\(\int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx\) [140]

Optimal result

Integrand size = 15, antiderivative size = 74 \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=-c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))} E\left (\left .\csc ^{-1}(c x)\right |-1\right )+c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))} \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right ) \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {5671, 5669, 342, 313, 227, 1195, 435} \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=c^3 x \sqrt {1-\frac {1}{c^4 x^4}} \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right ) \sqrt {\text {csch}(2 \log (c x))}-c^3 x \sqrt {1-\frac {1}{c^4 x^4}} E\left (\left .\csc ^{-1}(c x)\right |-1\right ) \sqrt {\text {csch}(2 \log (c x))} \]

[In]

Int[Sqrt[Csch[2*Log[c*x]]]/x^3,x]

[Out]

-(c^3*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]]*EllipticE[ArcCsc[c*x], -1]) + c^3*Sqrt[1 - 1/(c^4*x^4)]*x
*Sqrt[Csch[2*Log[c*x]]]*EllipticF[ArcCsc[c*x], -1]

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 313

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[-q^(-1), Int[1/Sqrt[a + b*x^4]
, x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 342

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] && IntegerQ[m]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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
]

Rule 1195

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[Sqrt[-c], Int
[(d + e*x^2)/(Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c, 0]

Rule 5669

Int[Csch[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[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]

Rule 5671

Int[Csch[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(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])

Rubi steps \begin{align*} \text {integral}& = c^2 \text {Subst}\left (\int \frac {\sqrt {\text {csch}(2 \log (x))}}{x^3} \, dx,x,c x\right ) \\ & = \left (c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {1}{x^4}} x^4} \, dx,x,c x\right ) \\ & = -\left (\left (c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )\right ) \\ & = \left (c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\right )-\left (c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}\right ) \text {Subst}\left (\int \frac {1+x^2}{\sqrt {1-x^4}} \, dx,x,\frac {1}{c x}\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 )-\left (c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {1-x^2}} \, dx,x,\frac {1}{c x}\right ) \\ & = -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 ) \\ \end{align*}

Mathematica [C] (verified)

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 \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=-\frac {\sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},c^4 x^4\right )}{x^2} \]

[In]

Integrate[Sqrt[Csch[2*Log[c*x]]]/x^3,x]

[Out]

-((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)

Maple [A] (verified)

Time = 0.68 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.70

method	result	size
risch	\(\frac {\left (c^{4} x^{4}-1\right ) \sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}{x^{2}}-\frac {c^{2} \sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {-c^{2}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {-c^{2}}, i\right )\right ) \sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}{\sqrt {-c^{2}}\, x}\)	\(126\)

[In]

int(csch(2*ln(c*x))^(1/2)/x^3,x,method=_RETURNVERBOSE)

[Out]

(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)*(Ell
ipticF(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

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=\frac {i \, \sqrt {2} c^{4} x^{2} E(\arcsin \left (c x\right )\,|\,-1) - i \, \sqrt {2} c^{4} x^{2} F(\arcsin \left (c x\right )\,|\,-1) + \sqrt {2} {\left (c^{4} x^{4} - 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}}}{x^{2}} \]

[In]

integrate(csch(2*log(c*x))^(1/2)/x^3,x, algorithm="fricas")

[Out]

(I*sqrt(2)*c^4*x^2*elliptic_e(arcsin(c*x), -1) - I*sqrt(2)*c^4*x^2*elliptic_f(arcsin(c*x), -1) + sqrt(2)*(c^4*
x^4 - 1)*sqrt(c^2*x^2/(c^4*x^4 - 1)))/x^2

Sympy [F]

\[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=\int \frac {\sqrt {\operatorname {csch}{\left (2 \log {\left (c x \right )} \right )}}}{x^{3}}\, dx \]

[In]

integrate(csch(2*ln(c*x))**(1/2)/x**3,x)

[Out]

Integral(sqrt(csch(2*log(c*x)))/x**3, x)

Maxima [F]

\[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=\int { \frac {\sqrt {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )}}{x^{3}} \,d x } \]

[In]

integrate(csch(2*log(c*x))^(1/2)/x^3,x, algorithm="maxima")

[Out]

integrate(sqrt(csch(2*log(c*x)))/x^3, x)

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=\text {Timed out} \]

[In]

integrate(csch(2*log(c*x))^(1/2)/x^3,x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {\text {csch}(2 \log (c x))}}{x^3} \, dx=\int \frac {\sqrt {\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}}}{x^3} \,d x \]

[In]

int((1/sinh(2*log(c*x)))^(1/2)/x^3,x)

[Out]

int((1/sinh(2*log(c*x)))^(1/2)/x^3, x)