\(\int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx\) [328]

Optimal result

Integrand size = 28, antiderivative size = 33 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=-\frac {2 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{27 c f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.09, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2815, 2752} \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=-\frac {2 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 c f} \]

[In]

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

[Out]

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

Rule 2752

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 - 1)/(f*g*(m - 1))), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

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}& = \frac {\int \sec ^4(e+f x) (c-c \sin (e+f x))^{5/2} \, dx}{a^2 c^2} \\ & = -\frac {2 \sec ^3(e+f x) (c-c \sin (e+f x))^{3/2}}{3 a^2 c f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(70\) vs. \(2(33)=66\).

Time = 0.50 (sec) , antiderivative size = 70, normalized size of antiderivative = 2.12 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=-\frac {2 \sqrt {c-c \sin (e+f x)}}{27 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3} \]

[In]

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

[Out]

(-2*Sqrt[c - c*Sin[e + f*x]])/(27*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2]
)^3)

Maple [A] (verified)

Time = 0.57 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48

method	result	size
default	\(\frac {2 c \left (\sin \left (f x +e \right )-1\right )}{3 a^{2} \left (\sin \left (f x +e \right )+1\right ) \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(49\)

[In]

int((c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=-\frac {2 \, \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, {\left (a^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a^{2} f \cos \left (f x + e\right )\right )}} \]

[In]

integrate((c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=\frac {\int \frac {\sqrt {- c \sin {\left (e + f x \right )} + c}}{\sin ^{2}{\left (e + f x \right )} + 2 \sin {\left (e + f x \right )} + 1}\, dx}{a^{2}} \]

[In]

integrate((c-c*sin(f*x+e))**(1/2)/(a+a*sin(f*x+e))**2,x)

[Out]

Integral(sqrt(-c*sin(e + f*x) + c)/(sin(e + f*x)**2 + 2*sin(e + f*x) + 1), x)/a**2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (32) = 64\).

Time = 0.39 (sec) , antiderivative size = 149, normalized size of antiderivative = 4.52 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=\frac {2 \, {\left (\sqrt {c} + \frac {2 \, \sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {\sqrt {c} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}\right )}}{3 \, {\left (a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )} f \sqrt {\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1}} \]

[In]

integrate((c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (32) = 64\).

Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 3.33 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=\frac {\sqrt {2} \sqrt {c} {\left (\frac {3 \, {\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )}^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{{\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{2}} + \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )}}{3 \, a^{2} f {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )}^{3}} \]

[In]

integrate((c-c*sin(f*x+e))^(1/2)/(a+a*sin(f*x+e))^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 9.35 (sec) , antiderivative size = 227, normalized size of antiderivative = 6.88 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{(3+3 \sin (e+f x))^2} \, dx=-\frac {4\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (\sin \left (2\,e+2\,f\,x\right )-4\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2-{\sin \left (e+f\,x\right )}^2\,2{}\mathrm {i}+2+2{}\mathrm {i}\right )}{3\,a^2\,f\,\left (-4\,{\sin \left (e+f\,x\right )}^2+\sin \left (e+f\,x\right )+\sin \left (3\,e+3\,f\,x\right )+4\right )}+\frac {4\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (-{\sin \left (e+f\,x\right )}^2\,4{}\mathrm {i}+\sin \left (e+f\,x\right )\,1{}\mathrm {i}-2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+2\,{\sin \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )}^2+2\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (3\,e+3\,f\,x\right )\,1{}\mathrm {i}+4{}\mathrm {i}\right )}{3\,a^2\,f\,\left (-8\,{\sin \left (e+f\,x\right )}^2+4\,\sin \left (e+f\,x\right )+2\,{\sin \left (2\,e+2\,f\,x\right )}^2+4\,\sin \left (3\,e+3\,f\,x\right )+8\right )} \]

[In]

int((c - c*sin(e + f*x))^(1/2)/(a + a*sin(e + f*x))^2,x)

[Out]

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