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

Optimal. Leaf size=39 \[ -\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)}{3 f \sqrt{c-c \sec (e+f x)}} \]

[Out]

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

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Rubi [A]  time = 0.0591537, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031, Rules used = {3953} \[ -\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)}{3 f \sqrt{c-c \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.175435, size = 51, normalized size = 1.31 \[ \frac{4 a \cos ^2\left (\frac{1}{2} (e+f x)\right ) \cot \left (\frac{1}{2} (e+f x)\right ) \sec (e+f x) \sqrt{c-c \sec (e+f x)}}{3 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.221, size = 53, normalized size = 1.4 \begin{align*}{\frac{2\,a \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{3\,f\cos \left ( fx+e \right ) \left ( -1+\cos \left ( fx+e \right ) \right ) ^{2}}\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)*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(1/2),x)

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [A]  time = 0.454288, size = 158, normalized size = 4.05 \begin{align*} \frac{2 \,{\left (a \cos \left (f x + e\right )^{2} + 2 \, a \cos \left (f x + e\right ) + a\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{3 \, f \cos \left (f x + e\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)*(a+a*sec(f*x+e))*(c-c*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.36078, size = 43, normalized size = 1.1 \begin{align*} \frac{4 \, \sqrt{2} a c^{2}}{3 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{3}{2}} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*sqrt(2)*a*c^2/((c*tan(1/2*f*x + 1/2*e)^2 - c)^(3/2)*f)