\(\int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx\) [399]

Optimal result

Integrand size = 30, antiderivative size = 42 \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{4 f (3+3 \sin (e+f x))^{5/2}} \]

[Out]

-1/4*cos(f*x+e)*(c-c*sin(f*x+e))^(3/2)/f/(a+a*sin(f*x+e))^(5/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2821} \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{4 f (a \sin (e+f x)+a)^{5/2}} \]

[In]

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

[Out]

-1/4*(Cos[e + f*x]*(c - c*Sin[e + f*x])^(3/2))/(f*(a + a*Sin[e + f*x])^(5/2))

Rule 2821

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

Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (e+f x) (c-c \sin (e+f x))^{3/2}}{4 f (a+a \sin (e+f x))^{5/2}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(92\) vs. \(2(42)=84\).

Time = 1.19 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.19 \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\frac {c \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x) \sqrt {c-c \sin (e+f x)}}{9 \sqrt {3} f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^{5/2}} \]

[In]

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

[Out]

(c*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])*Sin[e + f*x]*Sqrt[c - c*Sin[e + f*x]])/(9*Sqrt[3]*f*(Cos[(e + f*x)/2]
- Sin[(e + f*x)/2])*(1 + Sin[e + f*x])^(5/2))

Maple [A] (verified)

Time = 3.14 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.19

method	result	size
default	\(\frac {\tan \left (f x +e \right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c}{f \left (\sin \left (f x +e \right )+1\right ) \sqrt {a \left (\sin \left (f x +e \right )+1\right )}\, a^{2}}\)	\(50\)

[In]

int((c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 80 vs. \(2 (36) = 72\).

Time = 0.28 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.90 \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c} c \sin \left (f x + e\right )}{a^{3} f \cos \left (f x + e\right )^{3} - 2 \, a^{3} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a^{3} f \cos \left (f x + e\right )} \]

[In]

integrate((c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\int \frac {\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate((c-c*sin(f*x+e))**(3/2)/(a+a*sin(f*x+e))**(5/2),x)

[Out]

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

Maxima [F]

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

[In]

integrate((c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (36) = 72\).

Time = 0.54 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.21 \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=-\frac {{\left (2 \, \sqrt {a} c \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - \sqrt {a} c \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{4 \, a^{3} f \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]

[In]

integrate((c-c*sin(f*x+e))^(3/2)/(a+a*sin(f*x+e))^(5/2),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 8.05 (sec) , antiderivative size = 118, normalized size of antiderivative = 2.81 \[ \int \frac {(c-c \sin (e+f x))^{3/2}}{(3+3 \sin (e+f x))^{5/2}} \, dx=\frac {2\,c\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (-2\,{\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+2\,{\sin \left (\frac {3\,e}{2}+\frac {3\,f\,x}{2}\right )}^2+2\,\sin \left (2\,e+2\,f\,x\right )\right )}{a^2\,f\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\left (-8\,{\sin \left (e+f\,x\right )}^2+4\,\sin \left (e+f\,x\right )+2\,{\sin \left (2\,e+2\,f\,x\right )}^2+4\,\sin \left (3\,e+3\,f\,x\right )+8\right )} \]

[In]

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

[Out]

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