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

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Rubi [A]  time = 0.08, antiderivative size = 40, normalized size of antiderivative = 1.00, 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*}

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Mathematica [B]  time = 0.21, size = 84, normalized size = 2.10 \[ \frac {\sqrt {a (\sin (e+f x)+1)} \sqrt {c-c \sin (e+f x)}}{c^2 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )} \]

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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fricas [A]  time = 0.44, size = 59, normalized size = 1.48 \[ -\frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{c^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - c^{2} f \cos \left (f x + e\right )} \]

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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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

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]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (8*pi/x/2)>(-8*pi/x/2)-sqrt(2*a)/8*(tan(1/2*(1/2*f*x+1/4*(2*exp(1)-pi)))^2+1/tan(1/2
*(1/2*f*x+1/4*(2*exp(1)-pi)))^2)*sign(cos(1/2*(f*x+exp(1))-1/4*pi))/sqrt(2)/sqrt(c)/c/f/sign(tan(1/2*(1/2*f*x+
1/4*(2*exp(1)-pi)))^3+tan(1/2*(1/2*f*x+1/4*(2*exp(1)-pi))))

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maple [A]  time = 0.27, size = 68, normalized size = 1.70 \[ \frac {\sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sin \left (f x +e \right ) \left (-1+\cos \left (f x +e \right )+\sin \left (f x +e \right )\right )}{f \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \left (1-\cos \left (f x +e \right )+\sin \left (f x +e \right )\right )} \]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \sin \left (f x + e\right ) + a}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a+a\,\sin \left (e+f\,x\right )}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )}}{\left (- c \left (\sin {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}\, dx \]

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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