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*ln(c*x^n)))/b/n

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 20, 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} \[ -\frac {\tanh ^{-1}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

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 = 0.06, size = 54, normalized size = 2.70 \[ \frac {\log \left (\sinh \left (\frac {a}{2}+\frac {1}{2} b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\log \left (\cosh \left (\frac {a}{2}+\frac {1}{2} b \log \left (c x^n\right )\right )\right )}{b n} \]

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)

________________________________________________________________________________________

fricas [B]  time = 0.60, size = 65, normalized size = 3.25 \[ -\frac {\log \left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + 1\right ) - \log \left (\cosh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) + \sinh \left (b n \log \relax (x) + b \log \relax (c) + a\right ) - 1\right )}{b n} \]

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)

________________________________________________________________________________________

giac [B]  time = 0.58, size = 145, normalized size = 7.25 \[ -c^{b} {\left (\frac {c^{b} e^{\left (-a\right )} \log \left (\sqrt {2 \, x^{b n} {\left | c \right |}^{b} \cos \left (-\frac {1}{2} \, \pi b \mathrm {sgn}\relax (c) + \frac {1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n} {\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1}\right )}{b c^{2 \, b} n} - \frac {c^{b} e^{\left (-a\right )} \log \left (\sqrt {-2 \, x^{b n} {\left | c \right |}^{b} \cos \left (-\frac {1}{2} \, \pi b \mathrm {sgn}\relax (c) + \frac {1}{2} \, \pi b\right ) e^{a} + x^{2 \, b n} {\left | c \right |}^{2 \, b} e^{\left (2 \, a\right )} + 1}\right )}{b c^{2 \, b} n}\right )} e^{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-c^b*(c^b*e^(-a)*log(sqrt(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(sqrt(-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))*e^a

________________________________________________________________________________________

maple [A]  time = 0.09, size = 23, normalized size = 1.15 \[ \frac {\ln \left (\tanh \left (\frac {a}{2}+\frac {b \ln \left (c \,x^{n}\right )}{2}\right )\right )}{n b} \]

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)))

________________________________________________________________________________________

maxima [A]  time = 0.31, size = 22, normalized size = 1.10 \[ \frac {\log \left (\tanh \left (\frac {1}{2} \, b \log \left (c x^{n}\right ) + \frac {1}{2} \, a\right )\right )}{b n} \]

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)

________________________________________________________________________________________

mupad [B]  time = 1.62, size = 43, normalized size = 2.15 \[ -\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {-b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{\sqrt {-b^2\,n^2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*sinh(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 \[ \int \frac {\operatorname {csch}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

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)

________________________________________________________________________________________