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

Optimal. Leaf size=41 \[ \frac{2 c \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.100431, antiderivative size = 41, 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} \[ \frac{2 c \tan (e+f x)}{3 f (a \sec (e+f x)+a)^2 \sqrt{c-c \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.150477, size = 55, normalized size = 1.34 \[ -\frac{\cos ^2(e+f x) \csc \left (\frac{1}{2} (e+f x)\right ) \sec ^3\left (\frac{1}{2} (e+f x)\right ) \sqrt{c-c \sec (e+f x)}}{6 a^2 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cos[e + f*x]^2*Csc[(e + f*x)/2]*Sec[(e + f*x)/2]^3*Sqrt[c - c*Sec[e + f*x]])/(6*a^2*f)

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Maple [A]  time = 0.256, size = 53, normalized size = 1.3 \begin{align*}{\frac{ \left ( -2+2\,\cos \left ( fx+e \right ) \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}{3\,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 ) }}}} \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))^2,x)

[Out]

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

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Maxima [B]  time = 1.52617, size = 147, normalized size = 3.59 \begin{align*} -\frac{\sqrt{2} \sqrt{c} - \frac{2 \, \sqrt{2} \sqrt{c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac{\sqrt{2} \sqrt{c} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}}}{6 \, a^{2} 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))^2,x, algorithm="maxima")

[Out]

-1/6*(sqrt(2)*sqrt(c) - 2*sqrt(2)*sqrt(c)*sin(f*x + e)^2/(cos(f*x + e) + 1)^2 + sqrt(2)*sqrt(c)*sin(f*x + e)^4
/(cos(f*x + e) + 1)^4)/(a^2*f*sqrt(sin(f*x + e)/(cos(f*x + e) + 1) + 1)*sqrt(sin(f*x + e)/(cos(f*x + e) + 1) -
1))

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Fricas [A]  time = 0.459774, size = 142, normalized size = 3.46 \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 )^{2}}{3 \,{\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))^2,x, algorithm="fricas")

[Out]

-2/3*sqrt((c*cos(f*x + e) - c)/cos(f*x + e))*cos(f*x + e)^2/((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*} \frac{\int \frac{\sqrt{- c \sec{\left (e + f x \right )} + c} \sec{\left (e + f x \right )}}{\sec ^{2}{\left (e + f x \right )} + 2 \sec{\left (e + f x \right )} + 1}\, dx}{a^{2}} \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))**2,x)

[Out]

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

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Giac [A]  time = 1.39376, size = 89, normalized size = 2.17 \begin{align*} \frac{\sqrt{2}{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{3}{2}} \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 )}{6 \, a^{2} c 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))^2,x, algorithm="giac")

[Out]

1/6*sqrt(2)*(c*tan(1/2*f*x + 1/2*e)^2 - c)^(3/2)*sgn(tan(1/2*f*x + 1/2*e)^3 + tan(1/2*f*x + 1/2*e))*sgn(cos(f*
x + e))/(a^2*c*f)