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

Optimal. Leaf size=30 \[ \frac{x^7 \left (c^4-\frac{1}{x^4}\right )}{10 c^4 \text{csch}^{\frac{3}{2}}(2 \log (c x))} \]

[Out]

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

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Rubi [A]  time = 0.0408179, antiderivative size = 30, 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, 264} \[ \frac{x^7 \left (c^4-\frac{1}{x^4}\right )}{10 c^4 \text{csch}^{\frac{3}{2}}(2 \log (c x))} \]

Antiderivative was successfully verified.

[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 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 264

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]

Rubi steps

\begin{align*} \int \frac{x^6}{\text{csch}^{\frac{3}{2}}(2 \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^6}{\text{csch}^{\frac{3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^7}\\ &=\frac{\operatorname{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]  time = 0.0481718, size = 44, normalized size = 1.47 \[ \frac{\left (c^4 x^4-1\right )^3 \sqrt{\frac{c^2 x^2}{2 c^4 x^4-2}}}{20 c^8 x} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.03, size = 47, normalized size = 1.6 \begin{align*}{\frac{\sqrt{2}x \left ({c}^{8}{x}^{8}-2\,{c}^{4}{x}^{4}+1 \right ) }{40\,{c}^{6}}{\frac{1}{\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(x^6/csch(2*ln(c*x))^(3/2),x)

[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)

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Maxima [A]  time = 1.74628, size = 62, normalized size = 2.07 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [B]  time = 1.5711, size = 122, normalized size = 4.07 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\operatorname{csch}^{\frac{3}{2}}{\left (2 \log{\left (c x \right )} \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{6}}{\operatorname{csch}\left (2 \, \log \left (c x\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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