 8 \, a^{3} - {\left (a^{3} \cos \left (f x + e\right )^{3} + 4 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} \cos \left (f x + e\right ) - 8 \, a^{3}\right )} \sin \left (f x + e\right )}{7 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \]

[In]

integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x, algorithm="fricas")

[Out]

1/7*(a^3*cos(f*x + e)^4 - 3*a^3*cos(f*x + e)^3 - 8*a^3*cos(f*x + e)^2 + 4*a^3*cos(f*x + e) + 8*a^3 - (a^3*cos(
f*x + e)^3 + 4*a^3*cos(f*x + e)^2 - 4*a^3*cos(f*x + e) - 8*a^3)*sin(f*x + e))/(c^4*f*cos(f*x + e)^4 - 3*c^4*f*
cos(f*x + e)^3 - 8*c^4*f*cos(f*x + e)^2 + 4*c^4*f*cos(f*x + e) + 8*c^4*f + (c^4*f*cos(f*x + e)^3 + 4*c^4*f*cos
(f*x + e)^2 - 4*c^4*f*cos(f*x + e) - 8*c^4*f)*sin(f*x + e))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 619 vs. \(2 (29) = 58\).

Time = 14.26 (sec) , antiderivative size = 619, normalized size of antiderivative = 18.21 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\begin {cases} - \frac {14 a^{3} \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{7 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 49 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 147 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 147 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 49 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 7 c^{4} f} - \frac {70 a^{3} \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{7 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 49 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 147 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 147 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 49 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 7 c^{4} f} - \frac {42 a^{3} \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{7 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 49 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 147 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 147 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 49 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 7 c^{4} f} - \frac {2 a^{3}}{7 c^{4} f \tan ^{7}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 49 c^{4} f \tan ^{6}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 147 c^{4} f \tan ^{5}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 245 c^{4} f \tan ^{4}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 245 c^{4} f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - 147 c^{4} f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 49 c^{4} f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - 7 c^{4} f} & \text {for}\: f \neq 0 \\\frac {x \left (a \sin {\left (e \right )} + a\right )^{3}}{\left (- c \sin {\left (e \right )} + c\right )^{4}} & \text {otherwise} \end {cases} \]

[In]

integrate((a+a*sin(f*x+e))**3/(c-c*sin(f*x+e))**4,x)

[Out]

Piecewise((-14*a**3*tan(e/2 + f*x/2)**6/(7*c**4*f*tan(e/2 + f*x/2)**7 - 49*c**4*f*tan(e/2 + f*x/2)**6 + 147*c*
*4*f*tan(e/2 + f*x/2)**5 - 245*c**4*f*tan(e/2 + f*x/2)**4 + 245*c**4*f*tan(e/2 + f*x/2)**3 - 147*c**4*f*tan(e/
2 + f*x/2)**2 + 49*c**4*f*tan(e/2 + f*x/2) - 7*c**4*f) - 70*a**3*tan(e/2 + f*x/2)**4/(7*c**4*f*tan(e/2 + f*x/2
)**7 - 49*c**4*f*tan(e/2 + f*x/2)**6 + 147*c**4*f*tan(e/2 + f*x/2)**5 - 245*c**4*f*tan(e/2 + f*x/2)**4 + 245*c
**4*f*tan(e/2 + f*x/2)**3 - 147*c**4*f*tan(e/2 + f*x/2)**2 + 49*c**4*f*tan(e/2 + f*x/2) - 7*c**4*f) - 42*a**3*
tan(e/2 + f*x/2)**2/(7*c**4*f*tan(e/2 + f*x/2)**7 - 49*c**4*f*tan(e/2 + f*x/2)**6 + 147*c**4*f*tan(e/2 + f*x/2
)**5 - 245*c**4*f*tan(e/2 + f*x/2)**4 + 245*c**4*f*tan(e/2 + f*x/2)**3 - 147*c**4*f*tan(e/2 + f*x/2)**2 + 49*c
**4*f*tan(e/2 + f*x/2) - 7*c**4*f) - 2*a**3/(7*c**4*f*tan(e/2 + f*x/2)**7 - 49*c**4*f*tan(e/2 + f*x/2)**6 + 14
7*c**4*f*tan(e/2 + f*x/2)**5 - 245*c**4*f*tan(e/2 + f*x/2)**4 + 245*c**4*f*tan(e/2 + f*x/2)**3 - 147*c**4*f*ta
n(e/2 + f*x/2)**2 + 49*c**4*f*tan(e/2 + f*x/2) - 7*c**4*f), Ne(f, 0)), (x*(a*sin(e) + a)**3/(-c*sin(e) + c)**4
, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1045 vs. \(2 (33) = 66\).

Time = 0.22 (sec) , antiderivative size = 1045, normalized size of antiderivative = 30.74 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x, algorithm="maxima")

[Out]

2/35*(a^3*(91*sin(f*x + e)/(cos(f*x + e) + 1) - 168*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 280*sin(f*x + e)^3/(
cos(f*x + e) + 1)^3 - 175*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 105*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 13)/
(c^4 - 7*c^4*sin(f*x + e)/(cos(f*x + e) + 1) + 21*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 35*c^4*sin(f*x + e
)^3/(cos(f*x + e) + 1)^3 + 35*c^4*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 21*c^4*sin(f*x + e)^5/(cos(f*x + e) +
1)^5 + 7*c^4*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - c^4*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) - a^3*(49*sin(f*x
+ e)/(cos(f*x + e) + 1) - 147*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 210*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 -
210*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 + 105*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 - 35*sin(f*x + e)^6/(cos(f*x
+ e) + 1)^6 - 12)/(c^4 - 7*c^4*sin(f*x + e)/(cos(f*x + e) + 1) + 21*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 -
35*c^4*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 35*c^4*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 21*c^4*sin(f*x + e)
^5/(cos(f*x + e) + 1)^5 + 7*c^4*sin(f*x + e)^6/(cos(f*x + e) + 1)^6 - c^4*sin(f*x + e)^7/(cos(f*x + e) + 1)^7)
- 4*a^3*(14*sin(f*x + e)/(cos(f*x + e) + 1) - 42*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + 35*sin(f*x + e)^3/(cos
(f*x + e) + 1)^3 - 35*sin(f*x + e)^4/(cos(f*x + e) + 1)^4 - 2)/(c^4 - 7*c^4*sin(f*x + e)/(cos(f*x + e) + 1) +
21*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 35*c^4*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 35*c^4*sin(f*x + e)^
4/(cos(f*x + e) + 1)^4 - 21*c^4*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 7*c^4*sin(f*x + e)^6/(cos(f*x + e) + 1)^
6 - c^4*sin(f*x + e)^7/(cos(f*x + e) + 1)^7) + 2*a^3*(7*sin(f*x + e)/(cos(f*x + e) + 1) - 21*sin(f*x + e)^2/(c
os(f*x + e) + 1)^2 + 35*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 - 1)/(c^4 - 7*c^4*sin(f*x + e)/(cos(f*x + e) + 1)
+ 21*c^4*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 - 35*c^4*sin(f*x + e)^3/(cos(f*x + e) + 1)^3 + 35*c^4*sin(f*x + e
)^4/(cos(f*x + e) + 1)^4 - 21*c^4*sin(f*x + e)^5/(cos(f*x + e) + 1)^5 + 7*c^4*sin(f*x + e)^6/(cos(f*x + e) + 1
)^6 - c^4*sin(f*x + e)^7/(cos(f*x + e) + 1)^7))/f

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (33) = 66\).

Time = 0.33 (sec) , antiderivative size = 73, normalized size of antiderivative = 2.15 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=-\frac {2 \, {\left (7 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 35 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 21 \, a^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + a^{3}\right )}}{7 \, c^{4} f {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{7}} \]

[In]

integrate((a+a*sin(f*x+e))^3/(c-c*sin(f*x+e))^4,x, algorithm="giac")

[Out]

-2/7*(7*a^3*tan(1/2*f*x + 1/2*e)^6 + 35*a^3*tan(1/2*f*x + 1/2*e)^4 + 21*a^3*tan(1/2*f*x + 1/2*e)^2 + a^3)/(c^4
*f*(tan(1/2*f*x + 1/2*e) - 1)^7)

Mupad [B] (verification not implemented)

Time = 6.29 (sec) , antiderivative size = 116, normalized size of antiderivative = 3.41 \[ \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^4} \, dx=\frac {2\,a^3\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left ({\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+21\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+35\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+7\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\right )}{7\,c^4\,f\,{\left (\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )-\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}^7} \]

[In]

int((a + a*sin(e + f*x))^3/(c - c*sin(e + f*x))^4,x)

[Out]

(2*a^3*cos(e/2 + (f*x)/2)*(cos(e/2 + (f*x)/2)^6 + 7*sin(e/2 + (f*x)/2)^6 + 35*cos(e/2 + (f*x)/2)^2*sin(e/2 + (
f*x)/2)^4 + 21*cos(e/2 + (f*x)/2)^4*sin(e/2 + (f*x)/2)^2))/(7*c^4*f*(cos(e/2 + (f*x)/2) - sin(e/2 + (f*x)/2))^
7)