3.223 \(\int \sqrt{a-a \sec ^2(c+d x)} \, dx\)

Optimal. Leaf size=33 \[ -\frac{\cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]

[Out]

-((Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[-(a*Tan[c + d*x]^2)])/d)

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Rubi [A]  time = 0.0300599, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {4121, 3658, 3475} \[ -\frac{\cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a - a*Sec[c + d*x]^2],x]

[Out]

-((Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[-(a*Tan[c + d*x]^2)])/d)

Rule 4121

Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(b*tan[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3658

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Tan[e + f*x]^n)^FracPart[p])/(Tan[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0368466, size = 33, normalized size = 1. \[ -\frac{\cot (c+d x) \sqrt{-a \tan ^2(c+d x)} \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a - a*Sec[c + d*x]^2],x]

[Out]

-((Cot[c + d*x]*Log[Cos[c + d*x]]*Sqrt[-(a*Tan[c + d*x]^2)])/d)

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Maple [B]  time = 0.347, size = 108, normalized size = 3.3 \begin{align*} -{\frac{\cos \left ( dx+c \right ) }{d\sin \left ( dx+c \right ) } \left ( -\ln \left ( 2\, \left ( \cos \left ( dx+c \right ) +1 \right ) ^{-1} \right ) +\ln \left ({\frac{1-\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) +\ln \left ( -{\frac{-1+\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }} \right ) \right ) \sqrt{-{\frac{a \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{ \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.46501, size = 28, normalized size = 0.85 \begin{align*} \frac{\sqrt{-a} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.50536, size = 128, normalized size = 3.88 \begin{align*} -\frac{\sqrt{\frac{a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}} \cos \left (d x + c\right ) \log \left (-\cos \left (d x + c\right )\right )}{d \sin \left (d x + c\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.49128, size = 190, normalized size = 5.76 \begin{align*} -\frac{{\left (\log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right ) \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 ) - \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) \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 ) - \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) \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 )\right )} \sqrt{-a} \mathrm{sgn}\left (\cos \left (d x + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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