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

Optimal. Leaf size=58 \[ \frac{2 \, _2F_1\left (1,-\frac{1}{2 b d n};1-\frac{1}{2 b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{x}-\frac{1}{x} \]

[Out]

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

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Rubi [F]  time = 0.0313433, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[d*(a + b*Log[c*x^n])]/x^2,x]

[Out]

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

Rubi steps

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

Mathematica [B]  time = 3.82998, size = 197, normalized size = 3.4 \[ \frac{-\frac{e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1-\frac{1}{2 b d n};2-\frac{1}{2 b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )}{2 b d n-1}+\, _2F_1\left (1,-\frac{1}{2 b d n};1-\frac{1}{2 b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\coth \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+\sinh (b d n \log (x)) \text{csch}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \text{csch}\left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )}{x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Coth[d*(a + b*Log[c*x^n])] - Coth[d*(a - b*n*Log[x] + b*Log[c*x^n])] - (E^(2*d*(a + b*Log[c*x^n]))*Hypergeome
tric2F1[1, 1 - 1/(2*b*d*n), 2 - 1/(2*b*d*n), E^(2*d*(a + b*Log[c*x^n]))])/(-1 + 2*b*d*n) + Hypergeometric2F1[1
, -1/(2*b*d*n), 1 - 1/(2*b*d*n), E^(2*d*(a + b*Log[c*x^n]))] + Csch[d*(a + b*Log[c*x^n])]*Csch[d*(a - b*n*Log[
x] + b*Log[c*x^n])]*Sinh[b*d*n*Log[x]])/x

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Maple [F]  time = 1.026, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right )}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d*(a+b*ln(c*x^n)))/x^2,x)

[Out]

int(coth(d*(a+b*ln(c*x^n)))/x^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{x} - \int \frac{1}{c^{b d} x^{2} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} + x^{2}}\,{d x} + \int \frac{1}{c^{b d} x^{2} e^{\left (b d \log \left (x^{n}\right ) + a d\right )} - x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\coth \left (b d \log \left (c x^{n}\right ) + a d\right )}{x^{2}}, 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^2,x, algorithm="fricas")

[Out]

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

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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^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(d*(a+b*ln(c*x**n)))/x**2,x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth \left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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