3.2.16 \(\int \sec (c+d x) \sqrt {a-a \sec (c+d x)} \, dx\) [116]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 = {3877} \begin {gather*} -\frac {2 a \tan (c+d x)}{d \sqrt {a-a \sec (c+d x)}} \end {gather*}

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 3877

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*}

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Mathematica [A]
time = 0.12, size = 30, normalized size = 1.11 \begin {gather*} \frac {2 \cot \left (\frac {1}{2} (c+d x)\right ) \sqrt {a-a \sec (c+d x)}}{d} \end {gather*}

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 = 0.18, size = 42, normalized size = 1.56

method	result	size
default	\(-\frac {2 \sqrt {\frac {a \left (-1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )}{d \left (-1+\cos \left (d x +c \right )\right )}\)	\(42\)

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,method=_RETURNVERBOSE)

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 2.81, size = 44, normalized size = 1.63 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {- a \left (\sec {\left (c + d x \right )} - 1\right )} \sec {\left (c + d x \right )}\, dx \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (25) = 50\).
time = 0.68, size = 57, normalized size = 2.11 \begin {gather*} -\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 {gather*}

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)

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Mupad [B]
time = 0.79, size = 36, normalized size = 1.33 \begin {gather*} \frac {\sin \left (c+d\,x\right )\,\sqrt {a-\frac {a}{\cos \left (c+d\,x\right )}}}{d\,{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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