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

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

[Out]

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

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Rubi [F]  time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}

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Mathematica [C]  time = 3.07, size = 163, normalized size = 3.98 \[ \frac {2}{105} e^{2 a} x^6 \left (e^{2 a} x^4+1\right )^2 \, _4F_3\left (\frac {3}{2},2,2,2;1,1,\frac {9}{2};e^{2 a} x^4\right )+\frac {e^{-4 a} \left (61 e^{6 a} x^{12}-181 e^{4 a} x^8-713 e^{2 a} x^4+\frac {3 \left (e^{8 a} x^{16}-52 e^{6 a} x^{12}-14 e^{4 a} x^8+196 e^{2 a} x^4+125\right ) \tanh ^{-1}\left (\sqrt {e^{2 a} x^4}\right )}{\sqrt {e^{2 a} x^4}}-375\right )}{96 x^6} \]

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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fricas [B]  time = 0.40, size = 74, normalized size = 1.80 \[ \frac {x^{6} e^{\left (3 \, a\right )} - 3 \, x^{2} e^{a} - {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x^{2} e^{a} + 1\right ) + {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x^{2} e^{a} - 1\right )}{2 \, {\left (x^{4} e^{\left (3 \, a\right )} - e^{a}\right )}} \]

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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giac [A]  time = 0.13, size = 54, normalized size = 1.32 \[ \frac {1}{2} \, x^{2} - \frac {1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac {1}{2} \, e^{\left (-a\right )} \log \left ({\left | x^{2} e^{a} - 1 \right |}\right ) - \frac {x^{2}}{x^{4} e^{\left (2 \, a\right )} - 1} \]

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)

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maple [A]  time = 0.09, size = 54, normalized size = 1.32 \[ \frac {x^{2}}{2}-\frac {x^{2}}{-1+{\mathrm e}^{2 a} x^{4}}+\frac {{\mathrm e}^{-a} \ln \left ({\mathrm e}^{a} x^{2}-1\right )}{2}-\frac {{\mathrm e}^{-a} \ln \left ({\mathrm e}^{a} x^{2}+1\right )}{2} \]

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/(-1+exp(2*a)*x^4)+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 = 0.32, size = 53, normalized size = 1.29 \[ \frac {1}{2} \, x^{2} - \frac {1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} + 1\right ) + \frac {1}{2} \, e^{\left (-a\right )} \log \left (x^{2} e^{a} - 1\right ) - \frac {x^{2}}{x^{4} e^{\left (2 \, a\right )} - 1} \]

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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mupad [B]  time = 1.23, size = 42, normalized size = 1.02 \[ \frac {x^2}{2}-\frac {x^2}{x^4\,{\mathrm {e}}^{2\,a}-1}-\frac {\mathrm {atanh}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )}{\sqrt {{\mathrm {e}}^{2\,a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*coth(a + 2*log(x))^2,x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x \coth ^{2}{\left (a + 2 \log {\relax (x )} \right )}\, dx \]

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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