3.162 \(\int \text{csch}^3(a+2 \log (\frac{c}{\sqrt{x}})) \, dx\)

Optimal. Leaf size=26 \[ \frac{2 e^{-3 a} c^2}{\left (e^{-2 a}-\frac{c^4}{x^2}\right )^2} \]

[Out]

(2*c^2)/(E^(3*a)*(E^(-2*a) - c^4/x^2)^2)

________________________________________________________________________________________

Rubi [A]  time = 0.0460046, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {5546, 5548, 263, 261} \[ \frac{2 e^{-3 a} c^2}{\left (e^{-2 a}-\frac{c^4}{x^2}\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + 2*Log[c/Sqrt[x]]]^3,x]

[Out]

(2*c^2)/(E^(3*a)*(E^(-2*a) - c^4/x^2)^2)

Rule 5546

Int[Csch[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[
x^(1/n - 1)*Csch[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n
, 1])

Rule 5548

Int[Csch[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[2^p/E^(a*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}, x] && IntegerQ[p]

Rule 263

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

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 \text{csch}^3\left (a+2 \log \left (\frac{c}{\sqrt{x}}\right )\right ) \, dx &=-\left (\left (2 c^2\right ) \operatorname{Subst}\left (\int \frac{\text{csch}^3(a+2 \log (x))}{x^3} \, dx,x,\frac{c}{\sqrt{x}}\right )\right )\\ &=-\left (\left (16 c^2 e^{-3 a}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-\frac{e^{-2 a}}{x^4}\right )^3 x^9} \, dx,x,\frac{c}{\sqrt{x}}\right )\right )\\ &=-\left (\left (16 c^2 e^{-3 a}\right ) \operatorname{Subst}\left (\int \frac{x^3}{\left (-e^{-2 a}+x^4\right )^3} \, dx,x,\frac{c}{\sqrt{x}}\right )\right )\\ &=\frac{2 c^2 e^{-3 a}}{\left (e^{-2 a}-\frac{c^4}{x^2}\right )^2}\\ \end{align*}

Mathematica [B]  time = 0.0982161, size = 65, normalized size = 2.5 \[ -\frac{2 c^6 (\sinh (2 a)+\cosh (2 a)) \left (\sinh (a) \left (c^4+2 x^2\right )+\cosh (a) \left (c^4-2 x^2\right )\right )}{\left (\cosh (a) \left (x^2-c^4\right )-\sinh (a) \left (c^4+x^2\right )\right )^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + 2*Log[c/Sqrt[x]]]^3,x]

[Out]

(-2*c^6*((c^4 - 2*x^2)*Cosh[a] + (c^4 + 2*x^2)*Sinh[a])*(Cosh[2*a] + Sinh[2*a]))/((-c^4 + x^2)*Cosh[a] - (c^4
+ x^2)*Sinh[a])^2

________________________________________________________________________________________

Maple [F]  time = 0.066, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm csch} \left (a+2\,\ln \left ({\frac{c}{\sqrt{x}}} \right ) \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(a+2*ln(c/x^(1/2)))^3,x)

[Out]

int(csch(a+2*ln(c/x^(1/2)))^3,x)

________________________________________________________________________________________

Maxima [A]  time = 1.06159, size = 66, normalized size = 2.54 \begin{align*} -\frac{2 \,{\left (c^{10} e^{\left (5 \, a\right )} - 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{c^{8} e^{\left (4 \, a\right )} - 2 \, c^{4} x^{2} e^{\left (2 \, a\right )} + x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+2*log(c/x^(1/2)))^3,x, algorithm="maxima")

[Out]

-2*(c^10*e^(5*a) - 2*c^6*x^2*e^(3*a))/(c^8*e^(4*a) - 2*c^4*x^2*e^(2*a) + x^4)

________________________________________________________________________________________

Fricas [A]  time = 1.45316, size = 107, normalized size = 4.12 \begin{align*} -\frac{2 \,{\left (c^{10} e^{\left (5 \, a\right )} - 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{c^{8} e^{\left (4 \, a\right )} - 2 \, c^{4} x^{2} e^{\left (2 \, a\right )} + x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+2*log(c/x^(1/2)))^3,x, algorithm="fricas")

[Out]

-2*(c^10*e^(5*a) - 2*c^6*x^2*e^(3*a))/(c^8*e^(4*a) - 2*c^4*x^2*e^(2*a) + x^4)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+2*ln(c/x**(1/2)))**3,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.17209, size = 53, normalized size = 2.04 \begin{align*} -\frac{2 \,{\left (c^{10} e^{\left (5 \, a\right )} - 2 \, c^{6} x^{2} e^{\left (3 \, a\right )}\right )}}{{\left (c^{4} e^{\left (2 \, a\right )} - x^{2}\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(a+2*log(c/x^(1/2)))^3,x, algorithm="giac")

[Out]

-2*(c^10*e^(5*a) - 2*c^6*x^2*e^(3*a))/(c^4*e^(2*a) - x^2)^2