3.42 \(\int \sqrt {\sec ^2(x)} \, dx\)

Optimal. Leaf size=3 \[ \sinh ^{-1}(\tan (x)) \]

[Out]

arcsinh(tan(x))

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Rubi [A]  time = 0.01, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4122, 215} \[ \sinh ^{-1}(\tan (x)) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Sec[x]^2],x]

[Out]

ArcSinh[Tan[x]]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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

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Mathematica [B]  time = 0.01, size = 44, normalized size = 14.67 \[ \cos (x) \sqrt {\sec ^2(x)} \left (\log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )-\log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Sec[x]^2],x]

[Out]

Cos[x]*(-Log[Cos[x/2] - Sin[x/2]] + Log[Cos[x/2] + Sin[x/2]])*Sqrt[Sec[x]^2]

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fricas [B]  time = 0.49, size = 17, normalized size = 5.67 \[ -\frac {1}{2} \, \log \left (\sin \relax (x) + 1\right ) + \frac {1}{2} \, \log \left (-\sin \relax (x) + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(sin(x) + 1) + 1/2*log(-sin(x) + 1)

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giac [B]  time = 0.73, size = 35, normalized size = 11.67 \[ \frac {\log \left ({\left | \frac {1}{\sin \relax (x)} + \sin \relax (x) + 2 \right |}\right )}{4 \, \mathrm {sgn}\left (\cos \relax (x)\right )} - \frac {\log \left ({\left | \frac {1}{\sin \relax (x)} + \sin \relax (x) - 2 \right |}\right )}{4 \, \mathrm {sgn}\left (\cos \relax (x)\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*log(abs(1/sin(x) + sin(x) + 2))/sgn(cos(x)) - 1/4*log(abs(1/sin(x) + sin(x) - 2))/sgn(cos(x))

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maple [B]  time = 0.37, size = 21, normalized size = 7.00 \[ -2 \cos \relax (x ) \sqrt {2}\, \sqrt {\frac {1}{\cos \left (2 x \right )+1}}\, \arctanh \left (\frac {-1+\cos \relax (x )}{\sin \relax (x )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*cos(x)*(1/cos(x)^2)^(1/2)*arctanh((-1+cos(x))/sin(x))

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maxima [A]  time = 0.46, size = 3, normalized size = 1.00 \[ \operatorname {arsinh}\left (\tan \relax (x)\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arcsinh(tan(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.33 \[ \int \sqrt {\frac {1}{{\cos \relax (x)}^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((1/cos(x)^2)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\sec ^{2}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(sec(x)**2), x)

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