\(\int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx\) [343]

Optimal result

Integrand size = 30, antiderivative size = 40 \[ \int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=-\frac {3 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {3+3 \sin (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2817} \[ \int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=-\frac {a \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}} \]

[In]

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

[Out]

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

Rule 2817

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[
-2*b*Cos[e + f*x]*((c + d*Sin[e + f*x])^n/(f*(2*n + 1)*Sqrt[a + b*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[n, -2^(-1)]

Rubi steps \begin{align*} \text {integral}& = -\frac {a \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(82\) vs. \(2(40)=80\).

Time = 0.06 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.05 \[ \int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\frac {\sqrt {3} \left (-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {1+\sin (e+f x)} \sqrt {c-c \sin (e+f x)}}{f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 3.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90

method	result	size
default	\(\frac {\tan \left (f x +e \right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}{f}\)	\(36\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.08 \[ \int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} \sin \left (f x + e\right )}{f \cos \left (f x + e\right )} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.28 \[ \int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 \, \sqrt {a} \sqrt {c} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}{f} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 6.76 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.18 \[ \int \sqrt {3+3 \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx=\frac {\sin \left (2\,e+2\,f\,x\right )\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}}{2\,f\,{\cos \left (e+f\,x\right )}^2} \]

[In]

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

[Out]

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