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

Optimal. Leaf size=39 \[ \frac{2 c \tan (e+f x)}{f (a \sec (e+f x)+a) \sqrt{c-c \sec (e+f x)}} \]

[Out]

(2*c*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])

________________________________________________________________________________________

Rubi [A]  time = 0.106548, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.029, Rules used = {3953} \[ \frac{2 c \tan (e+f x)}{f (a \sec (e+f x)+a) \sqrt{c-c \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*c*Tan[e + f*x])/(f*(a + a*Sec[e + f*x])*Sqrt[c - c*Sec[e + f*x]])

Rule 3953

Int[csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*Sqrt[csc[(e_.) + (f_.)*(x_)]*(d_.) +
(c_)], x_Symbol] :> Simp[(2*a*c*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(b*f*(2*m + 1)*Sqrt[c + d*Csc[e + f*x]]),
x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[m, -2^(-1)]

Rubi steps

\begin{align*} \int \frac{\sec (e+f x) \sqrt{c-c \sec (e+f x)}}{a+a \sec (e+f x)} \, dx &=\frac{2 c \tan (e+f x)}{f (a+a \sec (e+f x)) \sqrt{c-c \sec (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.131101, size = 29, normalized size = 0.74 \[ -\frac{2 \cot (e+f x) \sqrt{c-c \sec (e+f x)}}{a f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Cot[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a*f)

________________________________________________________________________________________

Maple [A]  time = 0.235, size = 43, normalized size = 1.1 \begin{align*} -2\,{\frac{\cos \left ( fx+e \right ) }{fa\sin \left ( fx+e \right ) }\sqrt{{\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/a/f*(c*(-1+cos(f*x+e))/cos(f*x+e))^(1/2)*cos(f*x+e)/sin(f*x+e)

________________________________________________________________________________________

Maxima [B]  time = 1.52542, size = 113, normalized size = 2.9 \begin{align*} -\frac{\sqrt{2} \sqrt{c} - \frac{\sqrt{2} \sqrt{c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{a f \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1} \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sqrt(2)*sqrt(c) - sqrt(2)*sqrt(c)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2)/(a*f*sqrt(sin(f*x + e)/(cos(f*x + e)
+ 1) + 1)*sqrt(sin(f*x + e)/(cos(f*x + e) + 1) - 1))

________________________________________________________________________________________

Fricas [A]  time = 0.456224, size = 103, normalized size = 2.64 \begin{align*} -\frac{2 \, \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{a f \sin \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*cos(f*x + e)/(a*f*sin(f*x + e))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.42032, size = 85, normalized size = 2.18 \begin{align*} -\frac{\sqrt{2} \sqrt{c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{3} + \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )\right ) \mathrm{sgn}\left (\cos \left (f x + e\right )\right )}{a f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(2)*sqrt(c*tan(1/2*f*x + 1/2*e)^2 - c)*sgn(tan(1/2*f*x + 1/2*e)^3 + tan(1/2*f*x + 1/2*e))*sgn(cos(f*x + e
))/(a*f)