3.82 \(\int \sec (e+f x) (a+a \sec (e+f x))^3 \sqrt{c-c \sec (e+f x)} \, dx\)

Optimal. Leaf size=41 \[ -\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^3}{7 f \sqrt{c-c \sec (e+f x)}} \]

[Out]

(-2*c*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(7*f*Sqrt[c - c*Sec[e + f*x]])

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Rubi [A]  time = 0.09561, 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} \[ -\frac{2 c \tan (e+f x) (a \sec (e+f x)+a)^3}{7 f \sqrt{c-c \sec (e+f x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*c*(a + a*Sec[e + f*x])^3*Tan[e + f*x])/(7*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))^3 \sqrt{c-c \sec (e+f x)} \, dx &=-\frac{2 c (a+a \sec (e+f x))^3 \tan (e+f x)}{7 f \sqrt{c-c \sec (e+f x)}}\\ \end{align*}

Mathematica [A]  time = 0.70759, size = 55, normalized size = 1.34 \[ \frac{16 a^3 \cos ^6\left (\frac{1}{2} (e+f x)\right ) \cot \left (\frac{1}{2} (e+f x)\right ) \sec ^3(e+f x) \sqrt{c-c \sec (e+f x)}}{7 f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*a^3*Cos[(e + f*x)/2]^6*Cot[(e + f*x)/2]*Sec[e + f*x]^3*Sqrt[c - c*Sec[e + f*x]])/(7*f)

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Maple [A]  time = 0.248, size = 55, normalized size = 1.3 \begin{align*}{\frac{2\,{a}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{7}}{7\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3} \left ( -1+\cos \left ( fx+e \right ) \right ) ^{4}}\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))^3*(c-c*sec(f*x+e))^(1/2),x)

[Out]

2/7*a^3/f*(c*(-1+cos(f*x+e))/cos(f*x+e))^(1/2)*sin(f*x+e)^7/cos(f*x+e)^3/(-1+cos(f*x+e))^4

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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))^3*(c-c*sec(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

Timed out

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Fricas [B]  time = 0.468582, size = 231, normalized size = 5.63 \begin{align*} \frac{2 \,{\left (a^{3} \cos \left (f x + e\right )^{4} + 4 \, a^{3} \cos \left (f x + e\right )^{3} + 6 \, a^{3} \cos \left (f x + e\right )^{2} + 4 \, a^{3} \cos \left (f x + e\right ) + a^{3}\right )} \sqrt{\frac{c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}}}{7 \, f \cos \left (f x + e\right )^{3} \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))^3*(c-c*sec(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

2/7*(a^3*cos(f*x + e)^4 + 4*a^3*cos(f*x + e)^3 + 6*a^3*cos(f*x + e)^2 + 4*a^3*cos(f*x + e) + a^3)*sqrt((c*cos(
f*x + e) - c)/cos(f*x + e))/(f*cos(f*x + e)^3*sin(f*x + e))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int \sqrt{- c \sec{\left (e + f x \right )} + c} \sec{\left (e + f x \right )}\, dx + \int 3 \sqrt{- c \sec{\left (e + f x \right )} + c} \sec ^{2}{\left (e + f x \right )}\, dx + \int 3 \sqrt{- c \sec{\left (e + f x \right )} + c} \sec ^{3}{\left (e + f x \right )}\, dx + \int \sqrt{- c \sec{\left (e + f x \right )} + c} \sec ^{4}{\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))**3*(c-c*sec(f*x+e))**(1/2),x)

[Out]

a**3*(Integral(sqrt(-c*sec(e + f*x) + c)*sec(e + f*x), x) + Integral(3*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)*
*2, x) + Integral(3*sqrt(-c*sec(e + f*x) + c)*sec(e + f*x)**3, x) + Integral(sqrt(-c*sec(e + f*x) + c)*sec(e +
f*x)**4, x))

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Giac [A]  time = 1.55148, size = 46, normalized size = 1.12 \begin{align*} \frac{16 \, \sqrt{2} a^{3} c^{4}}{7 \,{\left (c \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} - c\right )}^{\frac{7}{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))^3*(c-c*sec(f*x+e))^(1/2),x, algorithm="giac")

[Out]

16/7*sqrt(2)*a^3*c^4/((c*tan(1/2*f*x + 1/2*e)^2 - c)^(7/2)*f)