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

Optimal. Leaf size=225 \[ -\frac{2 \cot (e+f x) \sqrt{-\frac{d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt{\frac{d (\sec (e+f x)+1)}{c+d \sec (e+f x)}} (c+d \sec (e+f x)) \Pi \left (\frac{c}{c+d};\sin ^{-1}\left (\frac{\sqrt{c+d}}{\sqrt{c+d \sec (e+f x)}}\right )|\frac{c-d}{c+d}\right )}{a f \sqrt{c+d}}-\frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} E\left (\sin ^{-1}\left (\frac{\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac{c-d}{c+d}\right )}{a f \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.21073, antiderivative size = 225, 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 = {3925, 3780, 3968} \[ -\frac{2 \cot (e+f x) \sqrt{-\frac{d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt{\frac{d (\sec (e+f x)+1)}{c+d \sec (e+f x)}} (c+d \sec (e+f x)) \Pi \left (\frac{c}{c+d};\sin ^{-1}\left (\frac{\sqrt{c+d}}{\sqrt{c+d \sec (e+f x)}}\right )|\frac{c-d}{c+d}\right )}{a f \sqrt{c+d}}-\frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} E\left (\sin ^{-1}\left (\frac{\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac{c-d}{c+d}\right )}{a f \sqrt{\frac{c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3925

Int[Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)]/(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_)), x_Symbol] :> Dist[1/c,
Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[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] && (EqQ[a^2 - b^2, 0] || EqQ[c^2 - d^2, 0])

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]

Rule 3968

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

Rubi steps

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

Mathematica [A]  time = 8.51973, size = 180, normalized size = 0.8 \[ -\frac{4 \cos ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{c+d \sec (e+f x)} \left (2 (c-d) \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right ),\frac{c-d}{c+d}\right )+(c+d) E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{c-d}{c+d}\right )+4 c \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{c-d}{c+d}\right )\right )}{a f (c+d) (\cos (e+f x)+1)^2 \sqrt{\frac{c \cos (e+f x)+d}{(c+d) (\cos (e+f x)+1)}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.414, size = 285, normalized size = 1.3 \begin{align*} -{\frac{ \left ( 1+\cos \left ( fx+e \right ) \right ) ^{2} \left ( -1+\cos \left ( fx+e \right ) \right ) }{fa \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 ) }{1+\cos \left ( fx+e \right ) }}}\sqrt{{\frac{d+c\cos \left ( fx+e \right ) }{ \left ( c+d \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}} \left ( 2\,{\it EllipticF} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) c-2\,{\it EllipticF} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) d+c{\it EllipticE} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) +d{\it EllipticE} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{c-d}{c+d}}} \right ) -4\,c{\it EllipticPi} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},-1,\sqrt{{\frac{c-d}{c+d}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(d*sec(f*x + e) + c)/(a*sec(f*x + e) + a), 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}}{a \sec \left (f x + e\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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