3.249 \(\int \sqrt{a+b \sec (e+f x)} \, dx\)

Optimal. Leaf size=125 \[ -\frac{2 \cot (e+f x) \sqrt{-\frac{b (1-\sec (e+f x))}{a+b \sec (e+f x)}} \sqrt{\frac{b (\sec (e+f x)+1)}{a+b \sec (e+f x)}} (a+b \sec (e+f x)) \Pi \left (\frac{a}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b}}{\sqrt{a+b \sec (e+f x)}}\right )|\frac{a-b}{a+b}\right )}{f \sqrt{a+b}} \]

[Out]

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

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Rubi [A]  time = 0.0316882, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3780} \[ -\frac{2 \cot (e+f x) \sqrt{-\frac{b (1-\sec (e+f x))}{a+b \sec (e+f x)}} \sqrt{\frac{b (\sec (e+f x)+1)}{a+b \sec (e+f x)}} (a+b \sec (e+f x)) \Pi \left (\frac{a}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b}}{\sqrt{a+b \sec (e+f x)}}\right )|\frac{a-b}{a+b}\right )}{f \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sec[e + f*x]],x]

[Out]

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

Rule 3780

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

Rubi steps

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

Mathematica [A]  time = 0.276401, size = 153, normalized size = 1.22 \[ -\frac{4 \cos ^2\left (\frac{1}{2} (e+f x)\right ) \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}} \sqrt{a+b \sec (e+f x)} \left ((a-b) \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right ),\frac{a-b}{a+b}\right )+2 a \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{a-b}{a+b}\right )\right )}{f (a \cos (e+f x)+b)} \]

Antiderivative was successfully verified.

[In]

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

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Maple [A]  time = 0.258, size = 215, normalized size = 1.7 \begin{align*} -2\,{\frac{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{2} \left ( -1+\cos \left ( fx+e \right ) \right ) }{f \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-{\it EllipticF} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{a-b}{a+b}}} \right ) b-2\,a{\it EllipticPi} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},-1,\sqrt{{\frac{a-b}{a+b}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/f*(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-EllipticF((-1+cos
(f*x+e))/sin(f*x+e),((a-b)/(a+b))^(1/2))*b-2*a*EllipticPi((-1+cos(f*x+e))/sin(f*x+e),-1,((a-b)/(a+b))^(1/2)))*
(-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 \sqrt{b \sec \left (f x + e\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sec(f*x + e) + a), 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),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \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),x)

[Out]

Integral(sqrt(a + b*sec(e + f*x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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