\(\int \frac {x^6}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx\) [145]

Optimal result

Integrand size = 15, antiderivative size = 30 \[ \int \frac {x^6}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\left (c^4-\frac {1}{x^4}\right ) x^7}{10 c^4 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]

[Out]

1/10*(c^4-1/x^4)*x^7/c^4/csch(2*ln(c*x))^(3/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 30, 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, 270} \[ \int \frac {x^6}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {x^7 \left (c^4-\frac {1}{x^4}\right )}{10 c^4 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]

[In]

Int[x^6/Csch[2*Log[c*x]]^(3/2),x]

[Out]

((c^4 - x^(-4))*x^7)/(10*c^4*Csch[2*Log[c*x]]^(3/2))

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -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}& = \frac {\text {Subst}\left (\int \frac {x^6}{\text {csch}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^7} \\ & = \frac {\text {Subst}\left (\int \left (1-\frac {1}{x^4}\right )^{3/2} x^9 \, dx,x,c x\right )}{c^{10} \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ & = \frac {\left (c^4-\frac {1}{x^4}\right ) x^7}{10 c^4 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.47 \[ \int \frac {x^6}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\left (-1+c^4 x^4\right )^3 \sqrt {\frac {c^2 x^2}{-2+2 c^4 x^4}}}{20 c^8 x} \]

[In]

Integrate[x^6/Csch[2*Log[c*x]]^(3/2),x]

[Out]

((-1 + c^4*x^4)^3*Sqrt[(c^2*x^2)/(-2 + 2*c^4*x^4)])/(20*c^8*x)

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.57

method	result	size
risch	\(\frac {\sqrt {2}\, x \left (c^{8} x^{8}-2 c^{4} x^{4}+1\right )}{40 c^{6} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}\)	\(47\)

[In]

int(x^6/csch(2*ln(c*x))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/40*2^(1/2)/c^6*x/(c^2*x^2/(c^4*x^4-1))^(1/2)*(c^8*x^8-2*c^4*x^4+1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).

Time = 0.26 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.87 \[ \int \frac {x^6}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\sqrt {2} {\left (c^{12} x^{12} - 3 \, c^{8} x^{8} + 3 \, c^{4} x^{4} - 1\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}}}{40 \, c^{8} x} \]

[In]

integrate(x^6/csch(2*log(c*x))^(3/2),x, algorithm="fricas")

[Out]

1/40*sqrt(2)*(c^12*x^12 - 3*c^8*x^8 + 3*c^4*x^4 - 1)*sqrt(c^2*x^2/(c^4*x^4 - 1))/(c^8*x)

Sympy [F]

\[ \int \frac {x^6}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^{6}}{\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]

[In]

integrate(x**6/csch(2*ln(c*x))**(3/2),x)

[Out]

Integral(x**6/csch(2*log(c*x))**(3/2), x)

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.53 \[ \int \frac {x^6}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {{\left (\sqrt {2} c^{4} x^{4} - \sqrt {2}\right )} {\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )}^{\frac {3}{2}}}{40 \, c^{7}} \]

[In]

integrate(x^6/csch(2*log(c*x))^(3/2),x, algorithm="maxima")

[Out]

1/40*(sqrt(2)*c^4*x^4 - sqrt(2))*(c^2*x^2 + 1)^(3/2)*(c*x + 1)^(3/2)*(c*x - 1)^(3/2)/c^7

Giac [F(-1)]

Timed out. \[ \int \frac {x^6}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\text {Timed out} \]

[In]

integrate(x^6/csch(2*log(c*x))^(3/2),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 2.30 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int \frac {x^6}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {{\left (c^4\,x^4-1\right )}^3\,\sqrt {\frac {2\,c^2\,x^2}{c^4\,x^4-1}}}{40\,c^8\,x} \]

[In]

int(x^6/(1/sinh(2*log(c*x)))^(3/2),x)

[Out]

((c^4*x^4 - 1)^3*((2*c^2*x^2)/(c^4*x^4 - 1))^(1/2))/(40*c^8*x)