\(\int \sin ^3(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx\) [1]

Optimal result

Integrand size = 32, antiderivative size = 121 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {1}{16} a^2 c x-\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \cos ^5(e+f x)}{5 f}-\frac {a^2 c \cos (e+f x) \sin (e+f x)}{16 f}-\frac {a^2 c \cos (e+f x) \sin ^3(e+f x)}{24 f}+\frac {a^2 c \cos (e+f x) \sin ^5(e+f x)}{6 f} \]

[Out]

1/16*a^2*c*x-1/3*a^2*c*cos(f*x+e)^3/f+1/5*a^2*c*cos(f*x+e)^5/f-1/16*a^2*c*cos(f*x+e)*sin(f*x+e)/f-1/24*a^2*c*c
os(f*x+e)*sin(f*x+e)^3/f+1/6*a^2*c*cos(f*x+e)*sin(f*x+e)^5/f

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3045, 2713, 2715, 8} \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2 c \cos ^5(e+f x)}{5 f}-\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \sin ^5(e+f x) \cos (e+f x)}{6 f}-\frac {a^2 c \sin ^3(e+f x) \cos (e+f x)}{24 f}-\frac {a^2 c \sin (e+f x) \cos (e+f x)}{16 f}+\frac {1}{16} a^2 c x \]

[In]

Int[Sin[e + f*x]^3*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]

[Out]

(a^2*c*x)/16 - (a^2*c*Cos[e + f*x]^3)/(3*f) + (a^2*c*Cos[e + f*x]^5)/(5*f) - (a^2*c*Cos[e + f*x]*Sin[e + f*x])
/(16*f) - (a^2*c*Cos[e + f*x]*Sin[e + f*x]^3)/(24*f) + (a^2*c*Cos[e + f*x]*Sin[e + f*x]^5)/(6*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 3045

Int[sin[(e_.) + (f_.)*(x_)]^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.
)*(x_)]), x_Symbol] :> Int[ExpandTrig[sin[e + f*x]^n*(a + b*sin[e + f*x])^m*(A + B*sin[e + f*x]), x], x] /; Fr
eeQ[{a, b, e, f, A, B}, x] && EqQ[A*b + a*B, 0] && EqQ[a^2 - b^2, 0] && IntegerQ[m] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 c \sin ^3(e+f x)+a^2 c \sin ^4(e+f x)-a^2 c \sin ^5(e+f x)-a^2 c \sin ^6(e+f x)\right ) \, dx \\ & = \left (a^2 c\right ) \int \sin ^3(e+f x) \, dx+\left (a^2 c\right ) \int \sin ^4(e+f x) \, dx-\left (a^2 c\right ) \int \sin ^5(e+f x) \, dx-\left (a^2 c\right ) \int \sin ^6(e+f x) \, dx \\ & = -\frac {a^2 c \cos (e+f x) \sin ^3(e+f x)}{4 f}+\frac {a^2 c \cos (e+f x) \sin ^5(e+f x)}{6 f}+\frac {1}{4} \left (3 a^2 c\right ) \int \sin ^2(e+f x) \, dx-\frac {1}{6} \left (5 a^2 c\right ) \int \sin ^4(e+f x) \, dx-\frac {\left (a^2 c\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (e+f x)\right )}{f}+\frac {\left (a^2 c\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cos (e+f x)\right )}{f} \\ & = -\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \cos ^5(e+f x)}{5 f}-\frac {3 a^2 c \cos (e+f x) \sin (e+f x)}{8 f}-\frac {a^2 c \cos (e+f x) \sin ^3(e+f x)}{24 f}+\frac {a^2 c \cos (e+f x) \sin ^5(e+f x)}{6 f}+\frac {1}{8} \left (3 a^2 c\right ) \int 1 \, dx-\frac {1}{8} \left (5 a^2 c\right ) \int \sin ^2(e+f x) \, dx \\ & = \frac {3}{8} a^2 c x-\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \cos ^5(e+f x)}{5 f}-\frac {a^2 c \cos (e+f x) \sin (e+f x)}{16 f}-\frac {a^2 c \cos (e+f x) \sin ^3(e+f x)}{24 f}+\frac {a^2 c \cos (e+f x) \sin ^5(e+f x)}{6 f}-\frac {1}{16} \left (5 a^2 c\right ) \int 1 \, dx \\ & = \frac {1}{16} a^2 c x-\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \cos ^5(e+f x)}{5 f}-\frac {a^2 c \cos (e+f x) \sin (e+f x)}{16 f}-\frac {a^2 c \cos (e+f x) \sin ^3(e+f x)}{24 f}+\frac {a^2 c \cos (e+f x) \sin ^5(e+f x)}{6 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.64 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2 c (60 e+60 f x-120 \cos (e+f x)-20 \cos (3 (e+f x))+12 \cos (5 (e+f x))-15 \sin (2 (e+f x))-15 \sin (4 (e+f x))+5 \sin (6 (e+f x)))}{960 f} \]

[In]

Integrate[Sin[e + f*x]^3*(a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x]),x]

[Out]

(a^2*c*(60*e + 60*f*x - 120*Cos[e + f*x] - 20*Cos[3*(e + f*x)] + 12*Cos[5*(e + f*x)] - 15*Sin[2*(e + f*x)] - 1
5*Sin[4*(e + f*x)] + 5*Sin[6*(e + f*x)]))/(960*f)

Maple [A] (verified)

Time = 1.33 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.65

method	result	size
parallelrisch	\(-\frac {a^{2} c \left (-60 f x +120 \cos \left (f x +e \right )-5 \sin \left (6 f x +6 e \right )-12 \cos \left (5 f x +5 e \right )+15 \sin \left (4 f x +4 e \right )+20 \cos \left (3 f x +3 e \right )+15 \sin \left (2 f x +2 e \right )+128\right )}{960 f}\)	\(79\)
risch	\(\frac {a^{2} c x}{16}-\frac {a^{2} c \cos \left (f x +e \right )}{8 f}+\frac {a^{2} c \sin \left (6 f x +6 e \right )}{192 f}+\frac {a^{2} c \cos \left (5 f x +5 e \right )}{80 f}-\frac {a^{2} c \sin \left (4 f x +4 e \right )}{64 f}-\frac {a^{2} c \cos \left (3 f x +3 e \right )}{48 f}-\frac {a^{2} c \sin \left (2 f x +2 e \right )}{64 f}\)	\(114\)
derivativedivides	\(\frac {-a^{2} c \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {a^{2} c \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\)	\(147\)
default	\(\frac {-a^{2} c \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {a^{2} c \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\)	\(147\)
parts	\(-\frac {a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {a^{2} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}+\frac {a^{2} c \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}-\frac {a^{2} c \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}\)	\(155\)
norman	\(\frac {-\frac {4 a^{2} c}{15 f}+\frac {a^{2} c x}{16}-\frac {8 a^{2} c \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {8 a^{2} c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f}-\frac {4 a^{2} c \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {a^{2} c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}-\frac {17 a^{2} c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}+\frac {19 a^{2} c \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {19 a^{2} c \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {17 a^{2} c \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}+\frac {a^{2} c \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+\frac {3 a^{2} c x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}+\frac {15 a^{2} c x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {5 a^{2} c x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4}+\frac {15 a^{2} c x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}+\frac {3 a^{2} c x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8}+\frac {a^{2} c x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{16}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{6}}\)	\(320\)

[In]

int(sin(f*x+e)^3*(a+a*sin(f*x+e))^2*(c-c*sin(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

-1/960*a^2*c*(-60*f*x+120*cos(f*x+e)-5*sin(6*f*x+6*e)-12*cos(5*f*x+5*e)+15*sin(4*f*x+4*e)+20*cos(3*f*x+3*e)+15
*sin(2*f*x+2*e)+128)/f

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.75 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {48 \, a^{2} c \cos \left (f x + e\right )^{5} - 80 \, a^{2} c \cos \left (f x + e\right )^{3} + 15 \, a^{2} c f x + 5 \, {\left (8 \, a^{2} c \cos \left (f x + e\right )^{5} - 14 \, a^{2} c \cos \left (f x + e\right )^{3} + 3 \, a^{2} c \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]

[In]

integrate(sin(f*x+e)^3*(a+a*sin(f*x+e))^2*(c-c*sin(f*x+e)),x, algorithm="fricas")

[Out]

1/240*(48*a^2*c*cos(f*x + e)^5 - 80*a^2*c*cos(f*x + e)^3 + 15*a^2*c*f*x + 5*(8*a^2*c*cos(f*x + e)^5 - 14*a^2*c
*cos(f*x + e)^3 + 3*a^2*c*cos(f*x + e))*sin(f*x + e))/f

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 415 vs. \(2 (110) = 220\).

Time = 0.37 (sec) , antiderivative size = 415, normalized size of antiderivative = 3.43 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\begin {cases} - \frac {5 a^{2} c x \sin ^{6}{\left (e + f x \right )}}{16} - \frac {15 a^{2} c x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} + \frac {3 a^{2} c x \sin ^{4}{\left (e + f x \right )}}{8} - \frac {15 a^{2} c x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} + \frac {3 a^{2} c x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - \frac {5 a^{2} c x \cos ^{6}{\left (e + f x \right )}}{16} + \frac {3 a^{2} c x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {11 a^{2} c \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{16 f} + \frac {a^{2} c \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {5 a^{2} c \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} - \frac {5 a^{2} c \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {4 a^{2} c \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {a^{2} c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {5 a^{2} c \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} - \frac {3 a^{2} c \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac {8 a^{2} c \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {2 a^{2} c \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (a \sin {\left (e \right )} + a\right )^{2} \left (- c \sin {\left (e \right )} + c\right ) \sin ^{3}{\left (e \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(sin(f*x+e)**3*(a+a*sin(f*x+e))**2*(c-c*sin(f*x+e)),x)

[Out]

Piecewise((-5*a**2*c*x*sin(e + f*x)**6/16 - 15*a**2*c*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 3*a**2*c*x*sin(e
+ f*x)**4/8 - 15*a**2*c*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 3*a**2*c*x*sin(e + f*x)**2*cos(e + f*x)**2/4 -
5*a**2*c*x*cos(e + f*x)**6/16 + 3*a**2*c*x*cos(e + f*x)**4/8 + 11*a**2*c*sin(e + f*x)**5*cos(e + f*x)/(16*f) +
a**2*c*sin(e + f*x)**4*cos(e + f*x)/f + 5*a**2*c*sin(e + f*x)**3*cos(e + f*x)**3/(6*f) - 5*a**2*c*sin(e + f*x
)**3*cos(e + f*x)/(8*f) + 4*a**2*c*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - a**2*c*sin(e + f*x)**2*cos(e + f*x)
/f + 5*a**2*c*sin(e + f*x)*cos(e + f*x)**5/(16*f) - 3*a**2*c*sin(e + f*x)*cos(e + f*x)**3/(8*f) + 8*a**2*c*cos
(e + f*x)**5/(15*f) - 2*a**2*c*cos(e + f*x)**3/(3*f), Ne(f, 0)), (x*(a*sin(e) + a)**2*(-c*sin(e) + c)*sin(e)**
3, True))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.21 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {64 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} a^{2} c + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c - 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c}{960 \, f} \]

[In]

integrate(sin(f*x+e)^3*(a+a*sin(f*x+e))^2*(c-c*sin(f*x+e)),x, algorithm="maxima")

[Out]

1/960*(64*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*a^2*c + 320*(cos(f*x + e)^3 - 3*cos(f*x + e
))*a^2*c - 5*(4*sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9*sin(4*f*x + 4*e) - 48*sin(2*f*x + 2*e))*a^2*c + 30*(12*
f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*a^2*c)/f

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.93 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {1}{16} \, a^{2} c x + \frac {a^{2} c \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} - \frac {a^{2} c \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {a^{2} c \cos \left (f x + e\right )}{8 \, f} + \frac {a^{2} c \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} - \frac {a^{2} c \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} - \frac {a^{2} c \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]

[In]

integrate(sin(f*x+e)^3*(a+a*sin(f*x+e))^2*(c-c*sin(f*x+e)),x, algorithm="giac")

[Out]

1/16*a^2*c*x + 1/80*a^2*c*cos(5*f*x + 5*e)/f - 1/48*a^2*c*cos(3*f*x + 3*e)/f - 1/8*a^2*c*cos(f*x + e)/f + 1/19
2*a^2*c*sin(6*f*x + 6*e)/f - 1/64*a^2*c*sin(4*f*x + 4*e)/f - 1/64*a^2*c*sin(2*f*x + 2*e)/f

Mupad [B] (verification not implemented)

Time = 14.08 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.12 \[ \int \sin ^3(e+f x) (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2\,c\,\left (15\,e-30\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )-384\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-170\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+1140\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-640\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6-1140\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7-960\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+170\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9+30\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}+15\,f\,x+90\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (e+f\,x\right )+225\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (e+f\,x\right )+300\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (e+f\,x\right )+225\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (e+f\,x\right )+90\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (e+f\,x\right )+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (e+f\,x\right )-64\right )}{240\,f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^6} \]

[In]

int(sin(e + f*x)^3*(a + a*sin(e + f*x))^2*(c - c*sin(e + f*x)),x)

[Out]

(a^2*c*(15*e - 30*tan(e/2 + (f*x)/2) - 384*tan(e/2 + (f*x)/2)^2 - 170*tan(e/2 + (f*x)/2)^3 + 1140*tan(e/2 + (f
*x)/2)^5 - 640*tan(e/2 + (f*x)/2)^6 - 1140*tan(e/2 + (f*x)/2)^7 - 960*tan(e/2 + (f*x)/2)^8 + 170*tan(e/2 + (f*
x)/2)^9 + 30*tan(e/2 + (f*x)/2)^11 + 15*f*x + 90*tan(e/2 + (f*x)/2)^2*(e + f*x) + 225*tan(e/2 + (f*x)/2)^4*(e
+ f*x) + 300*tan(e/2 + (f*x)/2)^6*(e + f*x) + 225*tan(e/2 + (f*x)/2)^8*(e + f*x) + 90*tan(e/2 + (f*x)/2)^10*(e
+ f*x) + 15*tan(e/2 + (f*x)/2)^12*(e + f*x) - 64))/(240*f*(tan(e/2 + (f*x)/2)^2 + 1)^6)