\(\int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx\) [45]

Optimal result

Integrand size = 38, antiderivative size = 45 \[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt {a+a \sin (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {2920, 2817} \[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 c f \sqrt {a \sin (e+f x)+a}} \]

[In]

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

[Out]

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

Rule 2920

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

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

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.82 \[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c-c \sin (e+f x))^{3/2}}{3 f \sqrt {a (1+\sin (e+f x))}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(124\) vs. \(2(39)=78\).

Time = 0.16 (sec) , antiderivative size = 125, normalized size of antiderivative = 2.78

method	result	size
default	\(\frac {\left (1+\cos \left (f x +e \right )\right ) \left (\left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\left (\cos ^{3}\left (f x +e \right )\right )-3 \cos \left (f x +e \right ) \sin \left (f x +e \right )-2 \left (\cos ^{2}\left (f x +e \right )\right )-\sin \left (f x +e \right )+4 \cos \left (f x +e \right )-1\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, c}{3 f \left (-\cos \left (f x +e \right )+\sin \left (f x +e \right )-1\right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}\)	\(125\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.73 \[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {{\left (3 \, c \cos \left (f x + e\right )^{2} - {\left (c \cos \left (f x + e\right )^{2} - 4 \, c\right )} \sin \left (f x + e\right ) - 3 \, c\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, a f \cos \left (f x + e\right )} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.18 \[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=\frac {8 \, c^{\frac {3}{2}} \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6}}{3 \, \sqrt {a} f \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} \]

[In]

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

[Out]

8/3*c^(3/2)*sgn(sin(-1/4*pi + 1/2*f*x + 1/2*e))*sin(-1/4*pi + 1/2*f*x + 1/2*e)^6/(sqrt(a)*f*sgn(cos(-1/4*pi +
1/2*f*x + 1/2*e)))

Mupad [B] (verification not implemented)

Time = 10.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.84 \[ \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx=-\frac {c\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (6\,\cos \left (e+f\,x\right )+6\,\cos \left (3\,e+3\,f\,x\right )+14\,\sin \left (2\,e+2\,f\,x\right )-\sin \left (4\,e+4\,f\,x\right )\right )}{24\,f\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\left (\sin \left (e+f\,x\right )-1\right )} \]

[In]

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

[Out]

-(c*(-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
)))/(24*f*(a*(sin(e + f*x) + 1))^(1/2)*(sin(e + f*x) - 1))