\(\int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx\) [342]

Optimal result

Integrand size = 30, antiderivative size = 42 \[ \int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx=-\frac {3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {3+3 \sin (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 43, 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} \[ \int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx=-\frac {a \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a \sin (e+f x)+a}} \]

[In]

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

[Out]

-1/2*(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(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) (c-c \sin (e+f x))^{3/2}}{2 f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.50 \[ \int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx=\frac {\sqrt {3} c \sec (e+f x) \sqrt {1+\sin (e+f x)} (\cos (2 (e+f x))+4 \sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{4 f} \]

[In]

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

[Out]

(Sqrt[3]*c*Sec[e + f*x]*Sqrt[1 + Sin[e + f*x]]*(Cos[2*(e + f*x)] + 4*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]])/(
4*f)

Maple [A] (verified)

Time = 2.81 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.31

method	result	size
default	\(\frac {c \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \left (\cos \left (f x +e \right )+2 \tan \left (f x +e \right )-\sec \left (f x +e \right )\right )}{2 f}\)	\(55\)

[In]

int((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.45 \[ \int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx=\frac {{\left (c \cos \left (f x + e\right )^{2} + 2 \, c \sin \left (f x + e\right ) - c\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{2 \, f \cos \left (f x + e\right )} \]

[In]

integrate((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate((a+a*sin(f*x+e))**(1/2)*(c-c*sin(f*x+e))**(3/2),x)

[Out]

Integral(sqrt(a*(sin(e + f*x) + 1))*(-c*(sin(e + f*x) - 1))**(3/2), x)

Maxima [F]

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

[In]

integrate((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.21 \[ \int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx=\frac {2 \, \sqrt {a} c^{\frac {3}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4}}{f} \]

[In]

integrate((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

2*sqrt(a)*c^(3/2)*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)^4/f

Mupad [B] (verification not implemented)

Time = 0.95 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.69 \[ \int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx=\frac {c\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (\cos \left (e+f\,x\right )+\cos \left (3\,e+3\,f\,x\right )+4\,\sin \left (2\,e+2\,f\,x\right )\right )}{4\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]

[In]

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

[Out]

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