3.155 \(\int \frac{\text{csch}^{\frac{3}{2}}(2 \log (c x))}{x^4} \, dx\)

Optimal. Leaf size=69 \[ \frac{1}{2} c^6 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \csc ^{-1}\left (c^2 x^2\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))-\frac{1}{2} x \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x)) \]

[Out]

-((c^4 - x^(-4))*x*Csch[2*Log[c*x]]^(3/2))/2 + (c^6*(1 - 1/(c^4*x^4))^(3/2)*x^3*ArcCsc[c^2*x^2]*Csch[2*Log[c*x
]]^(3/2))/2

________________________________________________________________________________________

Rubi [A]  time = 0.0606238, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5552, 5550, 335, 275, 288, 216} \[ \frac{1}{2} c^6 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \csc ^{-1}\left (c^2 x^2\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))-\frac{1}{2} x \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x)) \]

Antiderivative was successfully verified.

[In]

Int[Csch[2*Log[c*x]]^(3/2)/x^4,x]

[Out]

-((c^4 - x^(-4))*x*Csch[2*Log[c*x]]^(3/2))/2 + (c^6*(1 - 1/(c^4*x^4))^(3/2)*x^3*ArcCsc[c^2*x^2]*Csch[2*Log[c*x
]]^(3/2))/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 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[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 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{\text{csch}^{\frac{3}{2}}(2 \log (c x))}{x^4} \, dx &=c^3 \operatorname{Subst}\left (\int \frac{\text{csch}^{\frac{3}{2}}(2 \log (x))}{x^4} \, dx,x,c x\right )\\ &=\left (c^6 \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^7} \, dx,x,c x\right )\\ &=-\left (\left (c^6 \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^5}{\left (1-x^4\right )^{3/2}} \, dx,x,\frac{1}{c x}\right )\right )\\ &=-\left (\frac{1}{2} \left (c^6 \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^2}{\left (1-x^2\right )^{3/2}} \, dx,x,\frac{1}{c^2 x^2}\right )\right )\\ &=-\frac{1}{2} \left (c^4-\frac{1}{x^4}\right ) x \text{csch}^{\frac{3}{2}}(2 \log (c x))+\frac{1}{2} \left (c^6 \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^2}} \, dx,x,\frac{1}{c^2 x^2}\right )\\ &=-\frac{1}{2} \left (c^4-\frac{1}{x^4}\right ) x \text{csch}^{\frac{3}{2}}(2 \log (c x))+\frac{1}{2} c^6 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \csc ^{-1}\left (c^2 x^2\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))\\ \end{align*}

Mathematica [C]  time = 0.104476, size = 53, normalized size = 0.77 \[ -\frac{\sqrt{2} c^2 \sqrt{\frac{c^2 x^2}{c^4 x^4-1}} \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};1-c^4 x^4\right )}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[2*Log[c*x]]^(3/2)/x^4,x]

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.035, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \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^4,x)

[Out]

int(csch(2*ln(c*x))^(3/2)/x^4,x)

________________________________________________________________________________________

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^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*log(c*x))^(3/2)/x^4,x, algorithm="maxima")

[Out]

integrate(csch(2*log(c*x))^(3/2)/x^4, x)

________________________________________________________________________________________

Fricas [A]  time = 1.63338, size = 159, normalized size = 2.3 \begin{align*} -\frac{\sqrt{2} c^{3} x \arctan \left (\frac{{\left (c^{4} x^{4} - 1\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} - 1}}}{c x}\right ) + \sqrt{2} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} - 1}} c^{2}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*log(c*x))^(3/2)/x^4,x, algorithm="fricas")

[Out]

-(sqrt(2)*c^3*x*arctan((c^4*x^4 - 1)*sqrt(c^2*x^2/(c^4*x^4 - 1))/(c*x)) + sqrt(2)*sqrt(c^2*x^2/(c^4*x^4 - 1))*
c^2)/x

________________________________________________________________________________________

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^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*ln(c*x))**(3/2)/x**4,x)

[Out]

Integral(csch(2*log(c*x))**(3/2)/x**4, x)

________________________________________________________________________________________

Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*log(c*x))^(3/2)/x^4,x, algorithm="giac")

[Out]

Timed out