\(\int \frac {1}{\sqrt {a \text {sech}^2(x)}} \, dx\) [35]

Optimal result

Integrand size = 10, antiderivative size = 13 \[ \int \frac {1}{\sqrt {a \text {sech}^2(x)}} \, dx=\frac {\tanh (x)}{\sqrt {a \text {sech}^2(x)}} \]

[Out]

tanh(x)/(a*sech(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4207, 197} \[ \int \frac {1}{\sqrt {a \text {sech}^2(x)}} \, dx=\frac {\tanh (x)}{\sqrt {a \text {sech}^2(x)}} \]

[In]

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

[Out]

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

Rule 197

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 4207

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*} \text {integral}& = a \text {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] (verified)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {a \text {sech}^2(x)}} \, dx=\frac {\tanh (x)}{\sqrt {a \text {sech}^2(x)}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(57\) vs. \(2(11)=22\).

Time = 0.16 (sec) , antiderivative size = 58, normalized size of antiderivative = 4.46

method	result	size
risch	\(\frac {{\mathrm e}^{2 x}}{2 \sqrt {\frac {{\mathrm e}^{2 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {1}{2 \left (1+{\mathrm e}^{2 x}\right ) \sqrt {\frac {{\mathrm e}^{2 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}}\)	\(58\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 79 vs. \(2 (11) = 22\).

Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 6.08 \[ \int \frac {1}{\sqrt {a \text {sech}^2(x)}} \, dx=\frac {{\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )} + 2 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \, {\left (a \cosh \left (x\right ) e^{x} + a e^{x} \sinh \left (x\right )\right )}} \]

[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] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {a \text {sech}^2(x)}} \, dx=\frac {\tanh {\left (x \right )}}{\sqrt {a \operatorname {sech}^{2}{\left (x \right )}}} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.31 \[ \int \frac {1}{\sqrt {a \text {sech}^2(x)}} \, dx=-\frac {e^{\left (-x\right )}}{2 \, \sqrt {a}} + \frac {e^{x}}{2 \, \sqrt {a}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt {a \text {sech}^2(x)}} \, dx=-\frac {e^{\left (-x\right )} - e^{x}}{2 \, \sqrt {a}} \]

[In]

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

[Out]

-1/2*(e^(-x) - e^x)/sqrt(a)

Mupad [B] (verification not implemented)

Time = 1.96 (sec) , antiderivative size = 33, normalized size of antiderivative = 2.54 \[ \int \frac {1}{\sqrt {a \text {sech}^2(x)}} \, dx=-\frac {\left (\frac {{\mathrm {e}}^{-2\,x}}{2}-\frac {{\mathrm {e}}^{2\,x}}{2}\right )\,\sqrt {\frac {1}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^2}}}{2\,\sqrt {a}} \]

[In]

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

[Out]

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