\(\int \frac {1}{\sqrt {a \text {sech}^4(x)}} \, dx\) [49]

Optimal result

Integrand size = 10, antiderivative size = 36 \[ \int \frac {1}{\sqrt {a \text {sech}^4(x)}} \, dx=\frac {x \text {sech}^2(x)}{2 \sqrt {a \text {sech}^4(x)}}+\frac {\tanh (x)}{2 \sqrt {a \text {sech}^4(x)}} \]

[Out]

1/2*x*sech(x)^2/(a*sech(x)^4)^(1/2)+1/2*tanh(x)/(a*sech(x)^4)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4208, 2715, 8} \[ \int \frac {1}{\sqrt {a \text {sech}^4(x)}} \, dx=\frac {x \text {sech}^2(x)}{2 \sqrt {a \text {sech}^4(x)}}+\frac {\tanh (x)}{2 \sqrt {a \text {sech}^4(x)}} \]

[In]

Int[1/Sqrt[a*Sech[x]^4],x]

[Out]

(x*Sech[x]^2)/(2*Sqrt[a*Sech[x]^4]) + Tanh[x]/(2*Sqrt[a*Sech[x]^4])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 4208

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[b^IntPart[p]*((b*(c*Sec[e + f*x])^n)^
FracPart[p]/(c*Sec[e + f*x])^(n*FracPart[p])), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {sech}^2(x) \int \cosh ^2(x) \, dx}{\sqrt {a \text {sech}^4(x)}} \\ & = \frac {\tanh (x)}{2 \sqrt {a \text {sech}^4(x)}}+\frac {\text {sech}^2(x) \int 1 \, dx}{2 \sqrt {a \text {sech}^4(x)}} \\ & = \frac {x \text {sech}^2(x)}{2 \sqrt {a \text {sech}^4(x)}}+\frac {\tanh (x)}{2 \sqrt {a \text {sech}^4(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {a \text {sech}^4(x)}} \, dx=\frac {x \text {sech}^2(x)+\tanh (x)}{2 \sqrt {a \text {sech}^4(x)}} \]

[In]

Integrate[1/Sqrt[a*Sech[x]^4],x]

[Out]

(x*Sech[x]^2 + Tanh[x])/(2*Sqrt[a*Sech[x]^4])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(88\) vs. \(2(28)=56\).

Time = 0.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.47

method	result	size
risch	\(\frac {{\mathrm e}^{2 x} x}{2 \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}+\frac {{\mathrm e}^{4 x}}{8 \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1+{\mathrm e}^{2 x}\right )^{2}}-\frac {1}{8 \left (1+{\mathrm e}^{2 x}\right )^{2} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}}\)	\(89\)

[In]

int(1/(sech(x)^4*a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2/(exp(4*x)*a/(1+exp(2*x))^4)^(1/2)/(1+exp(2*x))^2*exp(2*x)*x+1/8/(exp(4*x)*a/(1+exp(2*x))^4)^(1/2)/(1+exp(2
*x))^2*exp(4*x)-1/8/(1+exp(2*x))^2/(exp(4*x)*a/(1+exp(2*x))^4)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (28) = 56\).

Time = 0.26 (sec) , antiderivative size = 253, normalized size of antiderivative = 7.03 \[ \int \frac {1}{\sqrt {a \text {sech}^4(x)}} \, dx=\frac {{\left ({\left (e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{4} + \cosh \left (x\right )^{4} + 4 \, {\left (\cosh \left (x\right ) e^{\left (4 \, x\right )} + 2 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, x \cosh \left (x\right )^{2} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + {\left (3 \, \cosh \left (x\right )^{2} + 2 \, x\right )} e^{\left (4 \, x\right )} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 2 \, x\right )} e^{\left (2 \, x\right )} + 2 \, x\right )} \sinh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{4} + 4 \, x \cosh \left (x\right )^{2} - 1\right )} e^{\left (4 \, x\right )} + 2 \, {\left (\cosh \left (x\right )^{4} + 4 \, x \cosh \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )} + 4 \, {\left (\cosh \left (x\right )^{3} + 2 \, x \cosh \left (x\right ) + {\left (\cosh \left (x\right )^{3} + 2 \, x \cosh \left (x\right )\right )} e^{\left (4 \, x\right )} + 2 \, {\left (\cosh \left (x\right )^{3} + 2 \, x \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right ) - 1\right )} \sqrt {\frac {a}{e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x\right )} + 6 \, e^{\left (4 \, x\right )} + 4 \, e^{\left (2 \, x\right )} + 1}} e^{\left (2 \, x\right )}}{8 \, {\left (a \cosh \left (x\right )^{2} e^{\left (2 \, x\right )} + 2 \, a \cosh \left (x\right ) e^{\left (2 \, x\right )} \sinh \left (x\right ) + a e^{\left (2 \, x\right )} \sinh \left (x\right )^{2}\right )}} \]

[In]

integrate(1/(a*sech(x)^4)^(1/2),x, algorithm="fricas")

[Out]

1/8*((e^(4*x) + 2*e^(2*x) + 1)*sinh(x)^4 + cosh(x)^4 + 4*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2*x) + cosh(x))*sinh(
x)^3 + 4*x*cosh(x)^2 + 2*(3*cosh(x)^2 + (3*cosh(x)^2 + 2*x)*e^(4*x) + 2*(3*cosh(x)^2 + 2*x)*e^(2*x) + 2*x)*sin
h(x)^2 + (cosh(x)^4 + 4*x*cosh(x)^2 - 1)*e^(4*x) + 2*(cosh(x)^4 + 4*x*cosh(x)^2 - 1)*e^(2*x) + 4*(cosh(x)^3 +
2*x*cosh(x) + (cosh(x)^3 + 2*x*cosh(x))*e^(4*x) + 2*(cosh(x)^3 + 2*x*cosh(x))*e^(2*x))*sinh(x) - 1)*sqrt(a/(e^
(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x)/(a*cosh(x)^2*e^(2*x) + 2*a*cosh(x)*e^(2*x)*sinh(x) + a
*e^(2*x)*sinh(x)^2)

Sympy [F]

\[ \int \frac {1}{\sqrt {a \text {sech}^4(x)}} \, dx=\int \frac {1}{\sqrt {a \operatorname {sech}^{4}{\left (x \right )}}}\, dx \]

[In]

integrate(1/(a*sech(x)**4)**(1/2),x)

[Out]

Integral(1/sqrt(a*sech(x)**4), x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\sqrt {a \text {sech}^4(x)}} \, dx=-\frac {{\left (\sqrt {a} e^{\left (-4 \, x\right )} - \sqrt {a}\right )} e^{\left (2 \, x\right )}}{8 \, a} + \frac {x}{2 \, \sqrt {a}} \]

[In]

integrate(1/(a*sech(x)^4)^(1/2),x, algorithm="maxima")

[Out]

-1/8*(sqrt(a)*e^(-4*x) - sqrt(a))*e^(2*x)/a + 1/2*x/sqrt(a)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78 \[ \int \frac {1}{\sqrt {a \text {sech}^4(x)}} \, dx=-\frac {{\left (2 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-2 \, x\right )} - 4 \, x - e^{\left (2 \, x\right )}}{8 \, \sqrt {a}} \]

[In]

integrate(1/(a*sech(x)^4)^(1/2),x, algorithm="giac")

[Out]

-1/8*((2*e^(2*x) + 1)*e^(-2*x) - 4*x - e^(2*x))/sqrt(a)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {a \text {sech}^4(x)}} \, dx=\int \frac {1}{\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^4}}} \,d x \]

[In]

int(1/(a/cosh(x)^4)^(1/2),x)

[Out]

int(1/(a/cosh(x)^4)^(1/2), x)