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

Optimal. Leaf size=114 \[ \frac{\sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}{a c f}+\frac{\sqrt{g} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{\sqrt{2} \sqrt{a} c f} \]

[Out]

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

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Rubi [A]  time = 0.489969, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2936, 2782, 205, 2930, 12, 30} \[ \frac{\sec (e+f x) \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}{a c f}+\frac{\sqrt{g} \tan ^{-1}\left (\frac{\sqrt{a} \sqrt{g} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a} \sqrt{g \sin (e+f x)}}\right )}{\sqrt{2} \sqrt{a} c f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2936

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

Rule 2782

Int[1/(Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> D
ist[(-2*a)/f, Subst[Int[1/(2*b^2 - (a*c - b*d)*x^2), x], x, (b*Cos[e + f*x])/(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]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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

Mathematica [A]  time = 0.325987, size = 133, normalized size = 1.17 \[ \frac{\sqrt{\sin (e+f x)} \csc (2 (e+f x)) \sqrt{a (\sin (e+f x)+1)} \sqrt{g \sin (e+f x)} \left (2 \sqrt{c} \sqrt{\sin (e+f x)}-\sqrt{2} \sqrt{c-c \sin (e+f x)} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{\sin (e+f x)}}{\sqrt{c-c \sin (e+f x)}}\right )\right )}{a c^{3/2} f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.336, size = 187, normalized size = 1.6 \begin{align*} -{\frac{-1+\cos \left ( fx+e \right ) }{cf\sin \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) }\sqrt{g\sin \left ( fx+e \right ) } \left ( \sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\cos \left ( fx+e \right ) -2\,\arctan \left ( \sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}} \right ) \cos \left ( fx+e \right ) +\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) +\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}} \right ){\frac{1}{\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }}}{\frac{1}{\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.57961, size = 365, normalized size = 3.2 \begin{align*} \frac{\frac{4 \, \sqrt{2} \sqrt{g} \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )^{\frac{3}{2}}}{\sqrt{a} c + \frac{\sqrt{a} c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac{{\left (\sqrt{a} c + \frac{\sqrt{a} c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}} - \frac{2 \, \sqrt{2} \sqrt{g} \arctan \left (\sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}{\sqrt{a} c} + \frac{\sqrt{2} \sqrt{g} \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}} + \sqrt{2} \sqrt{g}}{\sqrt{a} c + \frac{\sqrt{a} c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}} + \frac{\sqrt{2} \sqrt{g} \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}} - \sqrt{2} \sqrt{g}}{\sqrt{a} c + \frac{\sqrt{a} c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}}{2 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 3.31889, size = 1006, normalized size = 8.82 \begin{align*} \left [\frac{\sqrt{2} a \sqrt{-\frac{g}{a}} \cos \left (f x + e\right ) \log \left (\frac{17 \, g \cos \left (f x + e\right )^{3} + 4 \, \sqrt{2}{\left (3 \, \cos \left (f x + e\right )^{2} +{\left (3 \, \cos \left (f x + e\right ) + 4\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 4\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{g \sin \left (f x + e\right )} \sqrt{-\frac{g}{a}} + 3 \, g \cos \left (f x + e\right )^{2} - 18 \, g \cos \left (f x + e\right ) +{\left (17 \, g \cos \left (f x + e\right )^{2} + 14 \, g \cos \left (f x + e\right ) - 4 \, g\right )} \sin \left (f x + e\right ) - 4 \, g}{\cos \left (f x + e\right )^{3} + 3 \, \cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) - 4\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 4}\right ) + 8 \, \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{g \sin \left (f x + e\right )}}{8 \, a c f \cos \left (f x + e\right )}, -\frac{\sqrt{2} a \sqrt{\frac{g}{a}} \arctan \left (\frac{\sqrt{2} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{g \sin \left (f x + e\right )} \sqrt{\frac{g}{a}}{\left (3 \, \sin \left (f x + e\right ) - 1\right )}}{4 \, g \cos \left (f x + e\right ) \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right ) - 4 \, \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{g \sin \left (f x + e\right )}}{4 \, a c f \cos \left (f x + e\right )}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(sqrt(2)*a*sqrt(-g/a)*cos(f*x + e)*log((17*g*cos(f*x + e)^3 + 4*sqrt(2)*(3*cos(f*x + e)^2 + (3*cos(f*x +
e) + 4)*sin(f*x + e) - cos(f*x + e) - 4)*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(-g/a) + 3*g*cos(f*
x + e)^2 - 18*g*cos(f*x + e) + (17*g*cos(f*x + e)^2 + 14*g*cos(f*x + e) - 4*g)*sin(f*x + e) - 4*g)/(cos(f*x +
e)^3 + 3*cos(f*x + e)^2 + (cos(f*x + e)^2 - 2*cos(f*x + e) - 4)*sin(f*x + e) - 2*cos(f*x + e) - 4)) + 8*sqrt(a
*sin(f*x + e) + a)*sqrt(g*sin(f*x + e)))/(a*c*f*cos(f*x + e)), -1/4*(sqrt(2)*a*sqrt(g/a)*arctan(1/4*sqrt(2)*sq
rt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e))*sqrt(g/a)*(3*sin(f*x + e) - 1)/(g*cos(f*x + e)*sin(f*x + e)))*cos(
f*x + e) - 4*sqrt(a*sin(f*x + e) + a)*sqrt(g*sin(f*x + e)))/(a*c*f*cos(f*x + e))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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