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

Optimal. Leaf size=119 \[ \frac{2 \text{EllipticF}\left (\csc ^{-1}(c x),-1\right )}{5 c^5 x \sqrt{1-\frac{1}{c^4 x^4}} \sqrt{\text{csch}(2 \log (c x))}}-\frac{2 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 x \sqrt{1-\frac{1}{c^4 x^4}} \sqrt{\text{csch}(2 \log (c x))}}-\frac{2}{5 c^4 \sqrt{\text{csch}(2 \log (c x))}}+\frac{x^4}{5 \sqrt{\text{csch}(2 \log (c x))}} \]

[Out]

-2/(5*c^4*Sqrt[Csch[2*Log[c*x]]]) + x^4/(5*Sqrt[Csch[2*Log[c*x]]]) - (2*EllipticE[ArcCsc[c*x], -1])/(5*c^5*Sqr
t[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]]) + (2*EllipticF[ArcCsc[c*x], -1])/(5*c^5*Sqrt[1 - 1/(c^4*x^4)]*x*S
qrt[Csch[2*Log[c*x]]])

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Rubi [A]  time = 0.0863124, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{2 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 x \sqrt{1-\frac{1}{c^4 x^4}} \sqrt{\text{csch}(2 \log (c x))}}-\frac{2 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 x \sqrt{1-\frac{1}{c^4 x^4}} \sqrt{\text{csch}(2 \log (c x))}}-\frac{2}{5 c^4 \sqrt{\text{csch}(2 \log (c x))}}+\frac{x^4}{5 \sqrt{\text{csch}(2 \log (c x))}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2/(5*c^4*Sqrt[Csch[2*Log[c*x]]]) + x^4/(5*Sqrt[Csch[2*Log[c*x]]]) - (2*EllipticE[ArcCsc[c*x], -1])/(5*c^5*Sqr
t[1 - 1/(c^4*x^4)]*x*Sqrt[Csch[2*Log[c*x]]]) + (2*EllipticF[ArcCsc[c*x], -1])/(5*c^5*Sqrt[1 - 1/(c^4*x^4)]*x*S
qrt[Csch[2*Log[c*x]]])

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^3}{\sqrt{\text{csch}(2 \log (c x))}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^3}{\sqrt{\text{csch}(2 \log (x))}} \, dx,x,c x\right )}{c^4}\\ &=\frac{\operatorname{Subst}\left (\int \sqrt{1-\frac{1}{x^4}} x^4 \, dx,x,c x\right )}{c^5 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{1-x^4}}{x^6} \, dx,x,\frac{1}{c x}\right )}{c^5 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}\\ &=\frac{x^4}{5 \sqrt{\text{csch}(2 \log (c x))}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}\\ &=-\frac{2}{5 c^4 \sqrt{\text{csch}(2 \log (c x))}}+\frac{x^4}{5 \sqrt{\text{csch}(2 \log (c x))}}-\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}\\ &=-\frac{2}{5 c^4 \sqrt{\text{csch}(2 \log (c x))}}+\frac{x^4}{5 \sqrt{\text{csch}(2 \log (c x))}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}-\frac{2 \operatorname{Subst}\left (\int \frac{1+x^2}{\sqrt{1-x^4}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}\\ &=-\frac{2}{5 c^4 \sqrt{\text{csch}(2 \log (c x))}}+\frac{x^4}{5 \sqrt{\text{csch}(2 \log (c x))}}+\frac{2 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}-\frac{2 \operatorname{Subst}\left (\int \frac{\sqrt{1+x^2}}{\sqrt{1-x^2}} \, dx,x,\frac{1}{c x}\right )}{5 c^5 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}\\ &=-\frac{2}{5 c^4 \sqrt{\text{csch}(2 \log (c x))}}+\frac{x^4}{5 \sqrt{\text{csch}(2 \log (c x))}}-\frac{2 E\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}+\frac{2 F\left (\left .\csc ^{-1}(c x)\right |-1\right )}{5 c^5 \sqrt{1-\frac{1}{c^4 x^4}} x \sqrt{\text{csch}(2 \log (c x))}}\\ \end{align*}

Mathematica [C]  time = 0.110618, size = 60, normalized size = 0.5 \[ \frac{x^4 \, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{7}{4};c^4 x^4\right )}{3 \sqrt{2-2 c^4 x^4} \sqrt{\frac{c^2 x^2}{c^4 x^4-1}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.039, size = 127, normalized size = 1.1 \begin{align*}{\frac{{x}^{4}\sqrt{2}}{10}{\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}}{{c}^{4}{x}^{4}-1}}}}}}-{\frac{\sqrt{2}x}{ \left ( 5\,{c}^{4}{x}^{4}-5 \right ){c}^{2}}\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^3/csch(2*ln(c*x))^(1/2),x)

[Out]

1/10*x^4*2^(1/2)/(c^2*x^2/(c^4*x^4-1))^(1/2)-1/5/(-c^2)^(1/2)*(c^2*x^2+1)^(1/2)*(-c^2*x^2+1)^(1/2)/(c^4*x^4-1)
/c^2*(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^{3}}{\sqrt{\operatorname{csch}\left (2 \, \log \left (c x\right )\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/csch(2*ln(c*x))**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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