3.143 \(\int \frac{x^8}{\text{csch}^{\frac{3}{2}}(2 \log (c x))} \, dx\)

Optimal. Leaf size=128 \[ -\frac{x^5}{16 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x}{32 c^4 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{c^4 x^4}}\right )}{32 c^{12} x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^9}{12 \text{csch}^{\frac{3}{2}}(2 \log (c x))} \]

[Out]

x/(32*c^4*(c^4 - x^(-4))*Csch[2*Log[c*x]]^(3/2)) - x^5/(16*(c^4 - x^(-4))*Csch[2*Log[c*x]]^(3/2)) + x^9/(12*Cs
ch[2*Log[c*x]]^(3/2)) + ArcTanh[Sqrt[1 - 1/(c^4*x^4)]]/(32*c^12*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(
3/2))

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Rubi [A]  time = 0.0799591, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467, Rules used = {5552, 5550, 266, 47, 51, 63, 206} \[ -\frac{x^5}{16 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x}{32 c^4 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{c^4 x^4}}\right )}{32 c^{12} x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^9}{12 \text{csch}^{\frac{3}{2}}(2 \log (c x))} \]

Antiderivative was successfully verified.

[In]

Int[x^8/Csch[2*Log[c*x]]^(3/2),x]

[Out]

x/(32*c^4*(c^4 - x^(-4))*Csch[2*Log[c*x]]^(3/2)) - x^5/(16*(c^4 - x^(-4))*Csch[2*Log[c*x]]^(3/2)) + x^9/(12*Cs
ch[2*Log[c*x]]^(3/2)) + ArcTanh[Sqrt[1 - 1/(c^4*x^4)]]/(32*c^12*(1 - 1/(c^4*x^4))^(3/2)*x^3*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 266

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{x^8}{\text{csch}^{\frac{3}{2}}(2 \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^8}{\text{csch}^{\frac{3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^9}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{1}{x^4}\right )^{3/2} x^{11} \, dx,x,c x\right )}{c^{12} \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{(1-x)^{3/2}}{x^4} \, dx,x,\frac{1}{c^4 x^4}\right )}{4 c^{12} \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x^9}{12 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1-x}}{x^3} \, dx,x,\frac{1}{c^4 x^4}\right )}{8 c^{12} \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{x^5}{16 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^9}{12 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x^2} \, dx,x,\frac{1}{c^4 x^4}\right )}{32 c^{12} \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x}{32 c^4 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{x^5}{16 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^9}{12 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,\frac{1}{c^4 x^4}\right )}{64 c^{12} \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x}{32 c^4 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{x^5}{16 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^9}{12 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-\frac{1}{c^4 x^4}}\right )}{32 c^{12} \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x}{32 c^4 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{x^5}{16 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^9}{12 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{\tanh ^{-1}\left (\sqrt{1-\frac{1}{c^4 x^4}}\right )}{32 c^{12} \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ \end{align*}

Mathematica [A]  time = 0.199421, size = 95, normalized size = 0.74 \[ \frac{c^3 x^3 \sqrt{1-c^4 x^4} \left (8 c^8 x^8-14 c^4 x^4+3\right )-3 c x \sin ^{-1}\left (c^2 x^2\right )}{192 c^9 \sqrt{2-2 c^4 x^4} \sqrt{\frac{c^2 x^2}{c^4 x^4-1}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8/Csch[2*Log[c*x]]^(3/2),x]

[Out]

(c^3*x^3*Sqrt[1 - c^4*x^4]*(3 - 14*c^4*x^4 + 8*c^8*x^8) - 3*c*x*ArcSin[c^2*x^2])/(192*c^9*Sqrt[2 - 2*c^4*x^4]*
Sqrt[(c^2*x^2)/(-1 + c^4*x^4)])

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Maple [A]  time = 0.038, size = 121, normalized size = 1. \begin{align*}{\frac{{x}^{3} \left ( 8\,{c}^{8}{x}^{8}-14\,{c}^{4}{x}^{4}+3 \right ) \sqrt{2}}{384\,{c}^{6}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}-1}}}}}}+{\frac{\sqrt{2}x}{128\,{c}^{6}}\ln \left ({{c}^{4}{x}^{2}{\frac{1}{\sqrt{{c}^{4}}}}}+\sqrt{{c}^{4}{x}^{4}-1} \right ){\frac{1}{\sqrt{{c}^{4}}}}{\frac{1}{\sqrt{{c}^{4}{x}^{4}-1}}}{\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^8/csch(2*ln(c*x))^(3/2),x)

[Out]

1/384*x^3*(8*c^8*x^8-14*c^4*x^4+3)/c^6*2^(1/2)/(c^2*x^2/(c^4*x^4-1))^(1/2)+1/128/c^6*ln(c^4*x^2/(c^4)^(1/2)+(c
^4*x^4-1)^(1/2))/(c^4)^(1/2)*2^(1/2)*x/(c^4*x^4-1)^(1/2)/(c^2*x^2/(c^4*x^4-1))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{\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^8/csch(2*log(c*x))^(3/2),x, algorithm="maxima")

[Out]

integrate(x^8/csch(2*log(c*x))^(3/2), x)

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Fricas [A]  time = 1.75048, size = 240, normalized size = 1.88 \begin{align*} \frac{2 \, \sqrt{2}{\left (8 \, c^{13} x^{13} - 22 \, c^{9} x^{9} + 17 \, c^{5} x^{5} - 3 \, c x\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} - 1}} + 3 \, \sqrt{2} \log \left (2 \, c^{4} x^{4} + 2 \,{\left (c^{5} x^{5} - c x\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} - 1}} - 1\right )}{768 \, c^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8/csch(2*log(c*x))^(3/2),x, algorithm="fricas")

[Out]

1/768*(2*sqrt(2)*(8*c^13*x^13 - 22*c^9*x^9 + 17*c^5*x^5 - 3*c*x)*sqrt(c^2*x^2/(c^4*x^4 - 1)) + 3*sqrt(2)*log(2
*c^4*x^4 + 2*(c^5*x^5 - c*x)*sqrt(c^2*x^2/(c^4*x^4 - 1)) - 1))/c^9

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{8}}{\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**8/csch(2*ln(c*x))**(3/2),x)

[Out]

Integral(x**8/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^{8}}{\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^8/csch(2*log(c*x))^(3/2),x, algorithm="giac")

[Out]

integrate(x^8/csch(2*log(c*x))^(3/2), x)