Integrand size = 15, antiderivative size = 74 \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=-c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))} E\left (\left .\csc ^{-1}(c x)\right |-1\right )+c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))} \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right ) \]
-c^3*x*EllipticE(1/c/x,I)*(1-1/c^4/x^4)^(1/2)*csch(2*ln(c*x))^(1/2)+c^3*x* EllipticF(1/c/x,I)*(1-1/c^4/x^4)^(1/2)*csch(2*ln(c*x))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=-\frac {\sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},c^4 x^4\right )}{x^2} \]
-((Sqrt[2 - 2*c^4*x^4]*Sqrt[(c^2*x^2)/(-1 + c^4*x^4)]*Hypergeometric2F1[-1 /4, 1/2, 3/4, c^4*x^4])/x^2)
Time = 0.31 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.73, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {6086, 6084, 858, 836, 762, 1388, 327}
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.
\(\displaystyle \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx\) |
\(\Big \downarrow \) 6086 |
\(\displaystyle c^2 \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{c^3 x^3}d(c x)\) |
\(\Big \downarrow \) 6084 |
\(\displaystyle c^3 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))} \int \frac {1}{c^4 \sqrt {1-\frac {1}{c^4 x^4}} x^4}d(c x)\) |
\(\Big \downarrow \) 858 |
\(\displaystyle -c^3 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))} \int \frac {c^2 x^2}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}\) |
\(\Big \downarrow \) 836 |
\(\displaystyle -c^3 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))} \left (\int \frac {c^2 x^2+1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}-\int \frac {1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}\right )\) |
\(\Big \downarrow \) 762 |
\(\displaystyle -c^3 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))} \left (\int \frac {c^2 x^2+1}{\sqrt {1-c^4 x^4}}d\frac {1}{c x}-\operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )\right )\) |
\(\Big \downarrow \) 1388 |
\(\displaystyle -c^3 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))} \left (\int \frac {\sqrt {c^2 x^2+1}}{\sqrt {1-c^2 x^2}}d\frac {1}{c x}-\operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )\right )\) |
\(\Big \downarrow \) 327 |
\(\displaystyle -c^3 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))} \left (E\left (\left .\arcsin \left (\frac {1}{c x}\right )\right |-1\right )-\operatorname {EllipticF}\left (\arcsin \left (\frac {1}{c x}\right ),-1\right )\right )\) |
-(c^3*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]]*(EllipticE[ArcSin[1/( c*x)], -1] - EllipticF[ArcSin[1/(c*x)], -1]))
3.2.40.3.1 Defintions of rubi rules used
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) )*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]
Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Simp[-q^(-1) Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q Int[(1 + q*x^2)/S qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
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[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
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])
Time = 0.68 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.70
method | result | size |
risch | \(\frac {\left (c^{4} x^{4}-1\right ) \sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}{x^{2}}-\frac {c^{2} \sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-c^{2}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-c^{2}}, i\right )\right ) \sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}{\sqrt {-c^{2}}\, x}\) | \(126\) |
(c^4*x^4-1)/x^2*2^(1/2)*(c^2*x^2/(c^4*x^4-1))^(1/2)-c^2/(-c^2)^(1/2)*(c^2* x^2+1)^(1/2)*(-c^2*x^2+1)^(1/2)*(EllipticF(x*(-c^2)^(1/2),I)-EllipticE(x*( -c^2)^(1/2),I))*2^(1/2)*(c^2*x^2/(c^4*x^4-1))^(1/2)/x
Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=\frac {i \, \sqrt {2} c^{4} x^{2} E(\arcsin \left (c x\right )\,|\,-1) - i \, \sqrt {2} c^{4} x^{2} F(\arcsin \left (c x\right )\,|\,-1) + \sqrt {2} {\left (c^{4} x^{4} - 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}}}{x^{2}} \]
(I*sqrt(2)*c^4*x^2*elliptic_e(arcsin(c*x), -1) - I*sqrt(2)*c^4*x^2*ellipti c_f(arcsin(c*x), -1) + sqrt(2)*(c^4*x^4 - 1)*sqrt(c^2*x^2/(c^4*x^4 - 1)))/ x^2
\[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=\int \frac {\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}{x^{3}}\, dx \]
\[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=\int { \frac {\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}{x^{3}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=\int \frac {\sqrt {\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}}}{x^3} \,d x \]