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

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Rubi [A]  time = 0.03, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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
]

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*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 1.00 \[ \frac {\tan (x)}{\sqrt {a \sec ^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.62, size = 16, normalized size = 1.23 \[ \frac {\sqrt {\frac {a}{\cos \relax (x)^{2}}} \cos \relax (x) \sin \relax (x)}{a} \]

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

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giac [A]  time = 0.40, size = 11, normalized size = 0.85 \[ \frac {\sin \relax (x)}{\sqrt {a} \mathrm {sgn}\left (\cos \relax (x)\right )} \]

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)))

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maple [A]  time = 0.40, size = 16, normalized size = 1.23 \[ \frac {\sin \relax (x )}{\sqrt {\frac {a}{\cos \relax (x )^{2}}}\, \cos \relax (x )} \]

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)

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maxima [A]  time = 0.92, size = 6, normalized size = 0.46 \[ \frac {\sin \relax (x)}{\sqrt {a}} \]

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)

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mupad [B]  time = 0.21, size = 15, normalized size = 1.15 \[ \frac {\sqrt {2}\,\sin \left (2\,x\right )}{2\,\sqrt {a}\,\sqrt {2\,{\cos \relax (x)}^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a/cos(x)^2)^(1/2),x)

[Out]

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

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sympy [A]  time = 0.51, size = 15, normalized size = 1.15 \[ \frac {\tan {\relax (x )}}{\sqrt {a} \sqrt {\sec ^{2}{\relax (x )}}} \]

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))

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