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

Optimal. Leaf size=95 \[ \frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{a+b \sec (e+f x)} E\left (\sin ^{-1}\left (\frac{\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac{a-b}{a+b}\right )}{c f \sqrt{\frac{a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}} \]

[Out]

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

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Rubi [A]  time = 0.119371, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {3968} \[ \frac{\sqrt{\frac{1}{\sec (e+f x)+1}} \sqrt{a+b \sec (e+f x)} E\left (\sin ^{-1}\left (\frac{\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac{a-b}{a+b}\right )}{c f \sqrt{\frac{a+b \sec (e+f x)}{(a+b) (\sec (e+f x)+1)}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [B]  time = 6.32919, size = 264, normalized size = 2.78 \[ \frac{\cos ^2\left (\frac{1}{2} (e+f x)\right ) \sqrt{\sec (e+f x)} \sqrt{a+b \sec (e+f x)} \left (\frac{2 \sqrt{\frac{\cos (e+f x)}{\cos (e+f x)+1}} \sqrt{\sec (e+f x)+1} \sec ^4\left (\frac{1}{2} (e+f x)\right ) E\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (e+f x)\right )\right )|\frac{a-b}{a+b}\right )}{\left (\frac{1}{\cos (e+f x)+1}\right )^{3/2} \sqrt{\frac{a \cos (e+f x)+b}{(a+b) (\cos (e+f x)+1)}}}+\frac{\left (\sin \left (\frac{3}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \sqrt{\sec (e+f x)+1} \sec ^5\left (\frac{1}{2} (e+f x)\right )}{\left (\frac{1}{\cos (e+f x)+1}\right )^{3/2}}-8 \sqrt{\sec (e+f x)} \left (\sin (e+f x)-\tan \left (\frac{1}{2} (e+f x)\right )\right )\right )}{4 c f (\sec (e+f x)+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cos[(e + f*x)/2]^2*Sqrt[Sec[e + f*x]]*Sqrt[a + b*Sec[e + f*x]]*((2*Sqrt[Cos[e + f*x]/(1 + Cos[e + f*x])]*Elli
pticE[ArcSin[Tan[(e + f*x)/2]], (a - b)/(a + b)]*Sec[(e + f*x)/2]^4*Sqrt[1 + Sec[e + f*x]])/(((1 + Cos[e + f*x
])^(-1))^(3/2)*Sqrt[(b + a*Cos[e + f*x])/((a + b)*(1 + Cos[e + f*x]))]) + (Sec[(e + f*x)/2]^5*Sqrt[1 + Sec[e +
f*x]]*(-Sin[(e + f*x)/2] + Sin[(3*(e + f*x))/2]))/((1 + Cos[e + f*x])^(-1))^(3/2) - 8*Sqrt[Sec[e + f*x]]*(Sin
[e + f*x] - Tan[(e + f*x)/2])))/(4*c*f*(1 + Sec[e + f*x]))

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Maple [A]  time = 0.306, size = 153, normalized size = 1.6 \begin{align*} -{\frac{ \left ( -a-b \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) ^{2}}{fc \left ( a\cos \left ( fx+e \right ) +b \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{\it EllipticE} \left ({\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }},\sqrt{{\frac{a-b}{a+b}}} \right ) \sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{ \left ( a+b \right ) \left ( 1+\cos \left ( fx+e \right ) \right ) }}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{1+\cos \left ( fx+e \right ) }}}\sqrt{{\frac{a\cos \left ( fx+e \right ) +b}{\cos \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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