3.35 \(\int \frac{1}{\sqrt{a \text{sech}^2(x)}} \, dx\)

Optimal. Leaf size=13 \[ \frac{\tanh (x)}{\sqrt{a \text{sech}^2(x)}} \]

[Out]

Tanh[x]/Sqrt[a*Sech[x]^2]

________________________________________________________________________________________

Rubi [A]  time = 0.0285025, 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} \[ \frac{\tanh (x)}{\sqrt{a \text{sech}^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Tanh[x]/Sqrt[a*Sech[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 \text{sech}^2(x)}} \, dx &=a \operatorname{Subst}\left (\int \frac{1}{\left (a-a x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh (x)}{\sqrt{a \text{sech}^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0064179, size = 13, normalized size = 1. \[ \frac{\tanh (x)}{\sqrt{a \text{sech}^2(x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Tanh[x]/Sqrt[a*Sech[x]^2]

________________________________________________________________________________________

Maple [B]  time = 0.072, size = 58, normalized size = 4.5 \begin{align*}{\frac{{{\rm e}^{2\,x}}}{2\,{{\rm e}^{2\,x}}+2}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}}-{\frac{1}{2\,{{\rm e}^{2\,x}}+2}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2/(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)/(exp(2*x)+1)*exp(2*x)-1/2/(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)/(exp(2*x)+1)

________________________________________________________________________________________

Maxima [A]  time = 1.87257, size = 23, normalized size = 1.77 \begin{align*} -\frac{e^{\left (-x\right )}}{2 \, \sqrt{a}} + \frac{e^{x}}{2 \, \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*e^(-x)/sqrt(a) + 1/2*e^x/sqrt(a)

________________________________________________________________________________________

Fricas [B]  time = 2.0295, size = 247, normalized size = 19. \begin{align*} \frac{{\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )} + 2 \,{\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )} \sqrt{\frac{a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \,{\left (a \cosh \left (x\right ) e^{x} + a e^{x} \sinh \left (x\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((e^(2*x) + 1)*sinh(x)^2 + cosh(x)^2 + (cosh(x)^2 - 1)*e^(2*x) + 2*(cosh(x)*e^(2*x) + cosh(x))*sinh(x) - 1
)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x/(a*cosh(x)*e^x + a*e^x*sinh(x))

________________________________________________________________________________________

Sympy [A]  time = 0.695431, size = 15, normalized size = 1.15 \begin{align*} \frac{\tanh{\left (x \right )}}{\sqrt{a} \sqrt{\operatorname{sech}^{2}{\left (x \right )}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

tanh(x)/(sqrt(a)*sqrt(sech(x)**2))

________________________________________________________________________________________

Giac [A]  time = 1.14099, size = 19, normalized size = 1.46 \begin{align*} -\frac{e^{\left (-x\right )} - e^{x}}{2 \, \sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(e^(-x) - e^x)/sqrt(a)