3.151 \(\int \frac{1}{\text{csch}^{\frac{3}{2}}(2 \log (c x))} \, dx\)

Optimal. Leaf size=96 \[ \frac{3}{4 x^3 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^4 x^4}}\right )}{4 c^4 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x}{4 \text{csch}^{\frac{3}{2}}(2 \log (c x))} \]

[Out]

3/(4*(c^4 - x^(-4))*x^3*Csch[2*Log[c*x]]^(3/2)) + x/(4*Csch[2*Log[c*x]]^(3/2)) - (3*ArcTanh[Sqrt[1 - 1/(c^4*x^
4)]])/(4*c^4*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2))

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Rubi [A]  time = 0.0464677, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636, Rules used = {5546, 5544, 266, 47, 50, 63, 206} \[ \frac{3}{4 x^3 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^4 x^4}}\right )}{4 c^4 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x}{4 \text{csch}^{\frac{3}{2}}(2 \log (c x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

3/(4*(c^4 - x^(-4))*x^3*Csch[2*Log[c*x]]^(3/2)) + x/(4*Csch[2*Log[c*x]]^(3/2)) - (3*ArcTanh[Sqrt[1 - 1/(c^4*x^
4)]])/(4*c^4*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2))

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 5544

Int[Csch[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), 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[1/(x^(b*d*p)*(1 - 1/(E^(2*a*d)*x^(2*b*d)))^p), x], x] /; FreeQ[{a, b, d, 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 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && 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{1}{\text{csch}^{\frac{3}{2}}(2 \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\text{csch}^{\frac{3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{1}{x^4}\right )^{3/2} x^3 \, dx,x,c x\right )}{c^4 \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^2} \, dx,x,\frac{1}{c^4 x^4}\right )}{4 c^4 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x}{4 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{3 \operatorname{Subst}\left (\int \frac{\sqrt{1-x}}{x} \, dx,x,\frac{1}{c^4 x^4}\right )}{8 c^4 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{3}{4 \left (c^4-\frac{1}{x^4}\right ) x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x}{4 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x} x} \, dx,x,\frac{1}{c^4 x^4}\right )}{8 c^4 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{3}{4 \left (c^4-\frac{1}{x^4}\right ) x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x}{4 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{1-\frac{1}{c^4 x^4}}\right )}{4 c^4 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{3}{4 \left (c^4-\frac{1}{x^4}\right ) x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x}{4 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{3 \tanh ^{-1}\left (\sqrt{1-\frac{1}{c^4 x^4}}\right )}{4 c^4 \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 [C]  time = 0.0889616, size = 63, normalized size = 0.66 \[ \frac{\, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};c^4 x^4\right )}{4 c^2 x \sqrt{2-2 c^4 x^4} \sqrt{\frac{c^2 x^2}{c^4 x^4-1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Hypergeometric2F1[-3/2, -1/2, 1/2, c^4*x^4]/(4*c^2*x*Sqrt[2 - 2*c^4*x^4]*Sqrt[(c^2*x^2)/(-1 + c^4*x^4)])

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Maple [A]  time = 0.036, size = 130, normalized size = 1.4 \begin{align*}{\frac{ \left ({c}^{8}{x}^{8}+{c}^{4}{x}^{4}-2 \right ) \sqrt{2}}{16\,x \left ({c}^{4}{x}^{4}-1 \right ){c}^{2}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}-1}}}}}}-{\frac{3\,{c}^{2}\sqrt{2}x}{16}\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(1/csch(2*ln(c*x))^(3/2),x)

[Out]

1/16*(c^8*x^8+c^4*x^4-2)/x/(c^4*x^4-1)*2^(1/2)/c^2/(c^2*x^2/(c^4*x^4-1))^(1/2)-3/16*c^2*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{1}{\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(1/csch(2*log(c*x))^(3/2),x, algorithm="maxima")

[Out]

integrate(csch(2*log(c*x))^(-3/2), x)

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Fricas [A]  time = 1.70339, size = 225, normalized size = 2.34 \begin{align*} \frac{3 \, \sqrt{2} c^{3} x^{3} \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 ) + 2 \, \sqrt{2}{\left (c^{8} x^{8} + c^{4} x^{4} - 2\right )} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} - 1}}}{32 \, c^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(3*sqrt(2)*c^3*x^3*log(2*c^4*x^4 - 2*(c^5*x^5 - c*x)*sqrt(c^2*x^2/(c^4*x^4 - 1)) - 1) + 2*sqrt(2)*(c^8*x^
8 + c^4*x^4 - 2)*sqrt(c^2*x^2/(c^4*x^4 - 1)))/(c^4*x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\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(1/csch(2*ln(c*x))**(3/2),x)

[Out]

Integral(csch(2*log(c*x))**(-3/2), x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError