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

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

[Out]

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

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Rubi [A]  time = 0.092265, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {2771} \[ -\frac{2 a \cos (e+f x)}{f (c+d) \sqrt{a \sin (e+f x)+a} \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2771

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(3/2), x_Symbol] :> Sim
p[(-2*b^2*Cos[e + f*x])/(f*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]), x] /; FreeQ[{a, b,
c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rubi steps

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

Mathematica [A]  time = 0.195997, size = 84, normalized size = 1.87 \[ -\frac{2 \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}{f (c+d) \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \sqrt{c+d \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.205, size = 99, normalized size = 2.2 \begin{align*} 2\,{\frac{\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{c+d\sin \left ( fx+e \right ) } \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}d+c\sin \left ( fx+e \right ) +d\sin \left ( fx+e \right ) -c-d \right ) }{f \left ( c+d \right ) \cos \left ( fx+e \right ) \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}{d}^{2}+{c}^{2}-{d}^{2} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.83052, size = 242, normalized size = 5.38 \begin{align*} -\frac{2 \,{\left (\sqrt{a} c - \frac{\sqrt{a}{\left (c - 2 \, d\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{\sqrt{a}{\left (c - 2 \, d\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac{\sqrt{a} c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}{\left (\frac{\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}}{{\left (c + d + \frac{{\left (c + d\right )} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}{\left (c + \frac{2 \, d \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}^{\frac{3}{2}} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.28305, size = 323, normalized size = 7.18 \begin{align*} \frac{2 \, \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{d \sin \left (f x + e\right ) + c}{\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{{\left (c d + d^{2}\right )} f \cos \left (f x + e\right )^{2} -{\left (c^{2} + c d\right )} f \cos \left (f x + e\right ) -{\left (c^{2} + 2 \, c d + d^{2}\right )} f -{\left ({\left (c d + d^{2}\right )} f \cos \left (f x + e\right ) +{\left (c^{2} + 2 \, c d + d^{2}\right )} f\right )} \sin \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+d*sin(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

2*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)*(cos(f*x + e) - sin(f*x + e) + 1)/((c*d + d^2)*f*cos(f*x +
e)^2 - (c^2 + c*d)*f*cos(f*x + e) - (c^2 + 2*c*d + d^2)*f - ((c*d + d^2)*f*cos(f*x + e) + (c^2 + 2*c*d + d^2)
*f)*sin(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 + d \sin{\left (e + f x \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+d*sin(f*x+e))**(3/2),x)

[Out]

Integral(sqrt(a*(sin(e + f*x) + 1))/(c + d*sin(e + f*x))**(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 (d \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+d*sin(f*x+e))^(3/2),x, algorithm="giac")

[Out]

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