3.186 \(\int x \coth ^2(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=130 \[ -\frac{2 x^2 \, _2F_1\left (1,\frac{1}{b d n};1+\frac{1}{b d n};e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{b d n}+\frac{x^2 \left (e^{2 a d} \left (c x^n\right )^{2 b d}+1\right )}{b d n \left (1-e^{2 a d} \left (c x^n\right )^{2 b d}\right )}+\frac{1}{2} x^2 \left (\frac{2}{b d n}+1\right ) \]

[Out]

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

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Rubi [F]  time = 0.0427788, 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 = {} \[ \int x \coth ^2\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*Coth[d*(a + b*Log[c*x^n])]^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 6.48103, size = 151, normalized size = 1.16 \[ \frac{x^2 \left ((b d n+1) \left (-2 \, _2F_1\left (1,\frac{1}{b d n};1+\frac{1}{b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )-2 \coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )+b d n\right )-2 e^{2 d \left (a+b \log \left (c x^n\right )\right )} \, _2F_1\left (1,1+\frac{1}{b d n};2+\frac{1}{b d n};e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{2 b d n (b d n+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.11, size = 0, normalized size = 0. \begin{align*} \int x \left ({\rm coth} \left (d \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{b c^{2 \, b d} d n x^{2} e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} -{\left (b d n + 4\right )} x^{2}}{2 \,{\left (b c^{2 \, b d} d n e^{\left (2 \, b d \log \left (x^{n}\right ) + 2 \, a d\right )} - b d n\right )}} - 2 \, \int \frac{x}{b c^{b d} d n e^{\left (b d \log \left (x^{n}\right ) + a d\right )} + b d n}\,{d x} + 2 \, \int \frac{x}{b c^{b d} d n e^{\left (b d \log \left (x^{n}\right ) + a d\right )} - b d n}\,{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)))^2,x, algorithm="maxima")

[Out]

1/2*(b*c^(2*b*d)*d*n*x^2*e^(2*b*d*log(x^n) + 2*a*d) - (b*d*n + 4)*x^2)/(b*c^(2*b*d)*d*n*e^(2*b*d*log(x^n) + 2*
a*d) - b*d*n) - 2*integrate(x/(b*c^(b*d)*d*n*e^(b*d*log(x^n) + a*d) + b*d*n), x) + 2*integrate(x/(b*c^(b*d)*d*
n*e^(b*d*log(x^n) + a*d) - b*d*n), 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 )^{2}, 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)))^2,x, algorithm="fricas")

[Out]

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

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

[Out]

Integral(x*coth(a*d + b*d*log(c*x**n))**2, 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 )^{2}\,{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)))^2,x, algorithm="giac")

[Out]

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