3.728 \(\int \frac{\sec (x) \tan (x)}{\sqrt{4+\sec ^2(x)}} \, dx\)

Optimal. Leaf size=5 \[ \text{csch}^{-1}(2 \cos (x)) \]

[Out]

ArcCsch[2*Cos[x]]

________________________________________________________________________________________

Rubi [A]  time = 0.0448581, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4339, 335, 215} \[ \text{csch}^{-1}(2 \cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[(Sec[x]*Tan[x])/Sqrt[4 + Sec[x]^2],x]

[Out]

ArcCsch[2*Cos[x]]

Rule 4339

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))], x_Symbol] :> With[{d = FreeFactors[Cos[c*(a + b*x)], x]}, -Dist[(b*
c)^(-1), Subst[Int[SubstFor[1/x, Cos[c*(a + b*x)]/d, u, x], x], x, Cos[c*(a + b*x)]/d], x] /; FunctionOfQ[Cos[
c*(a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && (EqQ[F, Tan] || EqQ[F, tan])

Rule 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

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]

Rubi steps

\begin{align*} \int \frac{\sec (x) \tan (x)}{\sqrt{4+\sec ^2(x)}} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{\sqrt{4+\frac{1}{x^2}} x^2} \, dx,x,\cos (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{4+x^2}} \, dx,x,\sec (x)\right )\\ &=\sinh ^{-1}\left (\frac{\sec (x)}{2}\right )\\ \end{align*}

Mathematica [B]  time = 0.0303004, size = 38, normalized size = 7.6 \[ \frac{\sqrt{2 \cos (2 x)+3} \sec (x) \tanh ^{-1}\left (\sqrt{4 \cos ^2(x)+1}\right )}{\sqrt{\sec ^2(x)+4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[x]*Tan[x])/Sqrt[4 + Sec[x]^2],x]

[Out]

(ArcTanh[Sqrt[1 + 4*Cos[x]^2]]*Sqrt[3 + 2*Cos[2*x]]*Sec[x])/Sqrt[4 + Sec[x]^2]

________________________________________________________________________________________

Maple [A]  time = 0.032, size = 6, normalized size = 1.2 \begin{align*}{\it Arcsinh} \left ({\frac{\sec \left ( x \right ) }{2}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)*tan(x)/(4+sec(x)^2)^(1/2),x)

[Out]

arcsinh(1/2*sec(x))

________________________________________________________________________________________

Maxima [B]  time = 0.953622, size = 45, normalized size = 9. \begin{align*} \frac{1}{2} \, \log \left (\sqrt{\frac{1}{\cos \left (x\right )^{2}} + 4} \cos \left (x\right ) + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{\frac{1}{\cos \left (x\right )^{2}} + 4} \cos \left (x\right ) - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)*tan(x)/(4+sec(x)^2)^(1/2),x, algorithm="maxima")

[Out]

1/2*log(sqrt(1/cos(x)^2 + 4)*cos(x) + 1) - 1/2*log(sqrt(1/cos(x)^2 + 4)*cos(x) - 1)

________________________________________________________________________________________

Fricas [B]  time = 2.71256, size = 80, normalized size = 16. \begin{align*} \log \left (-\frac{\sqrt{\frac{4 \, \cos \left (x\right )^{2} + 1}{\cos \left (x\right )^{2}}} \cos \left (x\right ) + 1}{\cos \left (x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)*tan(x)/(4+sec(x)^2)^(1/2),x, algorithm="fricas")

[Out]

log(-(sqrt((4*cos(x)^2 + 1)/cos(x)^2)*cos(x) + 1)/cos(x))

________________________________________________________________________________________

Sympy [A]  time = 0.950007, size = 5, normalized size = 1. \begin{align*} \operatorname{asinh}{\left (\frac{\sec{\left (x \right )}}{2} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)*tan(x)/(4+sec(x)**2)**(1/2),x)

[Out]

asinh(sec(x)/2)

________________________________________________________________________________________

Giac [B]  time = 1.12927, size = 55, normalized size = 11. \begin{align*} \frac{\log \left (\sqrt{4 \, \cos \left (x\right )^{2} + 1} + 1\right )}{2 \, \mathrm{sgn}\left (\cos \left (x\right )\right )} - \frac{\log \left (\sqrt{4 \, \cos \left (x\right )^{2} + 1} - 1\right )}{2 \, \mathrm{sgn}\left (\cos \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)*tan(x)/(4+sec(x)^2)^(1/2),x, algorithm="giac")

[Out]

1/2*log(sqrt(4*cos(x)^2 + 1) + 1)/sgn(cos(x)) - 1/2*log(sqrt(4*cos(x)^2 + 1) - 1)/sgn(cos(x))