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

Optimal. Leaf size=41 \[ -\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^2}{5 f \sqrt{c-c \sec (e+f x)}} \]

[Out]

(-2*c*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(5*f*Sqrt[c - c*Sec[e + f*x]])

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Rubi [A]  time = 0.0967487, 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} \[ -\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^2}{5 f \sqrt{c-c \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c*(a + a*Sec[e + f*x])^2*Tan[e + f*x])/(5*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))^2 \sqrt{c-c \sec (e+f x)} \, dx &=-\frac{2 c (a+a \sec (e+f x))^2 \tan (e+f x)}{5 f \sqrt{c-c \sec (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.42574, size = 55, normalized size = 1.34 \[ \frac{8 a^2 \cos ^4\left (\frac{1}{2} (e+f x)\right ) \cot \left (\frac{1}{2} (e+f x)\right ) \sec ^2(e+f x) \sqrt{c-c \sec (e+f x)}}{5 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*a^2*Cos[(e + f*x)/2]^4*Cot[(e + f*x)/2]*Sec[e + f*x]^2*Sqrt[c - c*Sec[e + f*x]])/(5*f)

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Maple [A]  time = 0.253, size = 55, normalized size = 1.3 \begin{align*} -{\frac{2\,{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{5}}{5\,f \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( -1+\cos \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)*(a+a*sec(f*x+e))^2*(c-c*sec(f*x+e))^(1/2),x)

[Out]

-2/5*a^2/f*(c*(-1+cos(f*x+e))/cos(f*x+e))^(1/2)*sin(f*x+e)^5/cos(f*x+e)^2/(-1+cos(f*x+e))^3

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

[Out]

Timed out

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Fricas [B]  time = 0.465069, size = 200, normalized size = 4.88 \begin{align*} \frac{2 \,{\left (a^{2} \cos \left (f x + e\right )^{3} + 3 \, a^{2} \cos \left (f x + e\right )^{2} + 3 \, a^{2} \cos \left (f x + e\right ) + a^{2}\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{5 \, f \cos \left (f x + e\right )^{2} \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))^2*(c-c*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

2/5*(a^2*cos(f*x + e)^3 + 3*a^2*cos(f*x + e)^2 + 3*a^2*cos(f*x + e) + a^2)*sqrt((c*cos(f*x + e) - c)/cos(f*x +
e))/(f*cos(f*x + e)^2*sin(f*x + e))

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

[Out]

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

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Giac [A]  time = 1.42619, size = 46, normalized size = 1.12 \begin{align*} -\frac{8 \, \sqrt{2} a^{2} c^{3}}{5 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{5}{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))^2*(c-c*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

-8/5*sqrt(2)*a^2*c^3/((c*tan(1/2*f*x + 1/2*e)^2 - c)^(5/2)*f)