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

Optimal. Leaf size=69 \[ \frac{1}{2} c^5 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x)) \text{EllipticF}\left (\csc ^{-1}(c x),-1\right )-\frac{1}{2} x^2 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x)) \]

[Out]

-((c^4 - x^(-4))*x^2*Csch[2*Log[c*x]]^(3/2))/2 + (c^5*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2)*Ellip
ticF[ArcCsc[c*x], -1])/2

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Rubi [A]  time = 0.0539875, antiderivative size = 69, 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, 288, 221} \[ \frac{1}{2} c^5 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} F\left (\left .\csc ^{-1}(c x)\right |-1\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))-\frac{1}{2} x^2 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x)) \]

Antiderivative was successfully verified.

[In]

Int[Csch[2*Log[c*x]]^(3/2)/x^3,x]

[Out]

-((c^4 - x^(-4))*x^2*Csch[2*Log[c*x]]^(3/2))/2 + (c^5*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2)*Ellip
ticF[ArcCsc[c*x], -1])/2

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 288

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*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 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{\text{csch}^{\frac{3}{2}}(2 \log (c x))}{x^3} \, dx &=c^2 \operatorname{Subst}\left (\int \frac{\text{csch}^{\frac{3}{2}}(2 \log (x))}{x^3} \, dx,x,c x\right )\\ &=\left (c^5 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{1}{x^4}\right )^{3/2} x^6} \, dx,x,c x\right )\\ &=-\left (\left (c^5 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))\right ) \operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^4\right )^{3/2}} \, dx,x,\frac{1}{c x}\right )\right )\\ &=-\frac{1}{2} \left (c^4-\frac{1}{x^4}\right ) x^2 \text{csch}^{\frac{3}{2}}(2 \log (c x))+\frac{1}{2} \left (c^5 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )\\ &=-\frac{1}{2} \left (c^4-\frac{1}{x^4}\right ) x^2 \text{csch}^{\frac{3}{2}}(2 \log (c x))+\frac{1}{2} c^5 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x)) F\left (\left .\csc ^{-1}(c x)\right |-1\right )\\ \end{align*}

Mathematica [C]  time = 0.107646, size = 66, normalized size = 0.96 \[ -\sqrt{2} c^2 \sqrt{\frac{c^2 x^2}{c^4 x^4-1}} \left (\sqrt{1-c^4 x^4} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};c^4 x^4\right )+1\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[2*Log[c*x]]^(3/2)/x^3,x]

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ({\rm csch} \left (2\,\ln \left ( cx \right ) \right ) \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{\frac{3}{2}}{\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))**(3/2)/x**3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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