3.160 \(\int x \coth ^2(a+2 \log (x)) \, dx\)

Optimal. Leaf size=41 \[ \frac{x^2}{1-e^{2 a} x^4}-e^{-a} \tanh ^{-1}\left (e^a x^2\right )+\frac{x^2}{2} \]

[Out]

x^2/2 + x^2/(1 - E^(2*a)*x^4) - ArcTanh[E^a*x^2]/E^a

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Rubi [F]  time = 0.0298197, 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 = {} \[ \int x \coth ^2(a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x*Coth[a + 2*Log[x]]^2,x]

[Out]

Defer[Int][x*Coth[a + 2*Log[x]]^2, x]

Rubi steps

\begin{align*} \int x \coth ^2(a+2 \log (x)) \, dx &=\int x \coth ^2(a+2 \log (x)) \, dx\\ \end{align*}

Mathematica [C]  time = 2.93267, size = 163, normalized size = 3.98 \[ \frac{2}{105} e^{2 a} x^6 \left (e^{2 a} x^4+1\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{3}{2},2,2,2\right \},\left \{1,1,\frac{9}{2}\right \},e^{2 a} x^4\right )+\frac{e^{-4 a} \left (61 e^{6 a} x^{12}-181 e^{4 a} x^8-713 e^{2 a} x^4+\frac{3 \left (e^{8 a} x^{16}-52 e^{6 a} x^{12}-14 e^{4 a} x^8+196 e^{2 a} x^4+125\right ) \tanh ^{-1}\left (\sqrt{e^{2 a} x^4}\right )}{\sqrt{e^{2 a} x^4}}-375\right )}{96 x^6} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x*Coth[a + 2*Log[x]]^2,x]

[Out]

(-375 - 713*E^(2*a)*x^4 - 181*E^(4*a)*x^8 + 61*E^(6*a)*x^12 + (3*(125 + 196*E^(2*a)*x^4 - 14*E^(4*a)*x^8 - 52*
E^(6*a)*x^12 + E^(8*a)*x^16)*ArcTanh[Sqrt[E^(2*a)*x^4]])/Sqrt[E^(2*a)*x^4])/(96*E^(4*a)*x^6) + (2*E^(2*a)*x^6*
(1 + E^(2*a)*x^4)^2*HypergeometricPFQ[{3/2, 2, 2, 2}, {1, 1, 9/2}, E^(2*a)*x^4])/105

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Maple [A]  time = 0.02, size = 54, normalized size = 1.3 \begin{align*}{\frac{{x}^{2}}{2}}-{\frac{{x}^{2}}{{{\rm e}^{2\,a}}{x}^{4}-1}}+{\frac{{{\rm e}^{-a}}\ln \left ({{\rm e}^{a}}{x}^{2}-1 \right ) }{2}}-{\frac{{{\rm e}^{-a}}\ln \left ({{\rm e}^{a}}{x}^{2}+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*coth(a+2*ln(x))^2,x)

[Out]

1/2*x^2-x^2/(exp(2*a)*x^4-1)+1/2*exp(-a)*ln(exp(a)*x^2-1)-1/2*exp(-a)*ln(exp(a)*x^2+1)

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Maxima [A]  time = 1.06055, size = 72, normalized size = 1.76 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac{1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} - 1\right ) - \frac{x^{2}}{x^{4} e^{\left (2 \, a\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*coth(a+2*log(x))^2,x, algorithm="maxima")

[Out]

1/2*x^2 - 1/2*e^(-a)*log(x^2*e^a + 1) + 1/2*e^(-a)*log(x^2*e^a - 1) - x^2/(x^4*e^(2*a) - 1)

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Fricas [B]  time = 2.57195, size = 169, normalized size = 4.12 \begin{align*} \frac{x^{6} e^{\left (3 \, a\right )} - 3 \, x^{2} e^{a} -{\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x^{2} e^{a} + 1\right ) +{\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x^{2} e^{a} - 1\right )}{2 \,{\left (x^{4} e^{\left (3 \, a\right )} - e^{a}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*coth(a+2*log(x))^2,x, algorithm="fricas")

[Out]

1/2*(x^6*e^(3*a) - 3*x^2*e^a - (x^4*e^(2*a) - 1)*log(x^2*e^a + 1) + (x^4*e^(2*a) - 1)*log(x^2*e^a - 1))/(x^4*e
^(3*a) - e^a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \coth ^{2}{\left (a + 2 \log{\left (x \right )} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*coth(a+2*ln(x))**2,x)

[Out]

Integral(x*coth(a + 2*log(x))**2, x)

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Giac [A]  time = 1.11119, size = 73, normalized size = 1.78 \begin{align*} \frac{1}{2} \, x^{2} - \frac{1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac{1}{2} \, e^{\left (-a\right )} \log \left ({\left | x^{2} e^{a} - 1 \right |}\right ) - \frac{x^{2}}{x^{4} e^{\left (2 \, a\right )} - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*coth(a+2*log(x))^2,x, algorithm="giac")

[Out]

1/2*x^2 - 1/2*e^(-a)*log(x^2*e^a + 1) + 1/2*e^(-a)*log(abs(x^2*e^a - 1)) - x^2/(x^4*e^(2*a) - 1)