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

Optimal. Leaf size=56 \[ \sinh (2 \log (c x)) \sqrt {\text {sech}(2 \log (c x))}+i \sqrt {\text {sech}(2 \log (c x))} \sqrt {\cosh (2 \log (c x))} E(i \log (c x)|2) \]

[Out]

sinh(2*ln(c*x))*sech(2*ln(c*x))^(1/2)+I*((1/2*c*x+1/2/c/x)^2)^(1/2)/(1/2*c*x+1/2/c/x)*EllipticE(I*(1/2*c*x-1/2
/c/x),2^(1/2))*cosh(2*ln(c*x))^(1/2)*sech(2*ln(c*x))^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3768, 3771, 2639} \[ \sinh (2 \log (c x)) \sqrt {\text {sech}(2 \log (c x))}+i \sqrt {\text {sech}(2 \log (c x))} \sqrt {\cosh (2 \log (c x))} E(i \log (c x)|2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

I*Sqrt[Cosh[2*Log[c*x]]]*EllipticE[I*Log[c*x], 2]*Sqrt[Sech[2*Log[c*x]]] + Sqrt[Sech[2*Log[c*x]]]*Sinh[2*Log[c
*x]]

Rule 2639

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2*EllipticE[(1*(c - Pi/2 + d*x))/2, 2])/d, x] /; FreeQ[{
c, d}, x]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rubi steps

\begin {align*} \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx &=\operatorname {Subst}\left (\int \text {sech}^{\frac {3}{2}}(2 x) \, dx,x,\log (c x)\right )\\ &=\sqrt {\text {sech}(2 \log (c x))} \sinh (2 \log (c x))-\operatorname {Subst}\left (\int \frac {1}{\sqrt {\text {sech}(2 x)}} \, dx,x,\log (c x)\right )\\ &=\sqrt {\text {sech}(2 \log (c x))} \sinh (2 \log (c x))-\left (\sqrt {\cosh (2 \log (c x))} \sqrt {\text {sech}(2 \log (c x))}\right ) \operatorname {Subst}\left (\int \sqrt {\cosh (2 x)} \, dx,x,\log (c x)\right )\\ &=i \sqrt {\cosh (2 \log (c x))} E(i \log (c x)|2) \sqrt {\text {sech}(2 \log (c x))}+\sqrt {\text {sech}(2 \log (c x))} \sinh (2 \log (c x))\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 45, normalized size = 0.80 \[ \frac {\tanh (2 \log (c x))+\frac {i E(i \log (c x)|2)}{\sqrt {\cosh (2 \log (c x))}}}{\sqrt {\text {sech}(2 \log (c x))}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I*EllipticE[I*Log[c*x], 2])/Sqrt[Cosh[2*Log[c*x]]] + Tanh[2*Log[c*x]])/Sqrt[Sech[2*Log[c*x]]]

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sech(2*log(c*x))^(3/2)/x, x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 0.62, size = 127, normalized size = 2.27 \[ \frac {\sqrt {-2 \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}-1}\, \sqrt {-\left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}\, \EllipticE \left (\frac {c x}{2}+\frac {1}{2 c x}, \sqrt {2}\right )+2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right ) \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}{\left (\frac {c x}{2}-\frac {1}{2 c x}\right ) \sqrt {2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((-2*(1/2*c*x-1/2/c/x)^2-1)^(1/2)*(-(1/2*c*x-1/2/c/x)^2)^(1/2)*EllipticE(1/2*c*x+1/2/c/x,2^(1/2))+2*(1/2*c*x+1
/2/c/x)*(1/2*c*x-1/2/c/x)^2)/(1/2*c*x-1/2/c/x)/(2*(1/2*c*x+1/2/c/x)^2-1)^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1/cosh(2*log(c*x)))^(3/2)/x,x)

[Out]

int((1/cosh(2*log(c*x)))^(3/2)/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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