\(\int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx\) [154]

Optimal result

Integrand size = 15, antiderivative size = 69 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=-\frac {1}{2} \left (c^4-\frac {1}{x^4}\right ) x^2 \text {csch}^{\frac {3}{2}}(2 \log (c x))+\frac {1}{2} c^5 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x)) \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right ) \]

[Out]

-1/2*(c^4-1/x^4)*x^2*csch(2*ln(c*x))^(3/2)+1/2*c^5*(1-1/c^4/x^4)^(3/2)*x^3*csch(2*ln(c*x))^(3/2)*EllipticF(1/c
/x,I)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5671, 5669, 342, 294, 227} \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=\frac {1}{2} c^5 x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))-\frac {1}{2} x^2 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x)) \]

[In]

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

[Out]

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

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 294

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 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}& = c^2 \text {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 ) \text {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 ) \text {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 ) \text {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)) \operatorname {EllipticF}\left (\csc ^{-1}(c x),-1\right ) \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.09 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.96 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=-\sqrt {2} c^2 \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}} \left (1+\sqrt {1-c^4 x^4} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},c^4 x^4\right )\right ) \]

[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])
)

Maple [F]

\[\int \frac {\operatorname {csch}\left (2 \ln \left (c x \right )\right )^{\frac {3}{2}}}{x^{3}}d x\]

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.08 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.62 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=i \, \sqrt {2} c^{2} F(\arcsin \left (c x\right )\,|\,-1) - \sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} c^{2} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=\int \frac {\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=\int { \frac {\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x^{3}} \,d x } \]

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^3} \, dx=\int \frac {{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x^3} \,d x \]

[In]

int((1/sinh(2*log(c*x)))^(3/2)/x^3,x)

[Out]

int((1/sinh(2*log(c*x)))^(3/2)/x^3, x)