\(\int \frac {x}{\sqrt {\text {csch}(2 \log (c x))}} \, dx\) [136]

Optimal result

Integrand size = 13, antiderivative size = 60 \[ \int \frac {x}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {x^2}{3 \sqrt {\text {csch}(2 \log (c x))}}+\frac {2 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{3 c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(2 \log (c x))}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 60, 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 = {5671, 5669, 342, 283, 227} \[ \int \frac {x}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {2 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{3 c^3 x \sqrt {1-\frac {1}{c^4 x^4}} \sqrt {\text {csch}(2 \log (c x))}}+\frac {x^2}{3 \sqrt {\text {csch}(2 \log (c x))}} \]

[In]

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

[Out]

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

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 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 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}& = \frac {\text {Subst}\left (\int \frac {x}{\sqrt {\text {csch}(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 {csch}(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 {csch}(2 \log (c x))}} \\ & = \frac {x^2}{3 \sqrt {\text {csch}(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 {csch}(2 \log (c x))}} \\ & = \frac {x^2}{3 \sqrt {\text {csch}(2 \log (c x))}}+\frac {2 \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right )}{3 c^3 \sqrt {1-\frac {1}{c^4 x^4}} x \sqrt {\text {csch}(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 = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.95 \[ \int \frac {x}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {x^2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},c^4 x^4\right )}{\sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.82

method	result	size
risch	\(\frac {\sqrt {2}\, x^{2}}{6 \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}-\frac {\sqrt {c^{2} x^{2}+1}\, \sqrt {-c^{2} x^{2}+1}\, \operatorname {EllipticF}\left (x \sqrt {-c^{2}}, i\right ) \sqrt {2}\, x}{3 \sqrt {-c^{2}}\, \left (c^{4} x^{4}-1\right ) \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}\)	\(109\)

[In]

int(x/csch(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/(-c^2)^(1/2)*(c^2*x^2+1)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^4*x^4-1)*
EllipticF(x*(-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 = 63, normalized size of antiderivative = 1.05 \[ \int \frac {x}{\sqrt {\text {csch}(2 \log (c x))}} \, dx=\frac {\sqrt {2} {\left (c^{6} x^{4} - c^{2}\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} + 2 \, \sqrt {2} \sqrt {c^{4}} F(\arcsin \left (\frac {1}{c x}\right )\,|\,-1)}{6 \, c^{4}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(x/sqrt(csch(2*log(c*x))), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-2)]

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

[In]

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

[In]

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

[Out]

int(x/(1/sinh(2*log(c*x)))^(1/2), x)