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

Optimal. Leaf size=40 \[ \frac{a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0823106, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033, Rules used = {2738} \[ \frac{a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [B]  time = 0.208228, size = 84, normalized size = 2.1 \[ \frac{\sqrt{a (\sin (e+f x)+1)} \sqrt{c-c \sin (e+f x)}}{c^2 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.171, size = 68, normalized size = 1.7 \begin{align*}{\frac{ \left ( -1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) \sin \left ( fx+e \right ) }{f \left ( 1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) }\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) } \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a \sin \left (f x + e\right ) + a}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.04962, size = 146, normalized size = 3.65 \begin{align*} -\frac{\sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}}{c^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - c^{2} f \cos \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(a*sin(f*x + e) + a)*sqrt(-c*sin(f*x + e) + c)/(c^2*f*cos(f*x + e)*sin(f*x + e) - c^2*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 )}}{\left (- c \left (\sin{\left (e + f x \right )} - 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a*(sin(e + f*x) + 1))/(-c*(sin(e + f*x) - 1))**(3/2), 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}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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