3.188 \(\int \frac{\coth ^2(d (a+b \log (c x^n)))}{x} \, dx\)

Optimal. Leaf size=28 \[ \log (x)-\frac{\coth \left (a d+b d \log \left (c x^n\right )\right )}{b d n} \]

[Out]

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

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Rubi [A]  time = 0.0291035, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {3473, 8} \[ \log (x)-\frac{\coth \left (a d+b d \log \left (c x^n\right )\right )}{b d n} \]

Antiderivative was successfully verified.

[In]

Int[Coth[d*(a + b*Log[c*x^n])]^2/x,x]

[Out]

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

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \frac{\coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \coth ^2(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\coth \left (a d+b d \log \left (c x^n\right )\right )}{b d n}+\frac{\operatorname{Subst}\left (\int 1 \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\coth \left (a d+b d \log \left (c x^n\right )\right )}{b d n}+\log (x)\\ \end{align*}

Mathematica [C]  time = 0.103764, size = 49, normalized size = 1.75 \[ -\frac{\coth \left (a d+b d \log \left (c x^n\right )\right ) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\tanh ^2\left (a d+b \log \left (c x^n\right ) d\right )\right )}{b d n} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[d*(a + b*Log[c*x^n])]^2/x,x]

[Out]

-((Coth[a*d + b*d*Log[c*x^n]]*Hypergeometric2F1[-1/2, 1, 1/2, Tanh[a*d + b*d*Log[c*x^n]]^2])/(b*d*n))

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Maple [B]  time = 0.005, size = 80, normalized size = 2.9 \begin{align*} -{\frac{{\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right )}{dbn}}-{\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)))^2/x,x)

[Out]

-1/b/d/n*coth(d*(a+b*ln(c*x^n)))-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 = 1.30048, size = 50, normalized size = 1.79 \begin{align*} -\frac{2}{b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - 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)))^2/x,x, algorithm="maxima")

[Out]

-2/(b*c^(2*b*d)*d*n*e^(2*b*d*log(x^n) + 2*a*d) - b*d*n) + log(x)

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Fricas [B]  time = 2.51242, size = 197, normalized size = 7.04 \begin{align*} \frac{{\left (b d n \log \left (x\right ) + 1\right )} \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 )}{b d n \sinh \left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*(a+b*log(c*x^n)))^2/x,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth ^{2}{\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)))**2/x,x)

[Out]

Integral(coth(a*d + b*d*log(c*x**n))**2/x, x)

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Giac [A]  time = 1.33036, size = 50, normalized size = 1.79 \begin{align*} -\frac{2}{{\left (c^{2 \, b d} x^{2 \, b d n} e^{\left (2 \, a d\right )} - 1\right )} 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)))^2/x,x, algorithm="giac")

[Out]

-2/((c^(2*b*d)*x^(2*b*d*n)*e^(2*a*d) - 1)*b*d*n) + log(x)