\(\int (3+3 \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx\) [413]

Optimal result

Integrand size = 28, antiderivative size = 46 \[ \int (3+3 \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 c \cos (e+f x) (3+3 \sin (e+f x))^m}{f (1+2 m) \sqrt {c-c \sin (e+f x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {2817} \[ \int (3+3 \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 c \cos (e+f x) (a \sin (e+f x)+a)^m}{f (2 m+1) \sqrt {c-c \sin (e+f x)}} \]

[In]

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

[Out]

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

Mathematica [A] (verified)

Time = 0.41 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.87 \[ \int (3+3 \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx=\frac {2\ 3^m \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^m \sqrt {c-c \sin (e+f x)}}{f (1+2 m) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (a +a \sin \left (f x +e \right )\right )^{m} \sqrt {c -c \sin \left (f x +e \right )}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.65 \[ \int (3+3 \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx=\frac {2 \, \sqrt {-c \sin \left (f x + e\right ) + c} {\left (a \sin \left (f x + e\right ) + a\right )}^{m} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )}}{2 \, f m + {\left (2 \, f m + f\right )} \cos \left (f x + e\right ) - {\left (2 \, f m + f\right )} \sin \left (f x + e\right ) + f} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (44) = 88\).

Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.52 \[ \int (3+3 \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx=-\frac {2 \, {\left (a^{m} \sqrt {c} + \frac {a^{m} \sqrt {c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} e^{\left (2 \, m \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right ) - m \log \left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )\right )}}{f {\left (2 \, m + 1\right )} \sqrt {\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.51 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.15 \[ \int (3+3 \sin (e+f x))^m \sqrt {c-c \sin (e+f x)} \, dx=-\frac {2\,\cos \left (e+f\,x\right )\,{\left (a\,\left (\sin \left (e+f\,x\right )+1\right )\right )}^m\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}}{f\,\left (2\,m+1\right )\,\left (\sin \left (e+f\,x\right )-1\right )} \]

[In]

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

[Out]

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