Integrand size = 15, antiderivative size = 41 \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^2} \, dx=-\frac {1}{2} c^2 \sqrt {1-\frac {1}{c^4 x^4}} x \csc ^{-1}\left (c^2 x^2\right ) \sqrt {\text {csch}(2 \log (c x))} \] Output:
-1/2*c^2*(1-1/c^4/x^4)^(1/2)*x*arccsc(c^2*x^2)*csch(2*ln(c*x))^(1/2)
Time = 0.08 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.32 \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^2} \, dx=\frac {\sqrt {-1+c^4 x^4} \sqrt {\frac {c^2 x^2}{-2+2 c^4 x^4}} \arctan \left (\sqrt {-1+c^4 x^4}\right )}{x} \] Input:
Integrate[Sqrt[Csch[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)]*ArcTan[Sqrt[-1 + c^4* x^4]])/x
Time = 0.31 (sec) , antiderivative size = 41, 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 = {6086, 6084, 858, 807, 223}
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.
\(\displaystyle \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^2} \, dx\) |
\(\Big \downarrow \) 6086 |
\(\displaystyle c \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{c^2 x^2}d(c x)\) |
\(\Big \downarrow \) 6084 |
\(\displaystyle c^2 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))} \int \frac {1}{c^3 \sqrt {1-\frac {1}{c^4 x^4}} x^3}d(c x)\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -c^2 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))} \int \frac {1}{c x \sqrt {1-c^4 x^4}}d\frac {1}{c x}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle -\frac {1}{2} c^2 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))} \int \frac {1}{\sqrt {1-c^2 x^2}}d\left (c^2 x^2\right )\) |
\(\Big \downarrow \) 223 |
\(\displaystyle -\frac {1}{2} c^2 x \sqrt {1-\frac {1}{c^4 x^4}} \arcsin \left (c^2 x^2\right ) \sqrt {\text {csch}(2 \log (c x))}\) |
Input:
Int[Sqrt[Csch[2*Log[c*x]]]/x^2,x]
Output:
-1/2*(c^2*Sqrt[1 - 1/(c^4*x^4)]*x*ArcSin[c^2*x^2]*Sqrt[Csch[2*Log[c*x]]])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[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[Csch[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[Csch[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[Csch[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m _.), x_Symbol] :> Simp[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n)) Subst[Int[ x^((m + 1)/n - 1)*Csch[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])
\[\int \frac {\sqrt {\operatorname {csch}\left (2 \ln \left (x c \right )\right )}}{x^{2}}d x\]
Input:
int(csch(2*ln(x*c))^(1/2)/x^2,x)
Output:
int(csch(2*ln(x*c))^(1/2)/x^2,x)
Time = 0.10 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^2} \, dx=\frac {1}{2} \, \sqrt {2} c \arctan \left (\frac {{\left (c^{4} x^{4} - 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}}}{c x}\right ) \] Input:
integrate(csch(2*log(c*x))^(1/2)/x^2,x, algorithm="fricas")
Output:
1/2*sqrt(2)*c*arctan((c^4*x^4 - 1)*sqrt(c^2*x^2/(c^4*x^4 - 1))/(c*x))
\[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^2} \, dx=\int \frac {\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}{x^{2}}\, dx \] Input:
integrate(csch(2*ln(c*x))**(1/2)/x**2,x)
Output:
Integral(sqrt(csch(2*log(c*x)))/x**2, x)
\[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^2} \, dx=\int { \frac {\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}{x^{2}} \,d x } \] Input:
integrate(csch(2*log(c*x))^(1/2)/x^2,x, algorithm="maxima")
Output:
integrate(sqrt(csch(2*log(c*x)))/x^2, x)
Timed out. \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^2} \, dx=\text {Timed out} \] Input:
integrate(csch(2*log(c*x))^(1/2)/x^2,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^2} \, dx=\int \frac {\sqrt {\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}}}{x^2} \,d x \] Input:
int((1/sinh(2*log(c*x)))^(1/2)/x^2,x)
Output:
int((1/sinh(2*log(c*x)))^(1/2)/x^2, x)
\[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^2} \, dx=\int \frac {\sqrt {\mathrm {csch}\left (2 \,\mathrm {log}\left (c x \right )\right )}}{x^{2}}d x \] Input:
int(csch(2*log(c*x))^(1/2)/x^2,x)
Output:
int(sqrt(csch(2*log(c*x)))/x**2,x)