\(\int \frac {x}{\sqrt {\text {sech}(2 \log (c x))}} \, dx\) [162]

Optimal result

Integrand size = 13, antiderivative size = 87 \[ \int \frac {x}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {x^2}{3 \sqrt {\text {sech}(2 \log (c x))}}-\frac {\sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right )}{3 c \left (c^4+\frac {1}{x^4}\right ) x \sqrt {\text {sech}(2 \log (c x))}} \]

[Out]

1/3*x^2/sech(2*ln(c*x))^(1/2)-1/3*(c^2+1/x^2)*(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)/c/(c^4+1/x^4)/x/sech(2*ln(c*x))^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {5670, 5668, 342, 283, 226} \[ \int \frac {x}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {x^2}{3 \sqrt {\text {sech}(2 \log (c x))}}-\frac {\sqrt {\frac {c^4+\frac {1}{x^4}}{\left (c^2+\frac {1}{x^2}\right )^2}} \left (c^2+\frac {1}{x^2}\right ) \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right )}{3 c x \left (c^4+\frac {1}{x^4}\right ) \sqrt {\text {sech}(2 \log (c x))}} \]

[In]

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

[Out]

x^2/(3*Sqrt[Sech[2*Log[c*x]]]) - (Sqrt[(c^4 + x^(-4))/(c^2 + x^(-2))^2]*(c^2 + x^(-2))*EllipticF[2*ArcCot[c*x]
, 1/2])/(3*c*(c^4 + x^(-4))*x*Sqrt[Sech[2*Log[c*x]]])

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 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

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 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}& = \frac {\text {Subst}\left (\int \frac {x}{\sqrt {\text {sech}(2 \log (x))}} \, dx,x,c x\right )}{c^2} \\ & = \frac {\text {Subst}\left (\int \sqrt {1+\frac {1}{x^4}} x^2 \, dx,x,c x\right )}{c^3 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}} \\ & = -\frac {\text {Subst}\left (\int \frac {\sqrt {1+x^4}}{x^4} \, dx,x,\frac {1}{c x}\right )}{c^3 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}} \\ & = \frac {x^2}{3 \sqrt {\text {sech}(2 \log (c x))}}-\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^4}} \, dx,x,\frac {1}{c x}\right )}{3 c^3 \sqrt {1+\frac {1}{c^4 x^4}} x \sqrt {\text {sech}(2 \log (c x))}} \\ & = \frac {x^2}{3 \sqrt {\text {sech}(2 \log (c x))}}-\frac {\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 ) \operatorname {EllipticF}\left (2 \cot ^{-1}(c x),\frac {1}{2}\right )}{3 c \left (c^4+\frac {1}{x^4}\right ) x \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 = 0.19 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.67 \[ \int \frac {x}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {\sqrt {1+c^4 x^4} \sqrt {\frac {c^2 x^2}{2+2 c^4 x^4}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},-c^4 x^4\right )}{c^2} \]

[In]

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

[Out]

(Sqrt[1 + c^4*x^4]*Sqrt[(c^2*x^2)/(2 + 2*c^4*x^4)]*Hypergeometric2F1[-1/2, 1/4, 5/4, -(c^4*x^4)])/c^2

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.57 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.31

method	result	size
risch	\(\frac {\sqrt {2}\, x^{2}}{6 \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}+\frac {\sqrt {-i c^{2} x^{2}+1}\, \sqrt {i c^{2} x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {i c^{2}}, i\right ) \sqrt {2}\, x}{3 \sqrt {i c^{2}}\, \left (c^{4} x^{4}+1\right ) \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}+1}}}\)	\(114\)

[In]

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

[Out]

1/6*2^(1/2)*x^2/(c^2*x^2/(c^4*x^4+1))^(1/2)+1/3/(I*c^2)^(1/2)*(1-I*c^2*x^2)^(1/2)*(1+I*c^2*x^2)^(1/2)/(c^4*x^4
+1)*EllipticF(x*(I*c^2)^(1/2),I)*2^(1/2)*x/(c^2*x^2/(c^4*x^4+1))^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.82 \[ \int \frac {x}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\frac {2 \, \sqrt {2} \sqrt {c^{4}} c \left (-\frac {1}{c^{4}}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {1}{c^{4}}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + \sqrt {2} {\left (c^{4} x^{4} + 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}}}{6 \, c^{2}} \]

[In]

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

[Out]

1/6*(2*sqrt(2)*sqrt(c^4)*c*(-1/c^4)^(3/4)*elliptic_f(arcsin((-1/c^4)^(1/4)/x), -1) + sqrt(2)*(c^4*x^4 + 1)*sqr
t(c^2*x^2/(c^4*x^4 + 1)))/c^2

Sympy [F]

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

[In]

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

[Out]

Integral(x/sqrt(sech(2*log(c*x))), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x}{\sqrt {\text {sech}(2 \log (c x))}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly exception caught Unable to convert to real %%{poly1[1.0000000000000000000000000000000,0.0000000000
00000000000

Mupad [F(-1)]

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

[In]

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

[Out]

int(x/(1/cosh(2*log(c*x)))^(1/2), x)