3.167 \(\int \frac{\sqrt{\text{sech}(2 \log (c x))}}{x^4} \, dx\)

Optimal. Leaf size=23 \[ -\frac{1}{2} x \left (c^4+\frac{1}{x^4}\right ) \sqrt{\text{sech}(2 \log (c x))} \]

[Out]

-((c^4 + x^(-4))*x*Sqrt[Sech[2*Log[c*x]]])/2

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Rubi [A]  time = 0.0403995, antiderivative size = 23, 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 = {5551, 5549, 261} \[ -\frac{1}{2} x \left (c^4+\frac{1}{x^4}\right ) \sqrt{\text{sech}(2 \log (c x))} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Sech[2*Log[c*x]]]/x^4,x]

[Out]

-((c^4 + x^(-4))*x*Sqrt[Sech[2*Log[c*x]]])/2

Rule 5551

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

Rule 5549

Int[((e_.)*(x_))^(m_.)*Sech[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[(Sech[d*(a + b*Log[x])]^p*
(1 + 1/(E^(2*a*d)*x^(2*b*d)))^p)/x^(-(b*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, p}, 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 \frac{\sqrt{\text{sech}(2 \log (c x))}}{x^4} \, dx &=c^3 \operatorname{Subst}\left (\int \frac{\sqrt{\text{sech}(2 \log (x))}}{x^4} \, dx,x,c x\right )\\ &=\left (c^4 \sqrt{1+\frac{1}{c^4 x^4}} x \sqrt{\text{sech}(2 \log (c x))}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{1}{x^4}} x^5} \, dx,x,c x\right )\\ &=-\frac{1}{2} \left (c^4+\frac{1}{x^4}\right ) x \sqrt{\text{sech}(2 \log (c x))}\\ \end{align*}

Mathematica [A]  time = 0.0380192, size = 33, normalized size = 1.43 \[ -\frac{c^2}{2 x \sqrt{\frac{c^2 x^2}{2 c^4 x^4+2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Sech[2*Log[c*x]]]/x^4,x]

[Out]

-c^2/(2*x*Sqrt[(c^2*x^2)/(2 + 2*c^4*x^4)])

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Maple [A]  time = 0.037, size = 38, normalized size = 1.7 \begin{align*} -{\frac{\sqrt{2} \left ({c}^{4}{x}^{4}+1 \right ) }{2\,{x}^{3}}\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}+1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(2*ln(c*x))^(1/2)/x^4,x)

[Out]

-1/2*2^(1/2)*(c^2*x^2/(c^4*x^4+1))^(1/2)/x^3*(c^4*x^4+1)

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Maxima [B]  time = 1.60203, size = 57, normalized size = 2.48 \begin{align*} -\frac{1}{2} \, c^{3}{\left (\frac{\sqrt{2}}{\sqrt{\frac{1}{c^{4} x^{4}} + 1}} + \frac{\sqrt{2}}{c^{4} x^{4} \sqrt{\frac{1}{c^{4} x^{4}} + 1}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c^3*(sqrt(2)/sqrt(1/(c^4*x^4) + 1) + sqrt(2)/(c^4*x^4*sqrt(1/(c^4*x^4) + 1)))

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Fricas [A]  time = 2.96646, size = 81, normalized size = 3.52 \begin{align*} -\frac{\sqrt{2}{\left (c^{4} x^{4} + 1\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} + 1}}}{2 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*sqrt(2)*(c^4*x^4 + 1)*sqrt(c^2*x^2/(c^4*x^4 + 1))/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{sech}{\left (2 \log{\left (c x \right )} \right )}}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(2*ln(c*x))**(1/2)/x**4,x)

[Out]

Integral(sqrt(sech(2*log(c*x)))/x**4, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{sech}\left (2 \, \log \left (c x\right )\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sech(2*log(c*x)))/x^4, x)