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

Optimal. Leaf size=43 \[ \frac{2 \sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}{c f g} \]

[Out]

(2*Sec[e + f*x]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(c*f*g)

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Rubi [A]  time = 0.202931, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.075, Rules used = {2930, 12, 30} \[ \frac{2 \sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}{c f g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sec[e + f*x]*Sqrt[g*Sin[e + f*x]]*Sqrt[a + a*Sin[e + f*x]])/(c*f*g)

Rule 2930

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/(Sqrt[(g_.)*sin[(e_.) + (f_.)*(x_)]]*((c_) + (d_.)*sin[(e_.) +
(f_.)*(x_)])), x_Symbol] :> Dist[(-2*b)/f, Subst[Int[1/(b*c + a*d + c*g*x^2), x], x, (b*Cos[e + f*x])/(Sqrt[g*
Sin[e + f*x]]*Sqrt[a + b*Sin[e + f*x]])], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b*c - a*d, 0] && EqQ[a^
2 - b^2, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{g \sin (e+f x)} (c-c \sin (e+f x))} \, dx &=-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{c g x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\right )}{f}\\ &=-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}\right )}{c f g}\\ &=\frac{2 \sec (e+f x) \sqrt{g \sin (e+f x)} \sqrt{a+a \sin (e+f x)}}{c f g}\\ \end{align*}

Mathematica [A]  time = 0.220407, size = 40, normalized size = 0.93 \[ \frac{2 \tan (e+f x) \sqrt{a (\sin (e+f x)+1)}}{c f \sqrt{g \sin (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[a*(1 + Sin[e + f*x])]*Tan[e + f*x])/(c*f*Sqrt[g*Sin[e + f*x]])

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Maple [A]  time = 0.301, size = 45, normalized size = 1.1 \begin{align*} 2\,{\frac{\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }\sin \left ( fx+e \right ) }{cf\sqrt{g\sin \left ( fx+e \right ) }\cos \left ( fx+e \right ) }} \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))/(g*sin(f*x+e))^(1/2),x)

[Out]

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

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Maxima [B]  time = 1.57354, size = 417, normalized size = 9.7 \begin{align*} -\frac{\frac{4 \,{\left ({\left (\frac{3 \, \sqrt{2} \sqrt{a} \sqrt{g} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{\sqrt{2} \sqrt{a} \sqrt{g} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}} - \frac{2 \,{\left (\frac{3 \, \sqrt{2} \sqrt{a} \sqrt{g} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{\sqrt{2} \sqrt{a} \sqrt{g} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}}{\sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}}\right )}}{c g - \frac{c g \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}} - \frac{2 \, \sqrt{2} \sqrt{a} \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )^{\frac{3}{2}} + \frac{3 \, \sqrt{2} \sqrt{a} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}{c \sqrt{g}} - \frac{2 \, \sqrt{2} \sqrt{a} \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )^{\frac{3}{2}} - \frac{3 \, \sqrt{2} \sqrt{a} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}{c \sqrt{g}}}{12 \, f} \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))/(g*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

-1/12*(4*((3*sqrt(2)*sqrt(a)*sqrt(g)*sin(f*x + e)/(cos(f*x + e) + 1) - sqrt(2)*sqrt(a)*sqrt(g)*sin(f*x + e)^2/
(cos(f*x + e) + 1)^2)*sqrt(sin(f*x + e)/(cos(f*x + e) + 1)) - 2*(3*sqrt(2)*sqrt(a)*sqrt(g)*sin(f*x + e)/(cos(f
*x + e) + 1) + sqrt(2)*sqrt(a)*sqrt(g)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)/sqrt(sin(f*x + e)/(cos(f*x + e) +
1)))/(c*g - c*g*sin(f*x + e)/(cos(f*x + e) + 1)) - (2*sqrt(2)*sqrt(a)*(sin(f*x + e)/(cos(f*x + e) + 1))^(3/2)
+ 3*sqrt(2)*sqrt(a)*sin(f*x + e)/(cos(f*x + e) + 1))/(c*sqrt(g)) - (2*sqrt(2)*sqrt(a)*(sin(f*x + e)/(cos(f*x +
e) + 1))^(3/2) - 3*sqrt(2)*sqrt(a)*sin(f*x + e)/(cos(f*x + e) + 1))/(c*sqrt(g)))/f

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Fricas [A]  time = 1.99971, size = 95, normalized size = 2.21 \begin{align*} \frac{2 \, \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{g \sin \left (f x + e\right )}}{c f g \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))/(g*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

2*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))/(c*f*g*cos(f*x + e))

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

[Out]

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

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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 )} \sqrt{g \sin \left (f x + e\right )}}\,{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))/(g*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

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