3.142 \(\int \frac{\sqrt{\text{csch}(2 \log (c x))}}{x^5} \, dx\)

Optimal. Leaf size=64 \[ \frac{1}{3} \left (c^4-\frac{1}{x^4}\right ) \sqrt{\text{csch}(2 \log (c x))}-\frac{1}{3} c^5 x \sqrt{1-\frac{1}{c^4 x^4}} \sqrt{\text{csch}(2 \log (c x))} \text{EllipticF}\left (\csc ^{-1}(c x),-1\right ) \]

[Out]

((c^4 - x^(-4))*Sqrt[Csch[2*Log[c*x]]])/3 - (c^5*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]]*EllipticF[ArcC
sc[c*x], -1])/3

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Rubi [A]  time = 0.0546849, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5552, 5550, 335, 321, 221} \[ \frac{1}{3} \left (c^4-\frac{1}{x^4}\right ) \sqrt{\text{csch}(2 \log (c x))}-\frac{1}{3} c^5 x \sqrt{1-\frac{1}{c^4 x^4}} F\left (\left .\csc ^{-1}(c x)\right |-1\right ) \sqrt{\text{csch}(2 \log (c x))} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Csch[2*Log[c*x]]]/x^5,x]

[Out]

((c^4 - x^(-4))*Sqrt[Csch[2*Log[c*x]]])/3 - (c^5*Sqrt[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]]*EllipticF[ArcC
sc[c*x], -1])/3

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 321

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

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]

Rubi steps

\begin{align*} \int \frac{\sqrt{\text{csch}(2 \log (c x))}}{x^5} \, dx &=c^4 \operatorname{Subst}\left (\int \frac{\sqrt{\text{csch}(2 \log (x))}}{x^5} \, dx,x,c x\right )\\ &=\left (c^5 \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^6} \, dx,x,c x\right )\\ &=-\left (\left (c^5 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}\right ) \operatorname{Subst}\left (\int \frac{x^4}{\sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )\right )\\ &=\frac{1}{3} \left (c^4-\frac{1}{x^4}\right ) \sqrt{\text{csch}(2 \log (c x))}-\frac{1}{3} \left (c^5 \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 )\\ &=\frac{1}{3} \left (c^4-\frac{1}{x^4}\right ) \sqrt{\text{csch}(2 \log (c x))}-\frac{1}{3} c^5 \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 = 0.0937916, size = 60, normalized size = 0.94 \[ -\frac{\sqrt{2-2 c^4 x^4} \sqrt{\frac{c^2 x^2}{c^4 x^4-1}} \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{1}{4};c^4 x^4\right )}{3 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Csch[2*Log[c*x]]]/x^5,x]

[Out]

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

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Maple [A]  time = 0.036, size = 112, normalized size = 1.8 \begin{align*}{\frac{ \left ({c}^{4}{x}^{4}-1 \right ) \sqrt{2}}{3\,{x}^{4}}\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}-1}}}}+{\frac{{c}^{4}\sqrt{2}}{3\,x}\sqrt{{c}^{2}{x}^{2}+1}\sqrt{-{c}^{2}{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{-{c}^{2}},i \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^5,x)

[Out]

1/3*(c^4*x^4-1)/x^4*2^(1/2)*(c^2*x^2/(c^4*x^4-1))^(1/2)+1/3*c^4/(-c^2)^(1/2)*(c^2*x^2+1)^(1/2)*(-c^2*x^2+1)^(1
/2)*EllipticF(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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*log(c*x))^(1/2)/x^5,x, algorithm="maxima")

[Out]

integrate(sqrt(csch(2*log(c*x)))/x^5, 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^{5}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*log(c*x))^(1/2)/x^5,x, algorithm="fricas")

[Out]

integral(sqrt(csch(2*log(c*x)))/x^5, 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^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*ln(c*x))**(1/2)/x**5,x)

[Out]

Integral(sqrt(csch(2*log(c*x)))/x**5, 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^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*log(c*x))^(1/2)/x^5,x, algorithm="giac")

[Out]

integrate(sqrt(csch(2*log(c*x)))/x^5, x)