\(\int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx\) [153]

Optimal result

Integrand size = 15, antiderivative size = 27 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx=-\frac {1}{2} \left (c^4-\frac {1}{x^4}\right ) x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x)) \]

[Out]

-1/2*(c^4-1/x^4)*x^3*csch(2*ln(c*x))^(3/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5671, 5669, 267} \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx=-\frac {1}{2} x^3 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x)) \]

[In]

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

[Out]

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

Rule 267

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx=-\sqrt {2} c^2 x \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx=-\sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} c^{2} x \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (23) = 46\).

Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.22 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx=-c {\left (\frac {\sqrt {2}}{{\left (\frac {1}{c x} + 1\right )}^{\frac {3}{2}} {\left (-\frac {1}{c x} + 1\right )}^{\frac {3}{2}} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}}} - \frac {\sqrt {2}}{c^{4} x^{4} {\left (\frac {1}{c x} + 1\right )}^{\frac {3}{2}} {\left (-\frac {1}{c x} + 1\right )}^{\frac {3}{2}} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}}}\right )} \]

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 2.17 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\text {csch}^{\frac {3}{2}}(2 \log (c x))}{x^2} \, dx=-c^2\,x\,\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4-1}} \]

[In]

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

[Out]

-c^2*x*((2*c^2*x^2)/(c^4*x^4 - 1))^(1/2)