3.146 \(\int \frac{x^5}{\text{csch}^{\frac{3}{2}}(2 \log (c x))} \, dx\)

Optimal. Leaf size=162 \[ -\frac{4 \text{EllipticF}\left (\csc ^{-1}(c x),-1\right )}{15 c^9 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{2 x^2}{15 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{4}{15 c^4 x^2 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{4 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^6}{9 \text{csch}^{\frac{3}{2}}(2 \log (c x))} \]

[Out]

4/(15*c^4*(c^4 - x^(-4))*x^2*Csch[2*Log[c*x]]^(3/2)) - (2*x^2)/(15*(c^4 - x^(-4))*Csch[2*Log[c*x]]^(3/2)) + x^
6/(9*Csch[2*Log[c*x]]^(3/2)) + (4*EllipticE[ArcCsc[c*x], -1])/(15*c^9*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c
*x]]^(3/2)) - (4*EllipticF[ArcCsc[c*x], -1])/(15*c^9*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c*x]]^(3/2))

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Rubi [A]  time = 0.0999622, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {5552, 5550, 335, 277, 325, 307, 221, 1181, 424} \[ -\frac{2 x^2}{15 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{4}{15 c^4 x^2 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{4 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{4 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 x^3 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^6}{9 \text{csch}^{\frac{3}{2}}(2 \log (c x))} \]

Antiderivative was successfully verified.

[In]

Int[x^5/Csch[2*Log[c*x]]^(3/2),x]

[Out]

4/(15*c^4*(c^4 - x^(-4))*x^2*Csch[2*Log[c*x]]^(3/2)) - (2*x^2)/(15*(c^4 - x^(-4))*Csch[2*Log[c*x]]^(3/2)) + x^
6/(9*Csch[2*Log[c*x]]^(3/2)) + (4*EllipticE[ArcCsc[c*x], -1])/(15*c^9*(1 - 1/(c^4*x^4))^(3/2)*x^3*Csch[2*Log[c
*x]]^(3/2)) - (4*EllipticF[ArcCsc[c*x], -1])/(15*c^9*(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 325

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] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && 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^5}{\text{csch}^{\frac{3}{2}}(2 \log (c x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^5}{\text{csch}^{\frac{3}{2}}(2 \log (x))} \, dx,x,c x\right )}{c^6}\\ &=\frac{\operatorname{Subst}\left (\int \left (1-\frac{1}{x^4}\right )^{3/2} x^8 \, dx,x,c x\right )}{c^9 \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^{10}} \, dx,x,\frac{1}{c x}\right )}{c^9 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{x^6}{9 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{1-x^4}}{x^6} \, dx,x,\frac{1}{c x}\right )}{3 c^9 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=-\frac{2 x^2}{15 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^6}{9 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )}{15 c^9 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{4}{15 c^4 \left (c^4-\frac{1}{x^4}\right ) x^2 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{2 x^2}{15 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^6}{9 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{4 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )}{15 c^9 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{4}{15 c^4 \left (c^4-\frac{1}{x^4}\right ) x^2 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{2 x^2}{15 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^6}{9 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{4 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )}{15 c^9 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{4 \operatorname{Subst}\left (\int \frac{1+x^2}{\sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )}{15 c^9 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{4}{15 c^4 \left (c^4-\frac{1}{x^4}\right ) x^2 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{2 x^2}{15 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^6}{9 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{4 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{4 \operatorname{Subst}\left (\int \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} \, dx,x,\frac{1}{c x}\right )}{15 c^9 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}\\ &=\frac{4}{15 c^4 \left (c^4-\frac{1}{x^4}\right ) x^2 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{2 x^2}{15 \left (c^4-\frac{1}{x^4}\right ) \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{x^6}{9 \text{csch}^{\frac{3}{2}}(2 \log (c x))}+\frac{4 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 \left (1-\frac{1}{c^4 x^4}\right )^{3/2} x^3 \text{csch}^{\frac{3}{2}}(2 \log (c x))}-\frac{4 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{15 c^9 \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 = 0.120538, size = 63, normalized size = 0.39 \[ -\frac{x^4 \, _2F_1\left (-\frac{3}{2},\frac{3}{4};\frac{7}{4};c^4 x^4\right )}{6 c^2 \sqrt{2-2 c^4 x^4} \sqrt{\frac{c^2 x^2}{c^4 x^4-1}}} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5/Csch[2*Log[c*x]]^(3/2),x]

[Out]

-(x^4*Hypergeometric2F1[-3/2, 3/4, 7/4, c^4*x^4])/(6*c^2*Sqrt[2 - 2*c^4*x^4]*Sqrt[(c^2*x^2)/(-1 + c^4*x^4)])

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Maple [A]  time = 0.035, size = 140, normalized size = 0.9 \begin{align*}{\frac{{x}^{4} \left ( 5\,{c}^{4}{x}^{4}-11 \right ) \sqrt{2}}{180\,{c}^{2}}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}-1}}}}}}+{\frac{\sqrt{2}x}{ \left ( 15\,{c}^{4}{x}^{4}-15 \right ){c}^{4}}\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^5/csch(2*ln(c*x))^(3/2),x)

[Out]

1/180*x^4*(5*c^4*x^4-11)*2^(1/2)/c^2/(c^2*x^2/(c^4*x^4-1))^(1/2)+1/15/(-c^2)^(1/2)*(c^2*x^2+1)^(1/2)*(-c^2*x^2
+1)^(1/2)/(c^4*x^4-1)/c^4*(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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\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^5/csch(2*log(c*x))^(3/2),x, algorithm="maxima")

[Out]

integrate(x^5/csch(2*log(c*x))^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{5}}{\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^5/csch(2*log(c*x))^(3/2),x, algorithm="fricas")

[Out]

integral(x^5/csch(2*log(c*x))^(3/2), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5}}{\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**5/csch(2*ln(c*x))**(3/2),x)

[Out]

Integral(x**5/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^{5}}{\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^5/csch(2*log(c*x))^(3/2),x, algorithm="giac")

[Out]

integrate(x^5/csch(2*log(c*x))^(3/2), x)