3.116 \(\int \sec (c+d x) \sqrt{a-a \sec (c+d x)} \, dx\)

Optimal. Leaf size=27 \[ -\frac{2 a \tan (c+d x)}{d \sqrt{a-a \sec (c+d x)}} \]

[Out]

(-2*a*Tan[c + d*x])/(d*Sqrt[a - a*Sec[c + d*x]])

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Rubi [A]  time = 0.0298956, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {3792} \[ -\frac{2 a \tan (c+d x)}{d \sqrt{a-a \sec (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]*Sqrt[a - a*Sec[c + d*x]],x]

[Out]

(-2*a*Tan[c + d*x])/(d*Sqrt[a - a*Sec[c + d*x]])

Rule 3792

Int[csc[(e_.) + (f_.)*(x_)]*Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-2*b*Cot[e + f*x])/
(f*Sqrt[a + b*Csc[e + f*x]]), x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

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

Mathematica [A]  time = 0.110675, size = 30, normalized size = 1.11 \[ \frac{2 \cot \left (\frac{1}{2} (c+d x)\right ) \sqrt{a-a \sec (c+d x)}}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]*Sqrt[a - a*Sec[c + d*x]],x]

[Out]

(2*Cot[(c + d*x)/2]*Sqrt[a - a*Sec[c + d*x]])/d

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Maple [A]  time = 0.15, size = 42, normalized size = 1.6 \begin{align*} -2\,{\frac{\sin \left ( dx+c \right ) }{d \left ( -1+\cos \left ( dx+c \right ) \right ) }\sqrt{{\frac{a \left ( -1+\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)*(a-a*sec(d*x+c))^(1/2),x)

[Out]

-2/d*(a*(-1+cos(d*x+c))/cos(d*x+c))^(1/2)*sin(d*x+c)/(-1+cos(d*x+c))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{-a \sec \left (d x + c\right ) + a} \sec \left (d x + c\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-a*sec(d*x + c) + a)*sec(d*x + c), x)

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Fricas [A]  time = 1.935, size = 107, normalized size = 3.96 \begin{align*} \frac{2 \, \sqrt{\frac{a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )}}{\left (\cos \left (d x + c\right ) + 1\right )}}{d \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)*(a-a*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{- a \left (\sec{\left (c + d x \right )} - 1\right )} \sec{\left (c + d x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-a*(sec(c + d*x) - 1))*sec(c + d*x), x)

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Giac [B]  time = 1.5077, size = 77, normalized size = 2.85 \begin{align*} -\frac{2 \, \sqrt{2} a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right ) \mathrm{sgn}\left (\cos \left (d x + c\right )\right )}{\sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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