3.2.78 \(\int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx\) [178]

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

[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 = 0.03, 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 = {3853, 3856, 2719} \begin {gather*} \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) \end {gather*}

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 2719

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

Rule 3853

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 3856

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 &=\text {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))-\text {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 ) \text {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 = 0.09, size = 45, normalized size = 0.80 \begin {gather*} \frac {\frac {i E(i \log (c x)|2)}{\sqrt {\cosh (2 \log (c x))}}+\tanh (2 \log (c x))}{\sqrt {\text {sech}(2 \log (c x))}} \end {gather*}

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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Maple [A]
time = 2.62, size = 127, normalized size = 2.27

method	result	size
derivativedivides	\(\frac {2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right ) \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}+\sqrt {-2 \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}-1}\, \EllipticE \left (\frac {c x}{2}+\frac {1}{2 c x}, \sqrt {2}\right ) \sqrt {-\left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}}{\left (\frac {c x}{2}-\frac {1}{2 c x}\right ) \sqrt {2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}-1}}\)	\(127\)
default	\(\frac {2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right ) \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}+\sqrt {-2 \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}-1}\, \EllipticE \left (\frac {c x}{2}+\frac {1}{2 c x}, \sqrt {2}\right ) \sqrt {-\left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}}{\left (\frac {c x}{2}-\frac {1}{2 c x}\right ) \sqrt {2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}-1}}\)	\(127\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(1/2*c*x+1/2/c/x)*(1/2*c*x-1/2/c/x)^2+(-2*(1/2*c*x-1/2/c/x)^2-1)^(1/2)*EllipticE(1/2*c*x+1/2/c/x,2^(1/2))*(
-(1/2*c*x-1/2/c/x)^2)^(1/2))/(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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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

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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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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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