3.179 \(\int x \coth (d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=54 \[ \frac{x^2}{2}-x^2 \, _2F_1\left (1,\frac{1}{b d n};1+\frac{1}{b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right ) \]

[Out]

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

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Rubi [F]  time = 0.0239149, 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 = {} \[ \int x \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*Coth[d*(a + b*Log[c*x^n])],x]

[Out]

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

Rubi steps

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

Mathematica [B]  time = 6.93912, size = 193, normalized size = 3.57 \[ -\frac{x^2 \left (e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac{1}{b d n};2+\frac{1}{b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+(b d n+1) \left (\, _2F_1\left (1,\frac{1}{b d n};1+\frac{1}{b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\coth \left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )+\sinh (b d n \log (x)) \text{csch}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \text{csch}\left (d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )\right )\right )\right )}{2 b d n+2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 1., size = 0, normalized size = 0. \begin{align*} \int x{\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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