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

Optimal. Leaf size=130 \[ \frac{12 \text{EllipticF}\left (\csc ^{-1}(c x),-1\right )}{5 c^5 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{6}{5 x^2 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{12 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^2}{5 \text{csch}^{\frac{3}{2}}(2 \log (c x))} \]

[Out]

-6/(5*(c^4 - x^(-4))*x^2*Csch[2*Log[c*x]]^(3/2)) + x^2/(5*Csch[2*Log[c*x]]^(3/2)) - (12*EllipticE[ArcCsc[c*x],
-1])/(5*c^5*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2)) + (12*EllipticF[ArcCsc[c*x], -1])/(5*c^5*(1 -
1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0732644, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615, Rules used = {5552, 5550, 335, 277, 307, 221, 1181, 424} \[ -\frac{6}{5 x^2 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{12 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{12 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^2}{5 \text{csch}^{\frac{3}{2}}(2 \log (c x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-6/(5*(c^4 - x^(-4))*x^2*Csch[2*Log[c*x]]^(3/2)) + x^2/(5*Csch[2*Log[c*x]]^(3/2)) - (12*EllipticE[ArcCsc[c*x],
-1])/(5*c^5*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2)) + (12*EllipticF[ArcCsc[c*x], -1])/(5*c^5*(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 335

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 307

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(b/a), 2]}, -Dist[q^(-1), Int[1/Sqrt[a + b*x^
4], x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 1181

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[Sqrt[-c], Int
[(d + e*x^2)/(Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c, 0]

Rule 424

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]*EllipticE[ArcSin[Rt[-(d/c)
, 2]*x], (b*c)/(a*d)])/(Sqrt[c]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[
a, 0]

Rubi steps

\begin{align*} \int \frac{x}{\text{csch}^{\frac{3}{2}}(2 \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x}{\text{csch}^{\frac{3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^2}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{1}{x^4}\right )^{3/2} x^4 \, dx,x,c x\right )}{c^5 \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{\left (1-x^4\right )^{3/2}}{x^6} \, dx,x,\frac{1}{c x}\right )}{c^5 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x^2}{5 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{6 \operatorname{Subst}\left (\int \frac{\sqrt{1-x^4}}{x^2} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{6}{5 \left (c^4-\frac{1}{x^4}\right ) x^2 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^2}{5 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{12 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{6}{5 \left (c^4-\frac{1}{x^4}\right ) x^2 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^2}{5 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{12 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{12 \operatorname{Subst}\left (\int \frac{1+x^2}{\sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{6}{5 \left (c^4-\frac{1}{x^4}\right ) x^2 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^2}{5 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{12 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{12 \operatorname{Subst}\left (\int \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{6}{5 \left (c^4-\frac{1}{x^4}\right ) x^2 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^2}{5 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{12 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{12 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 \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 = 0.107892, size = 60, normalized size = 0.46 \[ \frac{\, _2F_1\left (-\frac{3}{2},-\frac{1}{4};\frac{3}{4};c^4 x^4\right )}{2 c^2 \sqrt{2-2 c^4 x^4} \sqrt{\frac{c^2 x^2}{c^4 x^4-1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.037, size = 152, normalized size = 1.2 \begin{align*}{\frac{ \left ({c}^{8}{x}^{8}+4\,{c}^{4}{x}^{4}-5 \right ) \sqrt{2}}{ \left ( 20\,{c}^{4}{x}^{4}-20 \right ){c}^{2}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}-1}}}}}}-{\frac{3\,\sqrt{2}x}{5\,{c}^{4}{x}^{4}-5}\sqrt{{c}^{2}{x}^{2}+1}\sqrt{-{c}^{2}{x}^{2}+1} \left ({\it EllipticF} \left ( x\sqrt{-{c}^{2}},i \right ) -{\it EllipticE} \left ( x\sqrt{-{c}^{2}},i \right ) \right ){\frac{1}{\sqrt{-{c}^{2}}}}{\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/csch(2*ln(c*x))^(3/2),x)

[Out]

1/20*(c^8*x^8+4*c^4*x^4-5)/(c^4*x^4-1)*2^(1/2)/c^2/(c^2*x^2/(c^4*x^4-1))^(1/2)-3/5/(-c^2)^(1/2)*(c^2*x^2+1)^(1
/2)*(-c^2*x^2+1)^(1/2)/(c^4*x^4-1)*(EllipticF(x*(-c^2)^(1/2),I)-EllipticE(x*(-c^2)^(1/2),I))*2^(1/2)*x/(c^2*x^
2/(c^4*x^4-1))^(1/2)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x/csch(2*log(c*x))^(3/2), x)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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