3.615 \(\int \frac{1}{\sqrt{c \cos (e+f x)} \sqrt{a+b \sin (e+f x)}} \, dx\)

Optimal. Leaf size=183 \[ \frac{2 \sqrt{2} \sqrt [4]{b-a} \sqrt{c \cos (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{a+b} \sqrt{\frac{\cos (e+f x)+\sin (e+f x)+1}{\cos (e+f x)-\sin (e+f x)+1}}}{\sqrt [4]{b-a}}\right )\right |-1\right )}{c f \sqrt [4]{a+b} \sqrt{\frac{\sin (e+f x)+\cos (e+f x)+1}{-\sin (e+f x)+\cos (e+f x)+1}} \sqrt{a+b \sin (e+f x)}} \]

[Out]

(2*Sqrt[2]*(-a + b)^(1/4)*Sqrt[c*Cos[e + f*x]]*EllipticF[ArcSin[((a + b)^(1/4)*Sqrt[(1 + Cos[e + f*x] + Sin[e
+ f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])])/(-a + b)^(1/4)], -1]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Sin[e
+ f*x]))])/((a + b)^(1/4)*c*f*Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]*Sqrt[
a + b*Sin[e + f*x]])

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Rubi [B]  time = 0.427507, antiderivative size = 374, normalized size of antiderivative = 2.04, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {2697, 220} \[ \frac{\sqrt{2} \sqrt [4]{a-b} \sqrt{c \cos (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}} \sqrt{\frac{a+b \sin (e+f x)}{(a-b) (\sin (e) (-\cos (f x))-\cos (e) \sin (f x)+1) \left (\frac{\sqrt{a+b} (\sin (e+f x)+\cos (e+f x)+1)}{\sqrt{a-b} (-\sin (e+f x)+\cos (e+f x)+1)}+1\right )^2}} \left (\frac{\sqrt{a+b} (\sin (e+f x)+\cos (e+f x)+1)}{\sqrt{a-b} (-\sin (e+f x)+\cos (e+f x)+1)}+1\right ) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b} \sqrt{\frac{\cos (e+f x)+\sin (e+f x)+1}{\cos (e+f x)-\sin (e+f x)+1}}}{\sqrt [4]{a-b}}\right )|\frac{1}{2}\right )}{c f \sqrt [4]{a+b} \sqrt{\frac{\sin (e+f x)+\cos (e+f x)+1}{-\sin (e+f x)+\cos (e+f x)+1}} \sqrt{a+b \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a-b) (\sin (e) (-\cos (f x))-\cos (e) \sin (f x)+1)}}} \]

Warning: Unable to verify antiderivative.

[In]

Int[1/(Sqrt[c*Cos[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]),x]

[Out]

(Sqrt[2]*(a - b)^(1/4)*Sqrt[c*Cos[e + f*x]]*EllipticF[2*ArcTan[((a + b)^(1/4)*Sqrt[(1 + Cos[e + f*x] + Sin[e +
f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])])/(a - b)^(1/4)], 1/2]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Sin[e
+ f*x]))]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Cos[f*x]*Sin[e] - Cos[e]*Sin[f*x])*(1 + (Sqrt[a + b]*(1 + Co
s[e + f*x] + Sin[e + f*x]))/(Sqrt[a - b]*(1 + Cos[e + f*x] - Sin[e + f*x])))^2)]*(1 + (Sqrt[a + b]*(1 + Cos[e
+ f*x] + Sin[e + f*x]))/(Sqrt[a - b]*(1 + Cos[e + f*x] - Sin[e + f*x]))))/((a + b)^(1/4)*c*f*Sqrt[(1 + Cos[e +
f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]*Sqrt[a + b*Sin[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((
a - b)*(1 - Cos[f*x]*Sin[e] - Cos[e]*Sin[f*x]))])

Rule 2697

Int[1/(Sqrt[cos[(e_.) + (f_.)*(x_)]*(g_.)]*Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]), x_Symbol] :> Dist[(2*S
qrt[2]*Sqrt[g*Cos[e + f*x]]*Sqrt[(a + b*Sin[e + f*x])/((a - b)*(1 - Sin[e + f*x]))])/(f*g*Sqrt[a + b*Sin[e + f
*x]]*Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]), Subst[Int[1/Sqrt[1 + ((a + b)
*x^4)/(a - b)], x], x, Sqrt[(1 + Cos[e + f*x] + Sin[e + f*x])/(1 + Cos[e + f*x] - Sin[e + f*x])]], x] /; FreeQ
[{a, b, e, f, g}, x] && NeQ[a^2 - b^2, 0]

Rule 220

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 4]}, Simp[((1 + q^2*x^2)*Sqrt[(a + b*x^4)/(a*(
1 + q^2*x^2)^2)]*EllipticF[2*ArcTan[q*x], 1/2])/(2*q*Sqrt[a + b*x^4]), x]] /; FreeQ[{a, b}, x] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{c \cos (e+f x)} \sqrt{a+b \sin (e+f x)}} \, dx &=\frac{\left (2 \sqrt{2} \sqrt{c \cos (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{(a+b) x^4}{a-b}}} \, dx,x,\sqrt{\frac{1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}}\right )}{c f \sqrt{\frac{1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}} \sqrt{a+b \sin (e+f x)}}\\ &=\frac{\sqrt{2} \sqrt [4]{a-b} \sqrt{c \cos (e+f x)} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{a+b} \sqrt{\frac{1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}}}{\sqrt [4]{a-b}}\right )|\frac{1}{2}\right ) \sqrt{\frac{a+b \sin (e+f x)}{(a-b) (1-\sin (e+f x))}} \sqrt{\frac{a+b \sin (e+f x)}{(a-b) (1-\cos (f x) \sin (e)-\cos (e) \sin (f x)) \left (1+\frac{\sqrt{a+b} (1+\cos (e+f x)+\sin (e+f x))}{\sqrt{a-b} (1+\cos (e+f x)-\sin (e+f x))}\right )^2}} \left (1+\frac{\sqrt{a+b} (1+\cos (e+f x)+\sin (e+f x))}{\sqrt{a-b} (1+\cos (e+f x)-\sin (e+f x))}\right )}{\sqrt [4]{a+b} c f \sqrt{\frac{1+\cos (e+f x)+\sin (e+f x)}{1+\cos (e+f x)-\sin (e+f x)}} \sqrt{a+b \sin (e+f x)} \sqrt{\frac{a+b \sin (e+f x)}{(a-b) (1-\cos (f x) \sin (e)-\cos (e) \sin (f x))}}}\\ \end{align*}

Mathematica [C]  time = 0.313366, size = 117, normalized size = 0.64 \[ -\frac{2 c (\sin (e+f x)-1) \left (\frac{(a+b) (\sin (e+f x)+1)}{(a-b) (\sin (e+f x)-1)}\right )^{3/4} \sqrt{a+b \sin (e+f x)} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{2 (a+b \sin (e+f x))}{(a-b) (\sin (e+f x)-1)}\right )}{f (a+b) (c \cos (e+f x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[c*Cos[e + f*x]]*Sqrt[a + b*Sin[e + f*x]]),x]

[Out]

(-2*c*Hypergeometric2F1[1/2, 3/4, 3/2, (-2*(a + b*Sin[e + f*x]))/((a - b)*(-1 + Sin[e + f*x]))]*(-1 + Sin[e +
f*x])*(((a + b)*(1 + Sin[e + f*x]))/((a - b)*(-1 + Sin[e + f*x])))^(3/4)*Sqrt[a + b*Sin[e + f*x]])/((a + b)*f*
(c*Cos[e + f*x])^(3/2))

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Maple [B]  time = 0.514, size = 442, normalized size = 2.4 \begin{align*} 4\,{\frac{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2} \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2} \left ( b+\sqrt{-{a}^{2}+{b}^{2}}+a \right ) \left ( 1+\sin \left ( fx+e \right ) \right ) }{f \left ( b+\sqrt{-{a}^{2}+{b}^{2}}-a \right ) \sqrt{a+b\sin \left ( fx+e \right ) } \left ( \sin \left ( fx+e \right ) \right ) ^{4}\sqrt{c\cos \left ( fx+e \right ) }}{\it EllipticF} \left ( \sqrt{{\frac{ \left ( b+\sqrt{-{a}^{2}+{b}^{2}}-a \right ) \left ( -1+\sin \left ( fx+e \right ) \right ) }{ \left ( b+\sqrt{-{a}^{2}+{b}^{2}}+a \right ) \cos \left ( fx+e \right ) }}},\sqrt{{\frac{ \left ( a-b+\sqrt{-{a}^{2}+{b}^{2}} \right ) \left ( b+\sqrt{-{a}^{2}+{b}^{2}}+a \right ) }{ \left ( -b+\sqrt{-{a}^{2}+{b}^{2}}-a \right ) \left ( b+\sqrt{-{a}^{2}+{b}^{2}}-a \right ) }}} \right ) \sqrt{{\frac{\cos \left ( fx+e \right ) \sqrt{-{a}^{2}+{b}^{2}}+a\sin \left ( fx+e \right ) +b\cos \left ( fx+e \right ) +\sqrt{-{a}^{2}+{b}^{2}}+b}{ \left ( b+\sqrt{-{a}^{2}+{b}^{2}}+a \right ) \left ( 1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) }}}\sqrt{{\frac{ \left ( b+\sqrt{-{a}^{2}+{b}^{2}}-a \right ) \left ( -1+\sin \left ( fx+e \right ) \right ) }{ \left ( b+\sqrt{-{a}^{2}+{b}^{2}}+a \right ) \cos \left ( fx+e \right ) }}}\sqrt{-{\frac{a\sin \left ( fx+e \right ) -\cos \left ( fx+e \right ) \sqrt{-{a}^{2}+{b}^{2}}+b\cos \left ( fx+e \right ) -\sqrt{-{a}^{2}+{b}^{2}}+b}{ \left ( -b+\sqrt{-{a}^{2}+{b}^{2}}-a \right ) \left ( 1+\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/f/(b+(-a^2+b^2)^(1/2)-a)*EllipticF(((b+(-a^2+b^2)^(1/2)-a)/(b+(-a^2+b^2)^(1/2)+a)*(-1+sin(f*x+e))/cos(f*x+e)
)^(1/2),((a-b+(-a^2+b^2)^(1/2))*(b+(-a^2+b^2)^(1/2)+a)/(-b+(-a^2+b^2)^(1/2)-a)/(b+(-a^2+b^2)^(1/2)-a))^(1/2))*
(1/(b+(-a^2+b^2)^(1/2)+a)*(cos(f*x+e)*(-a^2+b^2)^(1/2)+a*sin(f*x+e)+b*cos(f*x+e)+(-a^2+b^2)^(1/2)+b)/(1+cos(f*
x+e)+sin(f*x+e)))^(1/2)*((b+(-a^2+b^2)^(1/2)-a)/(b+(-a^2+b^2)^(1/2)+a)*(-1+sin(f*x+e))/cos(f*x+e))^(1/2)*(-1/(
-b+(-a^2+b^2)^(1/2)-a)*(a*sin(f*x+e)-cos(f*x+e)*(-a^2+b^2)^(1/2)+b*cos(f*x+e)-(-a^2+b^2)^(1/2)+b)/(1+cos(f*x+e
)+sin(f*x+e)))^(1/2)*(cos(f*x+e)+1)^2*(-1+cos(f*x+e))^2*(b+(-a^2+b^2)^(1/2)+a)*(1+sin(f*x+e))/(a+b*sin(f*x+e))
^(1/2)/sin(f*x+e)^4/(c*cos(f*x+e))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c \cos \left (f x + e\right )} \sqrt{b \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(c*cos(f*x + e))*sqrt(b*sin(f*x + e) + a)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{c \cos \left (f x + e\right )} \sqrt{b \sin \left (f x + e\right ) + a}}{b c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a c \cos \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(c*cos(f*x + e))*sqrt(b*sin(f*x + e) + a)/(b*c*cos(f*x + e)*sin(f*x + e) + a*c*cos(f*x + e)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*cos(f*x+e))**(1/2)/(a+b*sin(f*x+e))**(1/2),x)

[Out]

Integral(1/(sqrt(c*cos(e + f*x))*sqrt(a + b*sin(e + f*x))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{c \cos \left (f x + e\right )} \sqrt{b \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/(sqrt(c*cos(f*x + e))*sqrt(b*sin(f*x + e) + a)), x)