3.191 \(\int \frac {\text {sech}(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=19 \[ \frac {\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

[Out]

arctan(sinh(a+b*ln(c*x^n)))/b/n

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3770} \[ \frac {\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*Log[c*x^n]]/x,x]

[Out]

ArcTan[Sinh[a + b*Log[c*x^n]]]/(b*n)

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\text {sech}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \text {sech}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac {\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 19, normalized size = 1.00 \[ \frac {\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + b*Log[c*x^n]]/x,x]

[Out]

ArcTan[Sinh[a + b*Log[c*x^n]]]/(b*n)

________________________________________________________________________________________

fricas [A]  time = 0.42, size = 34, normalized size = 1.79 \[ \frac {2 \, \arctan \left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right )\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a))/(b*n)

________________________________________________________________________________________

giac [A]  time = 0.12, size = 27, normalized size = 1.42 \[ \frac {2 \, \arctan \left (\frac {c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*arctan(c^(2*b)*x^(b*n)*e^a/c^b)/(b*n)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 20, normalized size = 1.05 \[ \frac {\arctan \left (\sinh \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(a+b*ln(c*x^n))/x,x)

[Out]

arctan(sinh(a+b*ln(c*x^n)))/b/n

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 19, normalized size = 1.00 \[ \frac {\arctan \left (\sinh \left (b \log \left (c x^{n}\right ) + a\right )\right )}{b n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(sinh(b*log(c*x^n) + a))/(b*n)

________________________________________________________________________________________

mupad [B]  time = 1.41, size = 41, normalized size = 2.16 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{\sqrt {b^2\,n^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*cosh(a + b*log(c*x^n))),x)

[Out]

-(2*atan((exp(-a)*(b^2*n^2)^(1/2))/(b*n*(c*x^n)^b)))/(b^2*n^2)^(1/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________