3.36 \(\int \frac{1}{(a \text{sech}^2(x))^{3/2}} \, dx\)

Optimal. Leaf size=36 \[ \frac{2 \tanh (x)}{3 a \sqrt{a \text{sech}^2(x)}}+\frac{\tanh (x)}{3 \left (a \text{sech}^2(x)\right )^{3/2}} \]

[Out]

Tanh[x]/(3*(a*Sech[x]^2)^(3/2)) + (2*Tanh[x])/(3*a*Sqrt[a*Sech[x]^2])

________________________________________________________________________________________

Rubi [A]  time = 0.0201417, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {4122, 192, 191} \[ \frac{2 \tanh (x)}{3 a \sqrt{a \text{sech}^2(x)}}+\frac{\tanh (x)}{3 \left (a \text{sech}^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sech[x]^2)^(-3/2),x]

[Out]

Tanh[x]/(3*(a*Sech[x]^2)^(3/2)) + (2*Tanh[x])/(3*a*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 192

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1
], 0] && NeQ[p, -1]

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}{\left (a \text{sech}^2(x)\right )^{3/2}} \, dx &=a \operatorname{Subst}\left (\int \frac{1}{\left (a-a x^2\right )^{5/2}} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh (x)}{3 \left (a \text{sech}^2(x)\right )^{3/2}}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{\left (a-a x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )\\ &=\frac{\tanh (x)}{3 \left (a \text{sech}^2(x)\right )^{3/2}}+\frac{2 \tanh (x)}{3 a \sqrt{a \text{sech}^2(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0211334, size = 27, normalized size = 0.75 \[ \frac{(9 \sinh (x)+\sinh (3 x)) \text{sech}^3(x)}{12 \left (a \text{sech}^2(x)\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sech[x]^2)^(-3/2),x]

[Out]

(Sech[x]^3*(9*Sinh[x] + Sinh[3*x]))/(12*(a*Sech[x]^2)^(3/2))

________________________________________________________________________________________

Maple [B]  time = 0.059, size = 130, normalized size = 3.6 \begin{align*}{\frac{{{\rm e}^{4\,x}}}{24\,a \left ({{\rm e}^{2\,x}}+1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}}+{\frac{3\,{{\rm e}^{2\,x}}}{8\,a \left ({{\rm e}^{2\,x}}+1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}}-{\frac{3}{8\,a \left ({{\rm e}^{2\,x}}+1 \right ) }{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}}-{\frac{{{\rm e}^{-2\,x}}}{24\,a \left ({{\rm e}^{2\,x}}+1 \right ) }{\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)^(3/2),x)

[Out]

1/24/a*exp(4*x)/(exp(2*x)+1)/(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)+3/8/a*exp(2*x)/(exp(2*x)+1)/(a*exp(2*x)/(exp(2*
x)+1)^2)^(1/2)-3/8/a/(exp(2*x)+1)/(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)-1/24/a*exp(-2*x)/(exp(2*x)+1)/(a*exp(2*x)/
(exp(2*x)+1)^2)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.70525, size = 47, normalized size = 1.31 \begin{align*} \frac{e^{\left (3 \, x\right )}}{24 \, a^{\frac{3}{2}}} - \frac{3 \, e^{\left (-x\right )}}{8 \, a^{\frac{3}{2}}} - \frac{e^{\left (-3 \, x\right )}}{24 \, a^{\frac{3}{2}}} + \frac{3 \, e^{x}}{8 \, a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)^2)^(3/2),x, algorithm="maxima")

[Out]

1/24*e^(3*x)/a^(3/2) - 3/8*e^(-x)/a^(3/2) - 1/24*e^(-3*x)/a^(3/2) + 3/8*e^x/a^(3/2)

________________________________________________________________________________________

Fricas [B]  time = 2.19873, size = 856, normalized size = 23.78 \begin{align*} \frac{{\left ({\left (e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{6} + \cosh \left (x\right )^{6} + 6 \,{\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 3 \,{\left (5 \, \cosh \left (x\right )^{2} +{\left (5 \, \cosh \left (x\right )^{2} + 3\right )} e^{\left (2 \, x\right )} + 3\right )} \sinh \left (x\right )^{4} + 9 \, \cosh \left (x\right )^{4} + 4 \,{\left (5 \, \cosh \left (x\right )^{3} +{\left (5 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} + 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \,{\left (5 \, \cosh \left (x\right )^{4} + 18 \, \cosh \left (x\right )^{2} +{\left (5 \, \cosh \left (x\right )^{4} + 18 \, \cosh \left (x\right )^{2} - 3\right )} e^{\left (2 \, x\right )} - 3\right )} \sinh \left (x\right )^{2} - 9 \, \cosh \left (x\right )^{2} +{\left (\cosh \left (x\right )^{6} + 9 \, \cosh \left (x\right )^{4} - 9 \, \cosh \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )} + 6 \,{\left (\cosh \left (x\right )^{5} + 6 \, \cosh \left (x\right )^{3} +{\left (\cosh \left (x\right )^{5} + 6 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} - 3 \, \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}}{24 \,{\left (a^{2} \cosh \left (x\right )^{3} e^{x} + 3 \, a^{2} \cosh \left (x\right )^{2} e^{x} \sinh \left (x\right ) + 3 \, a^{2} \cosh \left (x\right ) e^{x} \sinh \left (x\right )^{2} + a^{2} e^{x} \sinh \left (x\right )^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)^2)^(3/2),x, algorithm="fricas")

[Out]

1/24*((e^(2*x) + 1)*sinh(x)^6 + cosh(x)^6 + 6*(cosh(x)*e^(2*x) + cosh(x))*sinh(x)^5 + 3*(5*cosh(x)^2 + (5*cosh
(x)^2 + 3)*e^(2*x) + 3)*sinh(x)^4 + 9*cosh(x)^4 + 4*(5*cosh(x)^3 + (5*cosh(x)^3 + 9*cosh(x))*e^(2*x) + 9*cosh(
x))*sinh(x)^3 + 3*(5*cosh(x)^4 + 18*cosh(x)^2 + (5*cosh(x)^4 + 18*cosh(x)^2 - 3)*e^(2*x) - 3)*sinh(x)^2 - 9*co
sh(x)^2 + (cosh(x)^6 + 9*cosh(x)^4 - 9*cosh(x)^2 - 1)*e^(2*x) + 6*(cosh(x)^5 + 6*cosh(x)^3 + (cosh(x)^5 + 6*co
sh(x)^3 - 3*cosh(x))*e^(2*x) - 3*cosh(x))*sinh(x) - 1)*sqrt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x/(a^2*cosh(x)^3*e^
x + 3*a^2*cosh(x)^2*e^x*sinh(x) + 3*a^2*cosh(x)*e^x*sinh(x)^2 + a^2*e^x*sinh(x)^3)

________________________________________________________________________________________

Sympy [A]  time = 1.53155, size = 37, normalized size = 1.03 \begin{align*} - \frac{2 \tanh ^{3}{\left (x \right )}}{3 a^{\frac{3}{2}} \left (\operatorname{sech}^{2}{\left (x \right )}\right )^{\frac{3}{2}}} + \frac{\tanh{\left (x \right )}}{a^{\frac{3}{2}} \left (\operatorname{sech}^{2}{\left (x \right )}\right )^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)**2)**(3/2),x)

[Out]

-2*tanh(x)**3/(3*a**(3/2)*(sech(x)**2)**(3/2)) + tanh(x)/(a**(3/2)*(sech(x)**2)**(3/2))

________________________________________________________________________________________

Giac [A]  time = 1.13785, size = 39, normalized size = 1.08 \begin{align*} -\frac{{\left (9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} - e^{\left (3 \, x\right )} - 9 \, e^{x}}{24 \, a^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*sech(x)^2)^(3/2),x, algorithm="giac")

[Out]

-1/24*((9*e^(2*x) + 1)*e^(-3*x) - e^(3*x) - 9*e^x)/a^(3/2)