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

Optimal. Leaf size=220 \[ \frac{2 (b c-a d) \tan (e+f x) \sqrt{\frac{a+b \sec (e+f x)}{a+b}} \Pi \left (\frac{2 d}{c+d};\sin ^{-1}\left (\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right )|\frac{2 b}{a+b}\right )}{c f (c+d) \sqrt{-\tan ^2(e+f x)} \sqrt{a+b \sec (e+f x)}}-\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{c f} \]

[Out]

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

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Rubi [A]  time = 0.243852, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {3926, 3784, 3973} \[ \frac{2 (b c-a d) \tan (e+f x) \sqrt{\frac{a+b \sec (e+f x)}{a+b}} \Pi \left (\frac{2 d}{c+d};\sin ^{-1}\left (\frac{\sqrt{1-\sec (e+f x)}}{\sqrt{2}}\right )|\frac{2 b}{a+b}\right )}{c f (c+d) \sqrt{-\tan ^2(e+f x)} \sqrt{a+b \sec (e+f x)}}-\frac{2 \sqrt{a+b} \cot (e+f x) \sqrt{\frac{b (1-\sec (e+f x))}{a+b}} \sqrt{-\frac{b (\sec (e+f x)+1)}{a-b}} \Pi \left (\frac{a+b}{a};\sin ^{-1}\left (\frac{\sqrt{a+b \sec (e+f x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{c f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3926

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Dist[a/c,
Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + Dist[(b*c - a*d)/c, Int[Csc[e + f*x]/(Sqrt[a + b*Csc[e + f*x]]*(c + d
*Csc[e + f*x])), 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 3784

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

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

Mathematica [A]  time = 4.20486, size = 229, normalized size = 1.04 \[ \frac{4 \cos ^2\left (\frac{1}{2} (e+f x)\right ) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \sqrt{a+b \sec (e+f x)} \left (c (-(a-b)) (c+d) \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right ),\frac{a-b}{a+b}\right )-2 a \left (c^2-d^2\right ) \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{a-b}{a+b}\right )+2 d (b c-a d) \Pi \left (\frac{c-d}{c+d};-\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{a-b}{a+b}\right )\right )}{c f (c-d) (c+d) (a \cos (e+f x)+b)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.335, size = 443, normalized size = 2. \begin{align*} -2\,{\frac{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{2} \left ( -1+\cos \left ( fx+e \right ) \right ) }{cf \left ( c-d \right ) \left ( c+d \right ) \left ( a\cos \left ( fx+e \right ) +b \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{1+\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 ) }}} \left ({\it EllipticF} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{a-b}{a+b}}} \right ) a{c}^{2}+{\it EllipticF} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{a-b}{a+b}}} \right ) acd-{\it EllipticF} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{a-b}{a+b}}} \right ) b{c}^{2}-{\it EllipticF} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{a-b}{a+b}}} \right ) bcd-2\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},-1,\sqrt{{\frac{a-b}{a+b}}} \right ) a{c}^{2}+2\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},-1,\sqrt{{\frac{a-b}{a+b}}} \right ) a{d}^{2}-2\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},{\frac{c-d}{c+d}},\sqrt{{\frac{a-b}{a+b}}} \right ) a{d}^{2}+2\,{\it EllipticPi} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},{\frac{c-d}{c+d}},\sqrt{{\frac{a-b}{a+b}}} \right ) bcd \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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