3.17 \(\int \frac{1}{(a+b \cos (e+f x)) \sqrt{c+d \sec (e+f x)}} \, dx\)

Optimal. Leaf size=102 \[ \frac{2 \tan (e+f x) \sqrt{\frac{c+d \sec (e+f x)}{c+d}} \Pi \left (\frac{2 a}{a+b};\sin ^{-1}\left (\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right )|\frac{2 d}{c+d}\right )}{f (a+b) \sqrt{-\tan ^2(e+f x)} \sqrt{c+d \sec (e+f x)}} \]

[Out]

(2*EllipticPi[(2*a)/(a + b), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*d)/(c + d)]*Sqrt[(c + d*Sec[e + f*x])/
(c + d)]*Tan[e + f*x])/((a + b)*f*Sqrt[c + d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])

________________________________________________________________________________________

Rubi [A]  time = 0.20217, antiderivative size = 102, normalized size of antiderivative = 1., 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 = {2829, 3973} \[ \frac{2 \tan (e+f x) \sqrt{\frac{c+d \sec (e+f x)}{c+d}} \Pi \left (\frac{2 a}{a+b};\sin ^{-1}\left (\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right )|\frac{2 d}{c+d}\right )}{f (a+b) \sqrt{-\tan ^2(e+f x)} \sqrt{c+d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*EllipticPi[(2*a)/(a + b), ArcSin[Sqrt[1 - Sec[e + f*x]]/Sqrt[2]], (2*d)/(c + d)]*Sqrt[(c + d*Sec[e + f*x])/
(c + d)]*Tan[e + f*x])/((a + b)*f*Sqrt[c + d*Sec[e + f*x]]*Sqrt[-Tan[e + f*x]^2])

Rule 2829

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Int
[((b + a*Csc[e + f*x])^m*(c + d*Csc[e + f*x])^n)/Csc[e + f*x]^m, x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !In
tegerQ[n] && IntegerQ[m]

Rule 3973

Int[csc[(e_.) + (f_.)*(x_)]/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))
), x_Symbol] :> Simp[(-2*Cot[e + f*x]*Sqrt[(a + b*Csc[e + f*x])/(a + b)]*EllipticPi[(2*d)/(c + d), ArcSin[Sqrt
[1 - Csc[e + f*x]]/Sqrt[2]], (2*b)/(a + b)])/(f*(c + d)*Sqrt[a + b*Csc[e + f*x]]*Sqrt[-Cot[e + f*x]^2]), x] /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(a+b \cos (e+f x)) \sqrt{c+d \sec (e+f x)}} \, dx &=\int \frac{\sec (e+f x)}{(b+a \sec (e+f x)) \sqrt{c+d \sec (e+f x)}} \, dx\\ &=\frac{2 \Pi \left (\frac{2 a}{a+b};\sin ^{-1}\left (\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right )|\frac{2 d}{c+d}\right ) \sqrt{\frac{c+d \sec (e+f x)}{c+d}} \tan (e+f x)}{(a+b) f \sqrt{c+d \sec (e+f x)} \sqrt{-\tan ^2(e+f x)}}\\ \end{align*}

Mathematica [A]  time = 4.40852, size = 189, normalized size = 1.85 \[ -\frac{2 \sqrt{\sec (e+f x)} \sqrt{\sec (e+f x)+1} \sqrt{\cos (e+f x) \sec ^2\left (\frac{1}{2} (e+f x)\right )} \sqrt{\frac{c \cos (e+f x)+d}{(c+d) (\cos (e+f x)+1)}} \left ((a+b) F\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{c-d}{c+d}\right )+2 a \Pi \left (\frac{b-a}{a+b};-\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{c-d}{c+d}\right )\right )}{f (a-b) (a+b) \sqrt{\sec ^2\left (\frac{1}{2} (e+f x)\right )} \sqrt{c+d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[(d + c*Cos[e + f*x])/((c + d)*(1 + Cos[e + f*x]))]*((a + b)*EllipticF[ArcSin[Tan[(e + f*x)/2]], (c -
d)/(c + d)] + 2*a*EllipticPi[(-a + b)/(a + b), -ArcSin[Tan[(e + f*x)/2]], (c - d)/(c + d)])*Sqrt[Cos[e + f*x]*
Sec[(e + f*x)/2]^2]*Sqrt[Sec[e + f*x]]*Sqrt[1 + Sec[e + f*x]])/((a - b)*(a + b)*f*Sqrt[Sec[(e + f*x)/2]^2]*Sqr
t[c + d*Sec[e + f*x]])

________________________________________________________________________________________

Maple [B]  time = 0.214, size = 237, normalized size = 2.3 \begin{align*} -2\,{\frac{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2} \left ( \cos \left ( fx+e \right ) -1 \right ) }{f \left ( a-b \right ) \left ( a+b \right ) \left ( d+c\cos \left ( fx+e \right ) \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{ \left ( c+d \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) }}} \left ( a{\it EllipticF} \left ({\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) +b{\it EllipticF} \left ({\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) -2\,a{\it EllipticPi} \left ({\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }},-{\frac{a-b}{a+b}},\sqrt{{\frac{c-d}{c+d}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/f/(a-b)/(a+b)*((d+c*cos(f*x+e))/cos(f*x+e))^(1/2)*(cos(f*x+e)/(cos(f*x+e)+1))^(1/2)*(1/(c+d)*(d+c*cos(f*x+e
))/(cos(f*x+e)+1))^(1/2)*(cos(f*x+e)+1)^2*(a*EllipticF((cos(f*x+e)-1)/sin(f*x+e),((c-d)/(c+d))^(1/2))+b*Ellipt
icF((cos(f*x+e)-1)/sin(f*x+e),((c-d)/(c+d))^(1/2))-2*a*EllipticPi((cos(f*x+e)-1)/sin(f*x+e),-(a-b)/(a+b),((c-d
)/(c+d))^(1/2)))*(cos(f*x+e)-1)/(d+c*cos(f*x+e))/sin(f*x+e)^2

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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