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*Log[c*x^n]]]/(b*n)

________________________________________________________________________________________

Rubi [A]  time = 0.0162663, antiderivative size = 19, normalized size of antiderivative = 1., 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 = 0.0480611, size = 19, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

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)

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 20, normalized size = 1.1 \begin{align*}{\frac{\arctan \left ( \sinh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) }{bn}} \end{align*}

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 = 1.07044, size = 26, normalized size = 1.37 \begin{align*} \frac{\arctan \left (\sinh \left (b \log \left (c x^{n}\right ) + a\right )\right )}{b n} \end{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 3.13184, size = 112, normalized size = 5.89 \begin{align*} \frac{2 \, \arctan \left (\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 )}{b n} \end{align*}

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)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}{\left (a + b \log{\left (c x^{n} \right )} \right )}}{x}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.12854, size = 36, normalized size = 1.89 \begin{align*} \frac{2 \, \arctan \left (\frac{c^{2 \, b} x^{b n} e^{a}}{c^{b}}\right )}{b n} \end{align*}

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)