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

Optimal. Leaf size=25 \[ \frac{1}{2} x \left (c^4-\frac{1}{x^4}\right ) \sqrt{\text{csch}(2 \log (c x))} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0409001, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5552, 5550, 261} \[ \frac{1}{2} x \left (c^4-\frac{1}{x^4}\right ) \sqrt{\text{csch}(2 \log (c x))} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[Csch[2*Log[c*x]]]/x^4,x]

[Out]

((c^4 - x^(-4))*x*Sqrt[Csch[2*Log[c*x]]])/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 261

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]

Rubi steps

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

Mathematica [A]  time = 0.0372616, size = 33, normalized size = 1.32 \[ \frac{c^2}{2 x \sqrt{\frac{c^2 x^2}{2 c^4 x^4-2}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[Csch[2*Log[c*x]]]/x^4,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.033, size = 38, normalized size = 1.5 \begin{align*}{\frac{\sqrt{2} \left ({c}^{4}{x}^{4}-1 \right ) }{2\,{x}^{3}}\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}-1}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(2*ln(c*x))^(1/2)/x^4,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.62694, size = 120, normalized size = 4.8 \begin{align*} \frac{1}{2} \, c^{3}{\left (\frac{\sqrt{2}}{\sqrt{\frac{1}{c x} + 1} \sqrt{-\frac{1}{c x} + 1} \sqrt{\frac{1}{c^{2} x^{2}} + 1}} - \frac{\sqrt{2}}{c^{4} x^{4} \sqrt{\frac{1}{c x} + 1} \sqrt{-\frac{1}{c x} + 1} \sqrt{\frac{1}{c^{2} x^{2}} + 1}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*log(c*x))^(1/2)/x^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.56276, size = 80, normalized size = 3.2 \begin{align*} \frac{\sqrt{2}{\left (c^{4} x^{4} - 1\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} - 1}}}{2 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*log(c*x))^(1/2)/x^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*ln(c*x))**(1/2)/x**4,x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(2*log(c*x))^(1/2)/x^4,x, algorithm="giac")

[Out]

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