Integrand size = 15, antiderivative size = 40 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^2} \, dx=-\frac {1}{2} c^2 \sqrt {1+\frac {1}{c^4 x^4}} x \text {csch}^{-1}\left (c^2 x^2\right ) \sqrt {\text {sech}(2 \log (c x))} \] Output:
-1/2*c^2*(1+1/c^4/x^4)^(1/2)*x*arccsch(c^2*x^2)*sech(2*ln(c*x))^(1/2)
Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^2} \, dx=-\frac {\sqrt {1+c^4 x^4} \sqrt {\frac {c^2 x^2}{2+2 c^4 x^4}} \text {arctanh}\left (\sqrt {1+c^4 x^4}\right )}{x} \] Input:
Integrate[Sqrt[Sech[2*Log[c*x]]]/x^2,x]
Output:
-((Sqrt[1 + c^4*x^4]*Sqrt[(c^2*x^2)/(2 + 2*c^4*x^4)]*ArcTanh[Sqrt[1 + c^4* x^4]])/x)
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6085, 6083, 858, 807, 222}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^2} \, dx\) |
\(\Big \downarrow \) 6085 |
\(\displaystyle c \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{c^2 x^2}d(c x)\) |
\(\Big \downarrow \) 6083 |
\(\displaystyle c^2 x \sqrt {\frac {1}{c^4 x^4}+1} \sqrt {\text {sech}(2 \log (c x))} \int \frac {1}{c^3 \sqrt {1+\frac {1}{c^4 x^4}} x^3}d(c x)\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -c^2 x \sqrt {\frac {1}{c^4 x^4}+1} \sqrt {\text {sech}(2 \log (c x))} \int \frac {1}{c x \sqrt {c^4 x^4+1}}d\frac {1}{c x}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle -\frac {1}{2} c^2 x \sqrt {\frac {1}{c^4 x^4}+1} \sqrt {\text {sech}(2 \log (c x))} \int \frac {1}{\sqrt {c^2 x^2+1}}d\left (c^2 x^2\right )\) |
\(\Big \downarrow \) 222 |
\(\displaystyle -\frac {1}{2} c^2 x \sqrt {\frac {1}{c^4 x^4}+1} \text {arcsinh}\left (c^2 x^2\right ) \sqrt {\text {sech}(2 \log (c x))}\) |
Input:
Int[Sqrt[Sech[2*Log[c*x]]]/x^2,x]
Output:
-1/2*(c^2*Sqrt[1 + 1/(c^4*x^4)]*x*ArcSinh[c^2*x^2]*Sqrt[Sech[2*Log[c*x]]])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
Int[((e_.)*(x_))^(m_.)*Sech[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Simp[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]
Int[((e_.)*(x_))^(m_.)*Sech[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p _.), x_Symbol] :> Simp[(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])
\[\int \frac {\sqrt {\operatorname {sech}\left (2 \ln \left (x c \right )\right )}}{x^{2}}d x\]
Input:
int(sech(2*ln(x*c))^(1/2)/x^2,x)
Output:
int(sech(2*ln(x*c))^(1/2)/x^2,x)
Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.42 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^2} \, dx=\frac {1}{4} \, \sqrt {2} c \log \left (\frac {c^{5} x^{5} + 2 \, c x - 2 \, {\left (c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{c x^{5}}\right ) \] Input:
integrate(sech(2*log(c*x))^(1/2)/x^2,x, algorithm="fricas")
Output:
1/4*sqrt(2)*c*log((c^5*x^5 + 2*c*x - 2*(c^4*x^4 + 1)*sqrt(c^2*x^2/(c^4*x^4 + 1)))/(c*x^5))
\[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^2} \, dx=\int \frac {\sqrt {\operatorname {sech}{\left (2 \log {\left (c x \right )} \right )}}}{x^{2}}\, dx \] Input:
integrate(sech(2*ln(c*x))**(1/2)/x**2,x)
Output:
Integral(sqrt(sech(2*log(c*x)))/x**2, x)
\[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^2} \, dx=\int { \frac {\sqrt {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )}}{x^{2}} \,d x } \] Input:
integrate(sech(2*log(c*x))^(1/2)/x^2,x, algorithm="maxima")
Output:
integrate(sqrt(sech(2*log(c*x)))/x^2, x)
Timed out. \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^2} \, dx=\text {Timed out} \] Input:
integrate(sech(2*log(c*x))^(1/2)/x^2,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^2} \, dx=\int \frac {\sqrt {\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}}}{x^2} \,d x \] Input:
int((1/cosh(2*log(c*x)))^(1/2)/x^2,x)
Output:
int((1/cosh(2*log(c*x)))^(1/2)/x^2, x)
\[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^2} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (2 \,\mathrm {log}\left (c x \right )\right )}}{x^{2}}d x \] Input:
int(sech(2*log(c*x))^(1/2)/x^2,x)
Output:
int(sqrt(sech(2*log(c*x)))/x**2,x)