Integrand size = 15, antiderivative size = 56 \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=i \sqrt {\cosh (2 \log (c x))} E(i \log (c x)|2) \sqrt {\text {sech}(2 \log (c x))}+\sqrt {\text {sech}(2 \log (c x))} \sinh (2 \log (c x)) \] Output:
I*cosh(2*ln(c*x))^(1/2)*EllipticE(I*(1/2*c*x-1/2/c/x),2^(1/2))*sech(2*ln(c *x))^(1/2)+sech(2*ln(c*x))^(1/2)*sinh(2*ln(c*x))
Time = 0.10 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.80 \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\frac {\frac {i E(i \log (c x)|2)}{\sqrt {\cosh (2 \log (c x))}}+\tanh (2 \log (c x))}{\sqrt {\text {sech}(2 \log (c x))}} \] Input:
Integrate[Sech[2*Log[c*x]]^(3/2)/x,x]
Output:
((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 |
\(\displaystyle \sinh (2 \log (c x)) \sqrt {\text {sech}(2 \log (c x))}+i \sqrt {\text {sech}(2 \log (c x))} \sqrt {\cosh (2 \log (c x))} E(i \log (c x)|2)\) |
Input:
Int[Sech[2*Log[c*x]]^(3/2)/x,x]
Output:
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]]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(126\) vs. \(2(59)=118\).
Time = 0.47 (sec) , antiderivative size = 127, normalized size of antiderivative = 2.27
method | result | size |
derivativedivides | \(\frac {2 \left (\frac {x c}{2}+\frac {1}{2 x c}\right ) \left (\frac {x c}{2}-\frac {1}{2 x c}\right )^{2}+\sqrt {-2 \left (\frac {x c}{2}-\frac {1}{2 x c}\right )^{2}-1}\, \sqrt {-\left (\frac {x c}{2}-\frac {1}{2 x c}\right )^{2}}\, \operatorname {EllipticE}\left (\frac {x c}{2}+\frac {1}{2 x c}, \sqrt {2}\right )}{\left (\frac {x c}{2}-\frac {1}{2 x c}\right ) \sqrt {2 \left (\frac {x c}{2}+\frac {1}{2 x c}\right )^{2}-1}}\) | \(127\) |
default | \(\frac {2 \left (\frac {x c}{2}+\frac {1}{2 x c}\right ) \left (\frac {x c}{2}-\frac {1}{2 x c}\right )^{2}+\sqrt {-2 \left (\frac {x c}{2}-\frac {1}{2 x c}\right )^{2}-1}\, \sqrt {-\left (\frac {x c}{2}-\frac {1}{2 x c}\right )^{2}}\, \operatorname {EllipticE}\left (\frac {x c}{2}+\frac {1}{2 x c}, \sqrt {2}\right )}{\left (\frac {x c}{2}-\frac {1}{2 x c}\right ) \sqrt {2 \left (\frac {x c}{2}+\frac {1}{2 x c}\right )^{2}-1}}\) | \(127\) |
Input:
int(sech(2*ln(x*c))^(3/2)/x,x,method=_RETURNVERBOSE)
Output:
(2*(1/2*x*c+1/2/x/c)*(1/2*x*c-1/2/x/c)^2+(-2*(1/2*x*c-1/2/x/c)^2-1)^(1/2)* (-(1/2*x*c-1/2/x/c)^2)^(1/2)*EllipticE(1/2*x*c+1/2/x/c,2^(1/2)))/(1/2*x*c- 1/2/x/c)/(2*(1/2*x*c+1/2/x/c)^2-1)^(1/2)
Time = 0.18 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.46 \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\frac {\sqrt {2} \sqrt {\frac {c^{2} x^{2}}{c^{4} x^{4} + 1}} c^{3} x^{2} + \sqrt {2} \left (-c^{4}\right )^{\frac {3}{4}} E(\arcsin \left (\left (-c^{4}\right )^{\frac {1}{4}} x\right )\,|\,-1) - \sqrt {2} \left (-c^{4}\right )^{\frac {3}{4}} F(\arcsin \left (\left (-c^{4}\right )^{\frac {1}{4}} x\right )\,|\,-1)}{c} \] Input:
integrate(sech(2*log(c*x))^(3/2)/x,x, algorithm="fricas")
Output:
(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
\[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int \frac {\operatorname {sech}^{\frac {3}{2}}{\left (2 \log {\left (c x \right )} \right )}}{x}\, dx \] Input:
integrate(sech(2*ln(c*x))**(3/2)/x,x)
Output:
Integral(sech(2*log(c*x))**(3/2)/x, x)
\[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int { \frac {\operatorname {sech}\left (2 \, \log \left (c x\right )\right )^{\frac {3}{2}}}{x} \,d x } \] Input:
integrate(sech(2*log(c*x))^(3/2)/x,x, algorithm="maxima")
Output:
integrate(sech(2*log(c*x))^(3/2)/x, x)
Timed out. \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\text {Timed out} \] Input:
integrate(sech(2*log(c*x))^(3/2)/x,x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int \frac {{\left (\frac {1}{\mathrm {cosh}\left (2\,\ln \left (c\,x\right )\right )}\right )}^{3/2}}{x} \,d x \] Input:
int((1/cosh(2*log(c*x)))^(3/2)/x,x)
Output:
int((1/cosh(2*log(c*x)))^(3/2)/x, x)
\[ \int \frac {\text {sech}^{\frac {3}{2}}(2 \log (c x))}{x} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (2 \,\mathrm {log}\left (c x \right )\right )}\, \mathrm {sech}\left (2 \,\mathrm {log}\left (c x \right )\right )}{x}d x \] Input:
int(sech(2*log(c*x))^(3/2)/x,x)
Output:
int((sqrt(sech(2*log(c*x)))*sech(2*log(c*x)))/x,x)