\(\int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx\) [320]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2815, 2752} \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=-\frac {2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{a f} \]

[In]

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

[Out]

(-2*Sec[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(a*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 ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx}{a c} \\ & = -\frac {2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{a f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.47 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=-\frac {2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{3 f} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39

method	result	size
default	\(\frac {2 c \left (\sin \left (f x +e \right )-1\right )}{a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\)	\(39\)

[In]

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

[Out]

2*c/a*(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 = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=-\frac {2 \, \sqrt {-c \sin \left (f x + e\right ) + c}}{a f \cos \left (f x + e\right )} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (27) = 54\).

Time = 0.33 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.71 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=\frac {2 \, {\left (\sqrt {c} + \frac {\sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}}{{\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} f \sqrt {\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1}} \]

[In]

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

[Out]

2*(sqrt(c) + sqrt(c)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)/((a + a*sin(f*x + e)/(cos(f*x + e) + 1))*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. 62 vs. \(2 (27) = 54\).

Time = 0.31 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.21 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=\frac {2 \, \sqrt {2} \sqrt {c} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{a f {\left (\frac {\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1}{\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1} + 1\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.43 \[ \int \frac {\sqrt {c-c \sin (e+f x)}}{3+3 \sin (e+f x)} \, dx=-\frac {4\,\cos \left (e+f\,x\right )\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}}{a\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]

[In]

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

[Out]

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