\(\int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx\) [341]

Optimal result

Integrand size = 30, antiderivative size = 40 \[ \int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{f \sqrt {3+3 \sin (e+f x)}} \]

[Out]

-1/3*a*cos(f*x+e)*(c-c*sin(f*x+e))^(5/2)/f/(a+a*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.08, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2817} \[ \int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=-\frac {a \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 f \sqrt {a \sin (e+f x)+a}} \]

[In]

Int[Sqrt[a + a*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2),x]

[Out]

-1/3*(a*Cos[e + f*x]*(c - c*Sin[e + f*x])^(5/2))/(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))^{5/2}}{3 f \sqrt {a+a \sin (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.90 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.92 \[ \int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\frac {c^2 \sec (e+f x) \sqrt {1+\sin (e+f x)} \sqrt {c-c \sin (e+f x)} (6 \cos (2 (e+f x))+15 \sin (e+f x)-\sin (3 (e+f x)))}{4 \sqrt {3} f} \]

[In]

Integrate[Sqrt[3 + 3*Sin[e + f*x]]*(c - c*Sin[e + f*x])^(5/2),x]

[Out]

(c^2*Sec[e + f*x]*Sqrt[1 + Sin[e + f*x]]*Sqrt[c - c*Sin[e + f*x]]*(6*Cos[2*(e + f*x)] + 15*Sin[e + f*x] - Sin[
3*(e + f*x)]))/(4*Sqrt[3]*f)

Maple [A] (verified)

Time = 3.05 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.80

method	result	size
default	\(-\frac {c^{2} \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \left (\sin \left (f x +e \right ) \cos \left (f x +e \right )-3 \cos \left (f x +e \right )-4 \tan \left (f x +e \right )+3 \sec \left (f x +e \right )\right )}{3 f}\)	\(72\)

[In]

int((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (37) = 74\).

Time = 0.28 (sec) , antiderivative size = 83, normalized size of antiderivative = 2.08 \[ \int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\frac {{\left (3 \, c^{2} \cos \left (f x + e\right )^{2} - 3 \, c^{2} - {\left (c^{2} \cos \left (f x + e\right )^{2} - 4 \, c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, f \cos \left (f x + e\right )} \]

[In]

integrate((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((a+a*sin(f*x+e))**(1/2)*(c-c*sin(f*x+e))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\int { \sqrt {a \sin \left (f x + e\right ) + a} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(f*x + e) + a)*(-c*sin(f*x + e) + c)^(5/2), x)

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.28 \[ \int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\frac {8 \, \sqrt {a} c^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}}{3 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^(1/2)*(c-c*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

8/3*sqrt(a)*c^(5/2)*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)^6/f

Mupad [B] (verification not implemented)

Time = 7.60 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.20 \[ \int \sqrt {3+3 \sin (e+f x)} (c-c \sin (e+f x))^{5/2} \, dx=\frac {c^2\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (6\,\cos \left (e+f\,x\right )+6\,\cos \left (3\,e+3\,f\,x\right )+14\,\sin \left (2\,e+2\,f\,x\right )-\sin \left (4\,e+4\,f\,x\right )\right )}{12\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]

[In]

int((a + a*sin(e + f*x))^(1/2)*(c - c*sin(e + f*x))^(5/2),x)

[Out]

(c^2*(a*(sin(e + f*x) + 1))^(1/2)*(-c*(sin(e + f*x) - 1))^(1/2)*(6*cos(e + f*x) + 6*cos(3*e + 3*f*x) + 14*sin(
2*e + 2*f*x) - sin(4*e + 4*f*x)))/(12*f*(cos(2*e + 2*f*x) + 1))