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

Optimal. Leaf size=398 \[ -\frac{2 b \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right ),\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{a f \sqrt{c+d} (b c-a d)}-\frac{2 \sqrt{c+d} \cot (e+f x) (a+b \sec (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \Pi \left (\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{a c f \sqrt{a+b}} \]

[Out]

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

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Rubi [A]  time = 0.436726, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {3938, 3936, 3984} \[ -\frac{2 b \sqrt{a+b} \cot (e+f x) (c+d \sec (e+f x)) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (\sec (e+f x)+1)}{(a-b) (c+d \sec (e+f x))}} F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )}{a f \sqrt{c+d} (b c-a d)}-\frac{2 \sqrt{c+d} \cot (e+f x) (a+b \sec (e+f x)) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (\sec (e+f x)+1)}{(c-d) (a+b \sec (e+f x))}} \Pi \left (\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right )}{a c f \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3938

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

Rule 3936

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)], x_Symbol] :> Simp[(
2*(a + b*Csc[e + f*x])*Sqrt[((b*c - a*d)*(1 + Csc[e + f*x]))/((c - d)*(a + b*Csc[e + f*x]))]*Sqrt[-(((b*c - a*
d)*(1 - Csc[e + f*x]))/((c + d)*(a + b*Csc[e + f*x])))]*EllipticPi[(a*(c + d))/(c*(a + b)), ArcSin[(Rt[(a + b)
/(c + d), 2]*Sqrt[c + d*Csc[e + f*x]])/Sqrt[a + b*Csc[e + f*x]]], ((a - b)*(c + d))/((a + b)*(c - d))])/(c*f*R
t[(a + b)/(c + d), 2]*Cot[e + f*x]), 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]

Rule 3984

Int[csc[(e_.) + (f_.)*(x_)]/(Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) + (
c_)]), x_Symbol] :> Simp[(-2*(c + d*Csc[e + f*x])*Sqrt[((b*c - a*d)*(1 - Csc[e + f*x]))/((a + b)*(c + d*Csc[e
+ f*x]))]*Sqrt[-(((b*c - a*d)*(1 + Csc[e + f*x]))/((a - b)*(c + d*Csc[e + f*x])))]*EllipticF[ArcSin[Rt[(c + d)
/(a + b), 2]*(Sqrt[a + b*Csc[e + f*x]]/Sqrt[c + d*Csc[e + f*x]])], ((a + b)*(c - d))/((a - b)*(c + d))])/(f*(b
*c - a*d)*Rt[(c + d)/(a + b), 2]*Cot[e + f*x]), 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}{\sqrt{a+b \sec (e+f x)} \sqrt{c+d \sec (e+f x)}} \, dx &=\frac{\int \frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{c+d \sec (e+f x)}} \, dx}{a}-\frac{b \int \frac{\sec (e+f x)}{\sqrt{a+b \sec (e+f x)} \sqrt{c+d \sec (e+f x)}} \, dx}{a}\\ &=-\frac{2 \sqrt{c+d} \cot (e+f x) \Pi \left (\frac{a (c+d)}{(a+b) c};\sin ^{-1}\left (\frac{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}\right )|\frac{(a-b) (c+d)}{(a+b) (c-d)}\right ) \sqrt{-\frac{(b c-a d) (1-\sec (e+f x))}{(c+d) (a+b \sec (e+f x))}} \sqrt{\frac{(b c-a d) (1+\sec (e+f x))}{(c-d) (a+b \sec (e+f x))}} (a+b \sec (e+f x))}{a \sqrt{a+b} c f}-\frac{2 b \sqrt{a+b} \cot (e+f x) F\left (\sin ^{-1}\left (\frac{\sqrt{c+d} \sqrt{a+b \sec (e+f x)}}{\sqrt{a+b} \sqrt{c+d \sec (e+f x)}}\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right ) \sqrt{\frac{(b c-a d) (1-\sec (e+f x))}{(a+b) (c+d \sec (e+f x))}} \sqrt{-\frac{(b c-a d) (1+\sec (e+f x))}{(a-b) (c+d \sec (e+f x))}} (c+d \sec (e+f x))}{a \sqrt{c+d} (b c-a d) f}\\ \end{align*}

Mathematica [C]  time = 1.98328, size = 249, normalized size = 0.63 \[ \frac{4 i \cos ^2\left (\frac{1}{2} (e+f x)\right ) \sec (e+f x) \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \sqrt{\frac{c \cos (e+f x)+d}{(c+d) (\cos (e+f x)+1)}} \left (\text{EllipticF}\left (i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (e+f x)\right )\right ),\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )-2 \Pi \left (-\frac{a+b}{a-b};i \sinh ^{-1}\left (\sqrt{\frac{b-a}{a+b}} \tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{(a+b) (c-d)}{(a-b) (c+d)}\right )\right )}{f \sqrt{\frac{b-a}{a+b}} \sqrt{a+b \sec (e+f x)} \sqrt{c+d \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*I)*Cos[(e + f*x)/2]^2*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]*Sqrt[(d + c*Cos[e + f*x])/((
c + d)*(1 + Cos[e + f*x]))]*(EllipticF[I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], ((a + b)*(c - d))/(
(a - b)*(c + d))] - 2*EllipticPi[-((a + b)/(a - b)), I*ArcSinh[Sqrt[(-a + b)/(a + b)]*Tan[(e + f*x)/2]], ((a +
b)*(c - d))/((a - b)*(c + d))])*Sec[e + f*x])/(Sqrt[(-a + b)/(a + b)]*f*Sqrt[a + b*Sec[e + f*x]]*Sqrt[c + d*S
ec[e + f*x]])

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Maple [A]  time = 0.372, size = 292, normalized size = 0.7 \begin{align*} -2\,{\frac{\cos \left ( fx+e \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{f \left ( -1+\cos \left ( fx+e \right ) \right ) \left ( d+c\cos \left ( fx+e \right ) \right ) \left ( a\cos \left ( fx+e \right ) +b \right ) } \left ({\it EllipticF} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }\sqrt{{\frac{a-b}{a+b}}}},\sqrt{{\frac{ \left ( c-d \right ) \left ( a+b \right ) }{ \left ( a-b \right ) \left ( c+d \right ) }}} \right ) -2\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }\sqrt{{\frac{a-b}{a+b}}}},-{\frac{a+b}{a-b}},{\sqrt{{\frac{c-d}{c+d}}}{\frac{1}{\sqrt{{\frac{a-b}{a+b}}}}}} \right ) \right ) \sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{ \left ( a+b \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{ \left ( c+d \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}}{\frac{1}{\sqrt{{\frac{a-b}{a+b}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sec \left (f x + e\right ) + a} \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/(c+d*sec(f*x+e))^(1/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sec \left (f x + e\right ) + a} \sqrt{d \sec \left (f x + e\right ) + c}}{b d \sec \left (f x + e\right )^{2} + a c +{\left (b c + 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/(c+d*sec(f*x+e))^(1/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(b*sec(f*x + e) + a)*sqrt(d*sec(f*x + e) + c)/(b*d*sec(f*x + e)^2 + a*c + (b*c + a*d)*sec(f*x + e
)), x)

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sec \left (f x + e\right ) + a} \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/(c+d*sec(f*x+e))^(1/2)/(a+b*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

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