3.181 \(\int \frac{\coth (d (a+b \log (c x^n)))}{x} \, dx\)

Optimal. Leaf size=25 \[ \frac{\log \left (\sinh \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \]

[Out]

Log[Sinh[a*d + b*d*Log[c*x^n]]]/(b*d*n)

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Rubi [A]  time = 0.0206705, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {3475} \[ \frac{\log \left (\sinh \left (a d+b d \log \left (c x^n\right )\right )\right )}{b d n} \]

Antiderivative was successfully verified.

[In]

Int[Coth[d*(a + b*Log[c*x^n])]/x,x]

[Out]

Log[Sinh[a*d + b*d*Log[c*x^n]]]/(b*d*n)

Rule 3475

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

Rubi steps

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

Mathematica [A]  time = 0.0569006, size = 40, normalized size = 1.6 \[ \frac{\log \left (\tanh \left (a d+b d \log \left (c x^n\right )\right )\right )+\log \left (\cosh \left (d \left (a+b \log \left (c x^n\right )\right )\right )\right )}{b d n} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[d*(a + b*Log[c*x^n])]/x,x]

[Out]

(Log[Cosh[d*(a + b*Log[c*x^n])]] + Log[Tanh[a*d + b*d*Log[c*x^n]]])/(b*d*n)

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Maple [B]  time = 0.004, size = 56, normalized size = 2.2 \begin{align*} -{\frac{\ln \left ({\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right )-1 \right ) }{2\,dbn}}-{\frac{\ln \left ({\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right )+1 \right ) }{2\,dbn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/b/d/n*ln(coth(d*(a+b*ln(c*x^n)))-1)-1/2/b/d/n*ln(coth(d*(a+b*ln(c*x^n)))+1)

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Maxima [A]  time = 1.15765, size = 32, normalized size = 1.28 \begin{align*} \frac{\log \left (\sinh \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )\right )}{b d n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(sinh((b*log(c*x^n) + a)*d))/(b*d*n)

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Fricas [B]  time = 2.91355, size = 205, normalized size = 8.2 \begin{align*} -\frac{b d n \log \left (x\right ) - \log \left (\frac{2 \, \sinh \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}{\cosh \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) - \sinh \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}\right )}{b d n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.24077, size = 99, normalized size = 3.96 \begin{align*} \frac{\log \left (-2 \, x^{2 \, b d n}{\left | c \right |}^{2 \, b d} \cos \left (\pi b d \mathrm{sgn}\left (c\right ) - \pi b d\right ) e^{\left (2 \, a d\right )} + x^{4 \, b d n}{\left | c \right |}^{4 \, b d} e^{\left (4 \, a d\right )} + 1\right )}{2 \, b d n} - \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(-2*x^(2*b*d*n)*abs(c)^(2*b*d)*cos(pi*b*d*sgn(c) - pi*b*d)*e^(2*a*d) + x^(4*b*d*n)*abs(c)^(4*b*d)*e^(4*
a*d) + 1)/(b*d*n) - log(x)