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

Optimal. Leaf size=92 \[ \frac{1}{2} x^2 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))-\frac{x^3 \left (c^4+\frac{1}{x^4}\right ) \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x)) \text{EllipticF}\left (2 \cot ^{-1}(c x),\frac{1}{2}\right )}{4 c} \]

[Out]

((c^4 + x^(-4))*x^2*Sech[2*Log[c*x]]^(3/2))/2 - ((c^4 + x^(-4))*Sqrt[(c^4 + x^(-4))/(c^2 + x^(-2))^2]*(c^2 + x
^(-2))*x^3*EllipticF[2*ArcCot[c*x], 1/2]*Sech[2*Log[c*x]]^(3/2))/(4*c)

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Rubi [A]  time = 0.0771525, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5551, 5549, 335, 288, 220} \[ \frac{1}{2} x^2 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x))-\frac{x^3 \left (c^4+\frac{1}{x^4}\right ) \sqrt{\frac{c^4+\frac{1}{x^4}}{\left (c^2+\frac{1}{x^2}\right )^2}} \left (c^2+\frac{1}{x^2}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x)) F\left (2 \cot ^{-1}(c x)|\frac{1}{2}\right )}{4 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c^4 + x^(-4))*x^2*Sech[2*Log[c*x]]^(3/2))/2 - ((c^4 + x^(-4))*Sqrt[(c^4 + x^(-4))/(c^2 + x^(-2))^2]*(c^2 + x
^(-2))*x^3*EllipticF[2*ArcCot[c*x], 1/2]*Sech[2*Log[c*x]]^(3/2))/(4*c)

Rule 5551

Int[((e_.)*(x_))^(m_.)*Sech[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[(e*x)^(m + 1
)/(e*n*(c*x^n)^((m + 1)/n)), Subst[Int[x^((m + 1)/n - 1)*Sech[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 5549

Int[((e_.)*(x_))^(m_.)*Sech[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[(Sech[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 288

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

Rule 220

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

Rubi steps

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

Mathematica [C]  time = 0.107889, size = 65, normalized size = 0.71 \[ \sqrt{2} c^2 \sqrt{\frac{c^2 x^2}{c^4 x^4+1}} \left (\sqrt{c^4 x^4+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-c^4 x^4\right )+1\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[2]*c^2*Sqrt[(c^2*x^2)/(1 + c^4*x^4)]*(1 + Sqrt[1 + c^4*x^4]*Hypergeometric2F1[1/4, 1/2, 5/4, -(c^4*x^4)])

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Maple [F]  time = 0.03, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3}} \left ({\rm sech} \left (2\,\ln \left ( cx \right ) \right ) \right ) ^{{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(2*ln(c*x))^(3/2)/x^3,x)

[Out]

int(sech(2*ln(c*x))^(3/2)/x^3,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(2*ln(c*x))**(3/2)/x**3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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