\(\int \frac {x^8}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx\) [143]

Optimal result

Integrand size = 15, antiderivative size = 128 \[ \int \frac {x^8}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {x}{32 c^4 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {x^5}{16 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^9}{12 \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{c^4 x^4}}\right )}{32 c^{12} \left (1-\frac {1}{c^4 x^4}\right )^{3/2} x^3 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]

[Out]

1/32*x/c^4/(c^4-1/x^4)/csch(2*ln(c*x))^(3/2)-1/16*x^5/(c^4-1/x^4)/csch(2*ln(c*x))^(3/2)+1/12*x^9/csch(2*ln(c*x
))^(3/2)+1/32*arctanh((1-1/c^4/x^4)^(1/2))/c^12/(1-1/c^4/x^4)^(3/2)/x^3/csch(2*ln(c*x))^(3/2)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {5671, 5669, 272, 43, 44, 65, 212} \[ \int \frac {x^8}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{c^4 x^4}}\right )}{32 c^{12} x^3 \left (1-\frac {1}{c^4 x^4}\right )^{3/2} \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x}{32 c^4 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}-\frac {x^5}{16 \left (c^4-\frac {1}{x^4}\right ) \text {csch}^{\frac {3}{2}}(2 \log (c x))}+\frac {x^9}{12 \text {csch}^{\frac {3}{2}}(2 \log (c x))} \]

[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 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

Rule 65

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 212

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

Rule 272

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 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^8}{\text {csch}^{\frac {3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^9} \\ & = \frac {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {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 {\text {arctanh}\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] (verified)

Time = 0.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.74 \[ \int \frac {x^8}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {c^3 x^3 \sqrt {1-c^4 x^4} \left (3-14 c^4 x^4+8 c^8 x^8\right )-3 c x \arcsin \left (c^2 x^2\right )}{192 c^9 \sqrt {2-2 c^4 x^4} \sqrt {\frac {c^2 x^2}{-1+c^4 x^4}}} \]

[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)])

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.95

method	result	size
risch	\(\frac {x^{3} \left (8 c^{8} x^{8}-14 c^{4} x^{4}+3\right ) \sqrt {2}}{384 c^{6} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}+\frac {\ln \left (\frac {c^{4} x^{2}}{\sqrt {c^{4}}}+\sqrt {c^{4} x^{4}-1}\right ) \sqrt {2}\, x}{128 c^{6} \sqrt {c^{4}}\, \sqrt {c^{4} x^{4}-1}\, \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4}-1}}}\)	\(121\)

[In]

int(x^8/csch(2*ln(c*x))^(3/2),x,method=_RETURNVERBOSE)

[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)

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.86 \[ \int \frac {x^8}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\frac {2 \, \sqrt {2} {\left (8 \, c^{13} x^{13} - 22 \, c^{9} x^{9} + 17 \, c^{5} x^{5} - 3 \, c x\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} + 3 \, \sqrt {2} \log \left (2 \, c^{4} x^{4} + 2 \, {\left (c^{5} x^{5} - c x\right )} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} - 1}} - 1\right )}{768 \, c^{9}} \]

[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

Sympy [F]

\[ \int \frac {x^8}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^{8}}{\operatorname {csch}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {x^8}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int { \frac {x^{8}}{\operatorname {csch}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}} \,d x } \]

[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)

Giac [F(-1)]

Timed out. \[ \int \frac {x^8}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {x^8}{\text {csch}^{\frac {3}{2}}(2 \log (c x))} \, dx=\int \frac {x^8}{{\left (\frac {1}{\mathrm {sinh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}} \,d x \]

[In]

int(x^8/(1/sinh(2*log(c*x)))^(3/2),x)

[Out]

int(x^8/(1/sinh(2*log(c*x)))^(3/2), x)