\(\int \frac {1}{x \sqrt {\text {sech}(a+b \log (c x^n))}} \, dx\) [199]

Optimal result

Integrand size = 19, antiderivative size = 58 \[ \int \frac {1}{x \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 i \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{b n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3856, 2719} \[ \int \frac {1}{x \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 i \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )} \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n} \]

[In]

Int[1/(x*Sqrt[Sech[a + b*Log[c*x^n]]]),x]

[Out]

((-2*I)*Sqrt[Cosh[a + b*Log[c*x^n]]]*EllipticE[(I/2)*(a + b*Log[c*x^n]), 2]*Sqrt[Sech[a + b*Log[c*x^n]]])/(b*n
)

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 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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\sqrt {\text {sech}(a+b x)}} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\left (\sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}\right ) \text {Subst}\left (\int \sqrt {\cosh (a+b x)} \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {2 i \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right ) \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}}{b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {2 i E\left (\left .\frac {1}{2} i \left (a+b \log \left (c x^n\right )\right )\right |2\right )}{b n \sqrt {\cosh \left (a+b \log \left (c x^n\right )\right )} \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \]

[In]

Integrate[1/(x*Sqrt[Sech[a + b*Log[c*x^n]]]),x]

[Out]

((-2*I)*EllipticE[(I/2)*(a + b*Log[c*x^n]), 2])/(b*n*Sqrt[Cosh[a + b*Log[c*x^n]]]*Sqrt[Sech[a + b*Log[c*x^n]]]
)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(182\) vs. \(2(90)=180\).

Time = 1.59 (sec) , antiderivative size = 183, normalized size of antiderivative = 3.16

method	result	size
derivativedivides	\(-\frac {2 \sqrt {\left (-1+2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+1}\, \operatorname {EllipticE}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{n \sqrt {2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {-1+2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, b}\)	\(183\)
default	\(-\frac {2 \sqrt {\left (-1+2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}\right ) {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sqrt {-2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}+1}\, \operatorname {EllipticE}\left (\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ), \sqrt {2}\right )}{n \sqrt {2 {\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{4}+{\sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, \sinh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right ) \sqrt {-1+2 {\cosh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )}^{2}}\, b}\)	\(183\)

[In]

int(1/x/sech(a+b*ln(c*x^n))^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/n*((-1+2*cosh(1/2*a+1/2*b*ln(c*x^n))^2)*sinh(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)*(-sinh(1/2*a+1/2*b*ln(c*x^n))^
2)^(1/2)*(-2*cosh(1/2*a+1/2*b*ln(c*x^n))^2+1)^(1/2)*EllipticE(cosh(1/2*a+1/2*b*ln(c*x^n)),2^(1/2))/(2*sinh(1/2
*a+1/2*b*ln(c*x^n))^4+sinh(1/2*a+1/2*b*ln(c*x^n))^2)^(1/2)/sinh(1/2*a+1/2*b*ln(c*x^n))/(-1+2*cosh(1/2*a+1/2*b*
ln(c*x^n))^2)^(1/2)/b

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.09 (sec) , antiderivative size = 248, normalized size of antiderivative = 4.28 \[ \int \frac {1}{x \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \, dx=-\frac {\sqrt {2} {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1\right )} \sqrt {\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 1}} + 2 \, {\left (\sqrt {2} \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sqrt {2} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )\right )}{b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \]

[In]

integrate(1/x/sech(a+b*log(c*x^n))^(1/2),x, algorithm="fricas")

[Out]

-(sqrt(2)*(cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) +
a) + sinh(b*n*log(x) + b*log(c) + a)^2 + 1)*sqrt((cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c)
+ a))/(cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)
+ sinh(b*n*log(x) + b*log(c) + a)^2 + 1)) + 2*(sqrt(2)*cosh(b*n*log(x) + b*log(c) + a) + sqrt(2)*sinh(b*n*log(
x) + b*log(c) + a))*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(b*n*log(x) + b*log(c) + a) + sinh(b
*n*log(x) + b*log(c) + a))))/(b*n*cosh(b*n*log(x) + b*log(c) + a) + b*n*sinh(b*n*log(x) + b*log(c) + a))

Sympy [F]

\[ \int \frac {1}{x \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {1}{x \sqrt {\operatorname {sech}{\left (a + b \log {\left (c x^{n} \right )} \right )}}}\, dx \]

[In]

integrate(1/x/sech(a+b*ln(c*x**n))**(1/2),x)

[Out]

Integral(1/(x*sqrt(sech(a + b*log(c*x**n)))), x)

Maxima [F]

\[ \int \frac {1}{x \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \, dx=\int { \frac {1}{x \sqrt {\operatorname {sech}\left (b \log \left (c x^{n}\right ) + a\right )}} \,d x } \]

[In]

integrate(1/x/sech(a+b*log(c*x^n))^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(x*sqrt(sech(b*log(c*x^n) + a))), x)

Giac [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \, dx=\text {Timed out} \]

[In]

integrate(1/x/sech(a+b*log(c*x^n))^(1/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x \sqrt {\text {sech}\left (a+b \log \left (c x^n\right )\right )}} \, dx=\int \frac {1}{x\,\sqrt {\frac {1}{\mathrm {cosh}\left (a+b\,\ln \left (c\,x^n\right )\right )}}} \,d x \]

[In]

int(1/(x*(1/cosh(a + b*log(c*x^n)))^(1/2)),x)

[Out]

int(1/(x*(1/cosh(a + b*log(c*x^n)))^(1/2)), x)