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

Optimal. Leaf size=25 \[ \frac{1}{2} x^3 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x)) \]

[Out]

((c^4 + x^(-4))*x^3*Sech[2*Log[c*x]]^(3/2))/2

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Rubi [A]  time = 0.0395768, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {5551, 5549, 261} \[ \frac{1}{2} x^3 \left (c^4+\frac{1}{x^4}\right ) \text{sech}^{\frac{3}{2}}(2 \log (c x)) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((c^4 + x^(-4))*x^3*Sech[2*Log[c*x]]^(3/2))/2

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 261

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

Rubi steps

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

Mathematica [A]  time = 0.0346873, size = 32, normalized size = 1.28 \[ \sqrt{2} c^2 x \sqrt{\frac{c^2 x^2}{c^4 x^4+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[2]*c^2*x*Sqrt[(c^2*x^2)/(1 + c^4*x^4)]

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \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^2,x)

[Out]

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

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Maxima [A]  time = 1.51017, size = 53, normalized size = 2.12 \begin{align*} c{\left (\frac{\sqrt{2}}{{\left (\frac{1}{c^{4} x^{4}} + 1\right )}^{\frac{3}{2}}} + \frac{\sqrt{2}}{c^{4} x^{4}{\left (\frac{1}{c^{4} x^{4}} + 1\right )}^{\frac{3}{2}}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(sqrt(2)/(1/(c^4*x^4) + 1)^(3/2) + sqrt(2)/(c^4*x^4*(1/(c^4*x^4) + 1)^(3/2)))

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Fricas [A]  time = 3.13494, size = 58, normalized size = 2.32 \begin{align*} \sqrt{2} \sqrt{\frac{c^{2} x^{2}}{c^{4} x^{4} + 1}} c^{2} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*sqrt(c^2*x^2/(c^4*x^4 + 1))*c^2*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^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sech(2*log(c*x))**(3/2)/x**2, 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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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