Integrand size = 15, antiderivative size = 56 \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=i \sqrt {\cosh (2 \log (c x))} E(i \log (c x)|2) \sqrt {\text {sech}(2 \log (c x))}+\sqrt {\text {sech}(2 \log (c x))} \sinh (2 \log (c x)) \]
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)*s ech(2*ln(c*x))^(1/2)
Time = 0.16 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.80 \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\frac {\frac {i E(i \log (c x)|2)}{\sqrt {\cosh (2 \log (c x))}}+\tanh (2 \log (c x))}{\sqrt {\text {sech}(2 \log (c x))}} \]
((I*EllipticE[I*Log[c*x], 2])/Sqrt[Cosh[2*Log[c*x]]] + Tanh[2*Log[c*x]])/S qrt[Sech[2*Log[c*x]]]
Time = 0.35 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3039, 3042, 4255, 3042, 4258, 3042, 3119}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \int \text {sech}^{\frac {3}{2}}(2 \log (c x))d\log (c x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \csc \left (\frac {\pi }{2}+2 i \log (c x)\right )^{3/2}d\log (c x)\) |
\(\Big \downarrow \) 4255 |
\(\displaystyle \sinh (2 \log (c x)) \sqrt {\text {sech}(2 \log (c x))}-\int \frac {1}{\sqrt {\text {sech}(2 \log (c x))}}d\log (c x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sinh (2 \log (c x)) \sqrt {\text {sech}(2 \log (c x))}-\int \frac {1}{\sqrt {\csc \left (2 i \log (c x)+\frac {\pi }{2}\right )}}d\log (c x)\) |
\(\Big \downarrow \) 4258 |
\(\displaystyle \sinh (2 \log (c x)) \sqrt {\text {sech}(2 \log (c x))}-\sqrt {\text {sech}(2 \log (c x))} \sqrt {\cosh (2 \log (c x))} \int \sqrt {\cosh (2 \log (c x))}d\log (c x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \sinh (2 \log (c x)) \sqrt {\text {sech}(2 \log (c x))}-\sqrt {\text {sech}(2 \log (c x))} \sqrt {\cosh (2 \log (c x))} \int \sqrt {\sin \left (2 i \log (c x)+\frac {\pi }{2}\right )}d\log (c x)\) |
\(\Big \downarrow \) 3119 |
\(\displaystyle \sinh (2 \log (c x)) \sqrt {\text {sech}(2 \log (c x))}+i \sqrt {\text {sech}(2 \log (c x))} \sqrt {\cosh (2 \log (c x))} E(i \log (c x)|2)\) |
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]]
3.2.78.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
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]
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] + Simp[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]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(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]
Time = 1.01 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.27
method | result | size |
derivativedivides | \(\frac {2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right ) \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}+\operatorname {EllipticE}\left (\frac {c x}{2}+\frac {1}{2 c x}, \sqrt {2}\right ) \sqrt {-2 \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}-1}\, \sqrt {-\left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}}{\left (\frac {c x}{2}-\frac {1}{2 c x}\right ) \sqrt {2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}-1}}\) | \(127\) |
default | \(\frac {2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right ) \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}+\operatorname {EllipticE}\left (\frac {c x}{2}+\frac {1}{2 c x}, \sqrt {2}\right ) \sqrt {-2 \left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}-1}\, \sqrt {-\left (\frac {c x}{2}-\frac {1}{2 c x}\right )^{2}}}{\left (\frac {c x}{2}-\frac {1}{2 c x}\right ) \sqrt {2 \left (\frac {c x}{2}+\frac {1}{2 c x}\right )^{2}-1}}\) | \(127\) |
(2*(1/2*c*x+1/2/c/x)*(1/2*c*x-1/2/c/x)^2+EllipticE(1/2*c*x+1/2/c/x,2^(1/2) )*(-2*(1/2*c*x-1/2/c/x)^2-1)^(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)
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.46 \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\frac {\sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}} c^{3} x^{2} + \sqrt {2} \left (-c^{4}\right )^{\frac {3}{4}} E(\arcsin \left (\left (-c^{4}\right )^{\frac {1}{4}} x\right )\,|\,-1) - \sqrt {2} \left (-c^{4}\right )^{\frac {3}{4}} F(\arcsin \left (\left (-c^{4}\right )^{\frac {1}{4}} x\right )\,|\,-1)}{c} \]
(sqrt(2)*sqrt(c^2*x^2/(c^4*x^4 + 1))*c^3*x^2 + sqrt(2)*(-c^4)^(3/4)*ellipt ic_e(arcsin((-c^4)^(1/4)*x), -1) - sqrt(2)*(-c^4)^(3/4)*elliptic_f(arcsin( (-c^4)^(1/4)*x), -1))/c
\[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int \frac {\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x}\, dx \]
\[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int { \frac {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x} \,d x } \]
Timed out. \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int \frac {{\left (\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x} \,d x \]