\(\int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^3} \, dx\) [166]

Optimal result

Integrand size = 15, antiderivative size = 137 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^3} \, dx=-\frac {\left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}}{c^2+\frac {1}{x^2}}+c \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) x E\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right ) \sqrt {\text {sech}(2 \log (c x))}-\frac {1}{2} c \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) x \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right ) \sqrt {\text {sech}(2 \log (c x))} \]

[Out]

-(c^4+1/x^4)*sech(2*ln(c*x))^(1/2)/(c^2+1/x^2)+c*(c^2+1/x^2)*x*(cos(2*arccot(c*x))^2)^(1/2)/cos(2*arccot(c*x))
*EllipticE(sin(2*arccot(c*x)),1/2*2^(1/2))*((c^4+1/x^4)/(c^2+1/x^2)^2)^(1/2)*sech(2*ln(c*x))^(1/2)-1/2*c*(c^2+
1/x^2)*x*(cos(2*arccot(c*x))^2)^(1/2)/cos(2*arccot(c*x))*EllipticF(sin(2*arccot(c*x)),1/2*2^(1/2))*((c^4+1/x^4
)/(c^2+1/x^2)^2)^(1/2)*sech(2*ln(c*x))^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5670, 5668, 342, 311, 226, 1210} \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^3} \, dx=-\frac {\left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}}{c^2+\frac {1}{x^2}}-\frac {1}{2} c x \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \sqrt {\text {sech}(2 \log (c x))} \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right )+c x \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \sqrt {\text {sech}(2 \log (c x))} E\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right ) \]

[In]

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

[Out]

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

Rule 226

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

Rule 311

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b/a, 2]}, Dist[1/q, 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] && PosQ[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 1210

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a +
c*x^4]/(a*(1 + q^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + c*x^4)/(a*(1 + q^2*x^2)^2)]/(q*Sqrt[a + c*x^4
]))*EllipticE[2*ArcTan[q*x], 1/2], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, c, d, e}, x] && PosQ[c/a]

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*} \text {integral}& = c^2 \text {Subst}\left (\int \frac {\sqrt {\text {sech}(2 \log (x))}}{x^3} \, dx,x,c x\right ) \\ & = \left (c^3 \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^4} \, dx,x,c x\right ) \\ & = -\left (\left (c^3 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(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 (\left (c^3 \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+x^4}} \, dx,x,\frac {1}{c x}\right )\right )+\left (c^3 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}\right ) \text {Subst}\left (\int \frac {1-x^2}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right ) \\ & = -\frac {\left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}}{c^2+\frac {1}{x^2}}+c \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) x E\left (2 \cot ^{-1}(c x)|\frac {1}{2}\right ) \sqrt {\text {sech}(2 \log (c x))}-\frac {1}{2} c \sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) x \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right ) \sqrt {\text {sech}(2 \log (c x))} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.15 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^3} \, dx=-\frac {c^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {1}{2},\frac {3}{4},-c^4 x^4\right )}{\sqrt {1+c^4 x^4} \sqrt {\frac {c^2 x^2}{2+2 c^4 x^4}}} \]

[In]

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

[Out]

-((c^2*Hypergeometric2F1[-1/4, 1/2, 3/4, -(c^4*x^4)])/(Sqrt[1 + c^4*x^4]*Sqrt[(c^2*x^2)/(2 + 2*c^4*x^4)]))

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.66 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.98

method	result	size
risch	\(-\frac {\left (c^{4} x^{4}+1\right ) \sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}{x^{2}}+\frac {i c^{2} \sqrt {-i c^{2} x^{2}+1}\, \sqrt {i c^{2} x^{2}+1}\, \left (\operatorname {EllipticF}\left (x \sqrt {i c^{2}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {i c^{2}}, i\right )\right ) \sqrt {2}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}{\sqrt {i c^{2}}\, x}\)	\(134\)

[In]

int(sech(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)+I*c^2/(I*c^2)^(1/2)*(1-I*c^2*x^2)^(1/2)*(1+I*c^2*x^2)^(1/
2)*(EllipticF(x*(I*c^2)^(1/2),I)-EllipticE(x*(I*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 = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {\text {sech}(2 \log (c x))}}{x^3} \, dx=-\frac {\sqrt {2} \left (-c^{4}\right )^{\frac {3}{4}} c x^{2} E(\arcsin \left (\left (-c^{4}\right )^{\frac {1}{4}} x\right )\,|\,-1) - \sqrt {2} \left (-c^{4}\right )^{\frac {3}{4}} c x^{2} F(\arcsin \left (\left (-c^{4}\right )^{\frac {1}{4}} x\right )\,|\,-1) + \sqrt {2} {\left (c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{x^{2}} \]

[In]

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

[Out]

-(sqrt(2)*(-c^4)^(3/4)*c*x^2*elliptic_e(arcsin((-c^4)^(1/4)*x), -1) - sqrt(2)*(-c^4)^(3/4)*c*x^2*elliptic_f(ar
csin((-c^4)^(1/4)*x), -1) + sqrt(2)*(c^4*x^4 + 1)*sqrt(c^2*x^2/(c^4*x^4 + 1)))/x^2

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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