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

Optimal. Leaf size=20 \[ -\frac{\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

[Out]

-(ArcTanh[Cosh[a + b*Log[c*x^n]]]/(b*n))

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Rubi [A]  time = 0.0175591, antiderivative size = 20, 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} \[ -\frac{\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Int[Csch[a + b*Log[c*x^n]]/x,x]

[Out]

-(ArcTanh[Cosh[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{csch}\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \text{csch}(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ \end{align*}

Mathematica [B]  time = 0.0638428, size = 54, normalized size = 2.7 \[ \frac{\log \left (\sinh \left (\frac{a}{2}+\frac{1}{2} b \log \left (c x^n\right )\right )\right )}{b n}-\frac{\log \left (\cosh \left (\frac{a}{2}+\frac{1}{2} b \log \left (c x^n\right )\right )\right )}{b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[a + b*Log[c*x^n]]/x,x]

[Out]

-(Log[Cosh[a/2 + (b*Log[c*x^n])/2]]/(b*n)) + Log[Sinh[a/2 + (b*Log[c*x^n])/2]]/(b*n)

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Maple [A]  time = 0.008, size = 23, normalized size = 1.2 \begin{align*}{\frac{1}{bn}\ln \left ( \tanh \left ({\frac{a}{2}}+{\frac{b\ln \left ( c{x}^{n} \right ) }{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/n/b*ln(tanh(1/2*a+1/2*b*ln(c*x^n)))

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Maxima [A]  time = 1.09665, size = 30, normalized size = 1.5 \begin{align*} \frac{\log \left (\tanh \left (\frac{1}{2} \, b \log \left (c x^{n}\right ) + \frac{1}{2} \, a\right )\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(tanh(1/2*b*log(c*x^n) + 1/2*a))/(b*n)

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Fricas [B]  time = 1.84175, size = 219, normalized size = 10.95 \begin{align*} -\frac{\log \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 ) + 1\right ) - \log \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 ) - 1\right )}{b n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(log(cosh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a) + 1) - log(cosh(b*n*log(x) + b*log(c)
+ a) + sinh(b*n*log(x) + b*log(c) + a) - 1))/(b*n)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}{\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(csch(a+b*ln(c*x**n))/x,x)

[Out]

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

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Giac [B]  time = 1.21073, size = 190, normalized size = 9.5 \begin{align*} -\frac{1}{2} \, c^{b}{\left (\frac{c^{b} e^{\left (-a\right )} \log \left (2 \, x^{b n}{\left | c \right |}^{b} \cos \left (-\frac{1}{2} \, \pi b \mathrm{sgn}\left (c\right ) + \frac{1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n}{\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1\right )}{b c^{2 \, b} n} - \frac{c^{b} e^{\left (-a\right )} \log \left (-2 \, x^{b n}{\left | c \right |}^{b} \cos \left (-\frac{1}{2} \, \pi b \mathrm{sgn}\left (c\right ) + \frac{1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n}{\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1\right )}{b c^{2 \, b} n}\right )} e^{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*c^b*(c^b*e^(-a)*log(2*x^(b*n)*abs(c)^b*cos(-1/2*pi*b*sgn(c) + 1/2*pi*b)*e^a + x^(2*b*n)*abs(c)^(2*b)*e^(2
*a) + 1)/(b*c^(2*b)*n) - c^b*e^(-a)*log(-2*x^(b*n)*abs(c)^b*cos(-1/2*pi*b*sgn(c) + 1/2*pi*b)*e^a + x^(2*b*n)*a
bs(c)^(2*b)*e^(2*a) + 1)/(b*c^(2*b)*n))*e^a