3.5 \(\int \frac{\cos ^2(e+f x) \sqrt{a+a \sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}} \, dx\)

Optimal. Leaf size=45 \[ \frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 a f \sqrt{c-c \sin (e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.283645, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {2841, 2738} \[ \frac{\cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 a f \sqrt{c-c \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2841

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]

Rule 2738

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

Mathematica [A]  time = 0.307567, size = 62, normalized size = 1.38 \[ -\frac{\sec (e+f x) \sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)} (\cos (2 (e+f x))-4 \sin (e+f x))}{4 c f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sec[e + f*x]*(Cos[2*(e + f*x)] - 4*Sin[e + f*x])*Sqrt[a*(1 + Sin[e + f*x])]*Sqrt[c - c*Sin[e + f*x]])/(4*c*f
)

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Maple [B]  time = 0.236, size = 94, normalized size = 2.1 \begin{align*}{\frac{ \left ( \sin \left ( fx+e \right ) \cos \left ( fx+e \right ) - \left ( \cos \left ( fx+e \right ) \right ) ^{2}+\sin \left ( fx+e \right ) +2\,\cos \left ( fx+e \right ) -1 \right ) \sin \left ( fx+e \right ) }{2\,f \left ( 1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) }\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }{\frac{1}{\sqrt{-c \left ( -1+\sin \left ( fx+e \right ) \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.73736, size = 522, normalized size = 11.6 \begin{align*} -\frac{\frac{2 \, \sqrt{a} \sqrt{c} + \frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{3 \, \sqrt{a} \sqrt{c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{c + \frac{2 \, c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{c \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} - \frac{2 \, \sqrt{a} \sqrt{c} - \frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}{c + \frac{2 \, c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{c \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}} + \frac{2 \,{\left (\frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{\sqrt{a} \sqrt{c} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}}{c + \frac{2 \, c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{c \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}}}{2 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.60429, size = 154, normalized size = 3.42 \begin{align*} -\frac{{\left (\cos \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) - 1\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{2 \, c f \cos \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )} \cos ^{2}{\left (e + f x \right )}}{\sqrt{- c \left (\sin{\left (e + f x \right )} - 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \sin \left (f x + e\right ) + a} \cos \left (f x + e\right )^{2}}{\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(cos(f*x+e)^2*(a+a*sin(f*x+e))^(1/2)/(c-c*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

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