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

Optimal. Leaf size=74 \[ c^3 x \sqrt{1-\frac{1}{c^4 x^4}} \sqrt{\text{csch}(2 \log (c x))} \text{EllipticF}\left (\csc ^{-1}(c x),-1\right )-c^3 x \sqrt{1-\frac{1}{c^4 x^4}} E\left (\left .\csc ^{-1}(c x)\right |-1\right ) \sqrt{\text{csch}(2 \log (c x))} \]

[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]

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Rubi [A]  time = 0.0656654, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {5552, 5550, 335, 307, 221, 1181, 424} \[ c^3 x \sqrt{1-\frac{1}{c^4 x^4}} F\left (\left .\csc ^{-1}(c x)\right |-1\right ) \sqrt{\text{csch}(2 \log (c x))}-c^3 x \sqrt{1-\frac{1}{c^4 x^4}} E\left (\left .\csc ^{-1}(c x)\right |-1\right ) \sqrt{\text{csch}(2 \log (c x))} \]

Antiderivative was successfully verified.

[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 5552

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])

Rule 5550

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/(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 335

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 307

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 221

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 1181

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 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{\text{csch}(2 \log (c x))}}{x^3} \, dx &=c^2 \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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))} F\left (\left .\csc ^{-1}(c x)\right |-1\right )-\left (c^3 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}\right ) \operatorname{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))} F\left (\left .\csc ^{-1}(c x)\right |-1\right )\\ \end{align*}

Mathematica [C]  time = 0.0936235, size = 58, normalized size = 0.78 \[ -\frac{\sqrt{2-2 c^4 x^4} \sqrt{\frac{c^2 x^2}{c^4 x^4-1}} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{4};c^4 x^4\right )}{x^2} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.036, size = 126, normalized size = 1.7 \begin{align*}{\frac{ \left ({c}^{4}{x}^{4}-1 \right ) \sqrt{2}}{{x}^{2}}\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}-1}}}}-{\frac{{c}^{2}\sqrt{2}}{x}\sqrt{{c}^{2}{x}^{2}+1}\sqrt{-{c}^{2}{x}^{2}+1} \left ({\it EllipticF} \left ( x\sqrt{-{c}^{2}},i \right ) -{\it EllipticE} \left ( x\sqrt{-{c}^{2}},i \right ) \right ) \sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}-1}}}{\frac{1}{\sqrt{-{c}^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{csch}\left (2 \, \log \left (c x\right )\right )}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{\operatorname{csch}\left (2 \, \log \left (c x\right )\right )}}{x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{csch}{\left (2 \log{\left (c x \right )} \right )}}}{x^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\operatorname{csch}\left (2 \, \log \left (c x\right )\right )}}{x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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