3.293 \(\int (a+a \sin (e+f x)) \sqrt{c-c \sin (e+f x)} \, dx\)

Optimal. Leaf size=34 \[ \frac{2 a c^2 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0918206, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2736, 2673} \[ \frac{2 a c^2 \cos ^3(e+f x)}{3 f (c-c \sin (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2736

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

Rule 2673

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m - 1)), x] /; FreeQ[{a, b, e, f, g, m, p}, x] && Eq
Q[a^2 - b^2, 0] && EqQ[2*m + p - 1, 0] && NeQ[m, 1]

Rubi steps

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

Mathematica [B]  time = 0.117942, size = 71, normalized size = 2.09 \[ \frac{2 a \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^3}{3 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])*Sqrt[c - c*Sin[e + f*x]],x]

[Out]

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

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Maple [A]  time = 0.375, size = 47, normalized size = 1.4 \begin{align*} -{\frac{ \left ( -2+2\,\sin \left ( fx+e \right ) \right ) c \left ( 1+\sin \left ( fx+e \right ) \right ) ^{2}a}{3\,f\cos \left ( fx+e \right ) }{\frac{1}{\sqrt{c-c\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )} \sqrt{-c \sin \left (f x + e\right ) + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.04304, size = 203, normalized size = 5.97 \begin{align*} -\frac{2 \,{\left (a \cos \left (f x + e\right )^{2} - a \cos \left (f x + e\right ) -{\left (a \cos \left (f x + e\right ) + 2 \, a\right )} \sin \left (f x + e\right ) - 2 \, a\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{3 \,{\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \sqrt{- c \sin{\left (e + f x \right )} + c} \sin{\left (e + f x \right )}\, dx + \int \sqrt{- c \sin{\left (e + f x \right )} + c}\, dx\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sin \left (f x + e\right ) + a\right )} \sqrt{-c \sin \left (f x + e\right ) + c}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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