3.51 \(\int \frac{1}{\sqrt{a \sec ^2(x)}} \, dx\)

Optimal. Leaf size=13 \[ \frac{\tan (x)}{\sqrt{a \sec ^2(x)}} \]

[Out]

Tan[x]/Sqrt[a*Sec[x]^2]

________________________________________________________________________________________

Rubi [A]  time = 0.0286669, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4122, 191} \[ \frac{\tan (x)}{\sqrt{a \sec ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Sec[x]^2],x]

[Out]

Tan[x]/Sqrt[a*Sec[x]^2]

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 191

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^(p + 1))/a, x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a \sec ^2(x)}} \, dx &=a \operatorname{Subst}\left (\int \frac{1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\tan (x)\right )\\ &=\frac{\tan (x)}{\sqrt{a \sec ^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0053793, size = 13, normalized size = 1. \[ \frac{\tan (x)}{\sqrt{a \sec ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Sec[x]^2],x]

[Out]

Tan[x]/Sqrt[a*Sec[x]^2]

________________________________________________________________________________________

Maple [A]  time = 0.069, size = 16, normalized size = 1.2 \begin{align*}{\frac{\sin \left ( x \right ) }{\cos \left ( x \right ) }{\frac{1}{\sqrt{{\frac{a}{ \left ( \cos \left ( x \right ) \right ) ^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*sec(x)^2)^(1/2),x)

[Out]

sin(x)/(a/cos(x)^2)^(1/2)/cos(x)

________________________________________________________________________________________

Maxima [A]  time = 1.88085, size = 8, normalized size = 0.62 \begin{align*} \frac{\sin \left (x\right )}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)/sqrt(a)

________________________________________________________________________________________

Fricas [A]  time = 1.44531, size = 46, normalized size = 3.54 \begin{align*} \frac{\sqrt{\frac{a}{\cos \left (x\right )^{2}}} \cos \left (x\right ) \sin \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a/cos(x)^2)*cos(x)*sin(x)/a

________________________________________________________________________________________

Sympy [A]  time = 0.53778, size = 15, normalized size = 1.15 \begin{align*} \frac{\tan{\left (x \right )}}{\sqrt{a} \sqrt{\sec ^{2}{\left (x \right )}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sec(x)**2)**(1/2),x)

[Out]

tan(x)/(sqrt(a)*sqrt(sec(x)**2))

________________________________________________________________________________________

Giac [A]  time = 1.22898, size = 15, normalized size = 1.15 \begin{align*} \frac{\sin \left (x\right )}{\sqrt{a} \mathrm{sgn}\left (\cos \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sin(x)/(sqrt(a)*sgn(cos(x)))