3.187 \(\int \text{sech}^3(a+2 \log (c \sqrt{x})) \, dx\)

Optimal. Leaf size=25 \[ \frac{2 e^{-a} c^6}{\left (\frac{e^{-2 a}}{x^2}+c^4\right )^2} \]

[Out]

(2*c^6)/(E^a*(c^4 + 1/(E^(2*a)*x^2))^2)

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Rubi [A]  time = 0.0389133, antiderivative size = 25, 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 = {5545, 5547, 261} \[ \frac{2 e^{-a} c^6}{\left (\frac{e^{-2 a}}{x^2}+c^4\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + 2*Log[c*Sqrt[x]]]^3,x]

[Out]

(2*c^6)/(E^a*(c^4 + 1/(E^(2*a)*x^2))^2)

Rule 5545

Int[Sech[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[
x^(1/n - 1)*Sech[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n
, 1])

Rule 5547

Int[((e_.)*(x_))^(m_.)*Sech[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[2^p/E^(a*d*p), Int[(e*x)^m
/(x^(b*d*p)*(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p), x], x] /; FreeQ[{a, b, d, e, m}, x] && IntegerQ[p]

Rule 261

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

Rubi steps

\begin{align*} \int \text{sech}^3\left (a+2 \log \left (c \sqrt{x}\right )\right ) \, dx &=\frac{2 \operatorname{Subst}\left (\int x \text{sech}^3(a+2 \log (x)) \, dx,x,c \sqrt{x}\right )}{c^2}\\ &=\frac{\left (16 e^{-3 a}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1+\frac{e^{-2 a}}{x^4}\right )^3 x^5} \, dx,x,c \sqrt{x}\right )}{c^2}\\ &=\frac{2 c^6 e^{-a}}{\left (c^4+\frac{e^{-2 a}}{x^2}\right )^2}\\ \end{align*}

Mathematica [B]  time = 0.115763, size = 62, normalized size = 2.48 \[ -\frac{2 (\cosh (a)-\sinh (a)) \left (\sinh ^2(a)+\cosh ^2(a)-2 \sinh (a) \cosh (a)+2 c^4 x^2\right )}{c^2 \left (\sinh (a) \left (c^4 x^2-1\right )+\cosh (a) \left (c^4 x^2+1\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + 2*Log[c*Sqrt[x]]]^3,x]

[Out]

(-2*(Cosh[a] - Sinh[a])*(2*c^4*x^2 + Cosh[a]^2 - 2*Cosh[a]*Sinh[a] + Sinh[a]^2))/(c^2*((1 + c^4*x^2)*Cosh[a] +
(-1 + c^4*x^2)*Sinh[a])^2)

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm sech} \left (a+2\,\ln \left ( c\sqrt{x} \right ) \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(a+2*ln(c*x^(1/2)))^3,x)

[Out]

int(sech(a+2*ln(c*x^(1/2)))^3,x)

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Maxima [B]  time = 1.10406, size = 100, normalized size = 4. \begin{align*} -\frac{2 \,{\left (\frac{2 \, c^{4} x^{2} e^{\left (2 \, a\right )}}{c^{8} x^{4} e^{\left (5 \, a\right )} + 2 \, c^{4} x^{2} e^{\left (3 \, a\right )} + e^{a}} + \frac{1}{c^{8} x^{4} e^{\left (5 \, a\right )} + 2 \, c^{4} x^{2} e^{\left (3 \, a\right )} + e^{a}}\right )}}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(2*c^4*x^2*e^(2*a)/(c^8*x^4*e^(5*a) + 2*c^4*x^2*e^(3*a) + e^a) + 1/(c^8*x^4*e^(5*a) + 2*c^4*x^2*e^(3*a) + e
^a))/c^2

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Fricas [B]  time = 2.87589, size = 104, normalized size = 4.16 \begin{align*} -\frac{2 \,{\left (2 \, c^{4} x^{2} e^{\left (2 \, a\right )} + 1\right )}}{c^{10} x^{4} e^{\left (5 \, a\right )} + 2 \, c^{6} x^{2} e^{\left (3 \, a\right )} + c^{2} e^{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*(2*c^4*x^2*e^(2*a) + 1)/(c^10*x^4*e^(5*a) + 2*c^6*x^2*e^(3*a) + c^2*e^a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}^{3}{\left (a + 2 \log{\left (c \sqrt{x} \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sech(a + 2*log(c*sqrt(x)))**3, x)

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Giac [A]  time = 1.13385, size = 51, normalized size = 2.04 \begin{align*} -\frac{2 \,{\left (2 \, c^{4} x^{2} e^{\left (2 \, a\right )} + 1\right )} e^{\left (-a\right )}}{{\left (c^{4} x^{2} e^{\left (2 \, a\right )} + 1\right )}^{2} c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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