3.2.67 \(\int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^4} \, dx\) [167]

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

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5670, 5668, 267} \begin {gather*} -\frac {1}{2} x \left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*((c^4 + x^(-4))*x*Sqrt[Sech[2*Log[c*x]]])

Rule 267

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]

Rule 5668

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*(1/(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 5670

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])

Rubi steps

\begin {align*} \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^4} \, dx &=c^3 \text {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 ) \text {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 = 0.03, size = 33, normalized size = 1.43 \begin {gather*} -\frac {c^2}{2 x \sqrt {\frac {c^2 x^2}{2+2 c^4 x^4}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*c^2/(x*Sqrt[(c^2*x^2)/(2 + 2*c^4*x^4)])

________________________________________________________________________________________

Maple [A]
time = 1.20, size = 38, normalized size = 1.65

method	result	size
risch	\(-\frac {\sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}\, \left (c^{4} x^{4}+1\right )}{2 x^{3}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (19) = 38\).
time = 0.47, size = 42, normalized size = 1.83 \begin {gather*} -\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 {gather*}

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)))

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 37, normalized size = 1.61 \begin {gather*} -\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 {gather*}

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

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\operatorname {sech}{\left (2 \log {\left (c x \right )} \right )}}}{x^{4}}\, dx \end {gather*}

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)

________________________________________________________________________________________

Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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]

Timed out

________________________________________________________________________________________

Mupad [B]
time = 1.35, size = 58, normalized size = 2.52 \begin {gather*} -\frac {\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4+1}}}{2\,x^3}-\frac {c^4\,x\,\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4+1}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/cosh(2*log(c*x)))^(1/2)/x^4,x)

[Out]

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

________________________________________________________________________________________