3.141 \(\int \frac{\sec (e+f x) \sqrt{c-c \sec (e+f x)}}{(a+a \sec (e+f x))^{3/2}} \, dx\)

Optimal. Leaf size=42 \[ \frac{\tan (e+f x) \sqrt{c-c \sec (e+f x)}}{2 f (a \sec (e+f x)+a)^{3/2}} \]

[Out]

(Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(2*f*(a + a*Sec[e + f*x])^(3/2))

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Rubi [A]  time = 0.134305, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.028, Rules used = {3950} \[ \frac{\tan (e+f x) \sqrt{c-c \sec (e+f x)}}{2 f (a \sec (e+f x)+a)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a + a*Sec[e + f*x])^(3/2),x]

[Out]

(Sqrt[c - c*Sec[e + f*x]]*Tan[e + f*x])/(2*f*(a + a*Sec[e + f*x])^(3/2))

Rule 3950

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

Rubi steps

\begin{align*} \int \frac{\sec (e+f x) \sqrt{c-c \sec (e+f x)}}{(a+a \sec (e+f x))^{3/2}} \, dx &=\frac{\sqrt{c-c \sec (e+f x)} \tan (e+f x)}{2 f (a+a \sec (e+f x))^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.183704, size = 42, normalized size = 1. \[ \frac{\csc (e+f x) \sqrt{c-c \sec (e+f x)}}{a f \sqrt{a (\sec (e+f x)+1)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a + a*Sec[e + f*x])^(3/2),x]

[Out]

(Csc[e + f*x]*Sqrt[c - c*Sec[e + f*x]])/(a*f*Sqrt[a*(1 + Sec[e + f*x])])

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Maple [A]  time = 0.284, size = 73, normalized size = 1.7 \begin{align*}{\frac{\cos \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2}}{2\,f{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}}\sqrt{{\frac{c \left ( -1+\cos \left ( fx+e \right ) \right ) }{\cos \left ( fx+e \right ) }}}\sqrt{{\frac{a \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))^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.52174, size = 73, normalized size = 1.74 \begin{align*} \frac{\sqrt{c}{\left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{\left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}}{2 \, \sqrt{-a} 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))^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 0.466875, size = 186, normalized size = 4.43 \begin{align*} \frac{\sqrt{\frac{a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{{\left (a^{2} f \cos \left (f x + e\right ) + a^{2} f\right )} \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))^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- c \left (\sec{\left (e + f x \right )} - 1\right )} \sec{\left (e + f x \right )}}{\left (a \left (\sec{\left (e + f x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \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))**(3/2),x)

[Out]

Integral(sqrt(-c*(sec(e + f*x) - 1))*sec(e + f*x)/(a*(sec(e + f*x) + 1))**(3/2), x)

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Giac [B]  time = 2.67175, size = 116, normalized size = 2.76 \begin{align*} -\frac{{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )} 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 )}{2 \, \sqrt{-a c} a f{\left | c \right |} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - 1\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))^(3/2),x, algorithm="giac")

[Out]

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