\(\int (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx\) [240]

Optimal result

Integrand size = 24, antiderivative size = 52 \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {1}{2} a^2 c x-\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \cos (e+f x) \sin (e+f x)}{2 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2815, 2748, 2715, 8} \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=-\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \sin (e+f x) \cos (e+f x)}{2 f}+\frac {1}{2} a^2 c x \]

[In]

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

[Out]

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

Rule 8

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

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 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x
] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 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}& = (a c) \int \cos ^2(e+f x) (a+a \sin (e+f x)) \, dx \\ & = -\frac {a^2 c \cos ^3(e+f x)}{3 f}+\left (a^2 c\right ) \int \cos ^2(e+f x) \, dx \\ & = -\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \cos (e+f x) \sin (e+f x)}{2 f}+\frac {1}{2} \left (a^2 c\right ) \int 1 \, dx \\ & = \frac {1}{2} a^2 c x-\frac {a^2 c \cos ^3(e+f x)}{3 f}+\frac {a^2 c \cos (e+f x) \sin (e+f x)}{2 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.83 \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=-\frac {a^2 c (3 \cos (e+f x)+\cos (3 (e+f x))-3 (2 f x+\sin (2 (e+f x))))}{12 f} \]

[In]

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

[Out]

-1/12*(a^2*c*(3*Cos[e + f*x] + Cos[3*(e + f*x)] - 3*(2*f*x + Sin[2*(e + f*x)])))/f

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.85

method	result	size
parallelrisch	\(-\frac {a^{2} c \left (-6 f x +\cos \left (3 f x +3 e \right )-3 \sin \left (2 f x +2 e \right )+3 \cos \left (f x +e \right )+4\right )}{12 f}\)	\(44\)
risch	\(\frac {a^{2} c x}{2}-\frac {a^{2} c \cos \left (f x +e \right )}{4 f}-\frac {a^{2} c \cos \left (3 f x +3 e \right )}{12 f}+\frac {a^{2} c \sin \left (2 f x +2 e \right )}{4 f}\)	\(60\)
derivativedivides	\(\frac {\frac {a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-a^{2} c \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{2} c \cos \left (f x +e \right )+a^{2} c \left (f x +e \right )}{f}\)	\(78\)
default	\(\frac {\frac {a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-a^{2} c \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{2} c \cos \left (f x +e \right )+a^{2} c \left (f x +e \right )}{f}\)	\(78\)
parts	\(a^{2} c x -\frac {a^{2} c \cos \left (f x +e \right )}{f}-\frac {a^{2} c \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}\)	\(79\)
norman	\(\frac {\frac {a^{2} c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {2 a^{2} c}{3 f}+\frac {a^{2} c x}{2}-\frac {2 a^{2} c \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {a^{2} c \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {3 a^{2} c x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\frac {3 a^{2} c x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\frac {a^{2} c x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}\)	\(145\)

[In]

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

[Out]

-1/12*a^2*c*(-6*f*x+cos(3*f*x+3*e)-3*sin(2*f*x+2*e)+3*cos(f*x+e)+4)/f

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.88 \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=-\frac {2 \, a^{2} c \cos \left (f x + e\right )^{3} - 3 \, a^{2} c f x - 3 \, a^{2} c \cos \left (f x + e\right ) \sin \left (f x + e\right )}{6 \, f} \]

[In]

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

[Out]

-1/6*(2*a^2*c*cos(f*x + e)^3 - 3*a^2*c*f*x - 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. 133 vs. \(2 (46) = 92\).

Time = 0.14 (sec) , antiderivative size = 133, normalized size of antiderivative = 2.56 \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\begin {cases} - \frac {a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} - \frac {a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + a^{2} c x + \frac {a^{2} c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {a^{2} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {2 a^{2} c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {a^{2} c \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (a \sin {\left (e \right )} + a\right )^{2} \left (- c \sin {\left (e \right )} + c\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-a**2*c*x*sin(e + f*x)**2/2 - a**2*c*x*cos(e + f*x)**2/2 + a**2*c*x + a**2*c*sin(e + f*x)**2*cos(e
+ f*x)/f + a**2*c*sin(e + f*x)*cos(e + f*x)/(2*f) + 2*a**2*c*cos(e + f*x)**3/(3*f) - a**2*c*cos(e + f*x)/f, Ne
(f, 0)), (x*(a*sin(e) + a)**2*(-c*sin(e) + c), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.48 \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=-\frac {4 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} a^{2} c + 3 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} a^{2} c - 12 \, {\left (f x + e\right )} a^{2} c + 12 \, a^{2} c \cos \left (f x + e\right )}{12 \, f} \]

[In]

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

[Out]

-1/12*(4*(cos(f*x + e)^3 - 3*cos(f*x + e))*a^2*c + 3*(2*f*x + 2*e - sin(2*f*x + 2*e))*a^2*c - 12*(f*x + e)*a^2
*c + 12*a^2*c*cos(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.13 \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {1}{2} \, a^{2} c x - \frac {a^{2} c \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {a^{2} c \cos \left (f x + e\right )}{4 \, f} + \frac {a^{2} c \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]

[In]

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

[Out]

1/2*a^2*c*x - 1/12*a^2*c*cos(3*f*x + 3*e)/f - 1/4*a^2*c*cos(f*x + e)/f + 1/4*a^2*c*sin(2*f*x + 2*e)/f

Mupad [B] (verification not implemented)

Time = 8.69 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.40 \[ \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x)) \, dx=\frac {a^2\,c\,x}{2}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {3\,a^2\,c\,\left (e+f\,x\right )}{2}-\frac {a^2\,c\,\left (9\,e+9\,f\,x-12\right )}{6}\right )-a^2\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+\frac {a^2\,c\,\left (e+f\,x\right )}{2}+a^2\,c\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5-\frac {a^2\,c\,\left (3\,e+3\,f\,x-4\right )}{6}}{f\,{\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}^3} \]

[In]

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

[Out]

(a^2*c*x)/2 - (tan(e/2 + (f*x)/2)^4*((3*a^2*c*(e + f*x))/2 - (a^2*c*(9*e + 9*f*x - 12))/6) - a^2*c*tan(e/2 + (
f*x)/2) + (a^2*c*(e + f*x))/2 + a^2*c*tan(e/2 + (f*x)/2)^5 - (a^2*c*(3*e + 3*f*x - 4))/6)/(f*(tan(e/2 + (f*x)/
2)^2 + 1)^3)