3.164 \(\int \frac {\coth ^2(a+2 \log (x))}{x^3} \, dx\)

Optimal. Leaf size=60 \[ e^a \tanh ^{-1}\left (e^a x^2\right )+\frac {3 e^{2 a} x^2}{2 \left (1-e^{2 a} x^4\right )}-\frac {1}{2 x^2 \left (1-e^{2 a} x^4\right )} \]

[Out]

-1/2/x^2/(1-exp(2*a)*x^4)+3/2*exp(2*a)*x^2/(1-exp(2*a)*x^4)+exp(a)*arctanh(exp(a)*x^2)

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Rubi [F]  time = 0.05, 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 = {} \[ \int \frac {\coth ^2(a+2 \log (x))}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + 2*Log[x]]^2/x^3,x]

[Out]

Defer[Int][Coth[a + 2*Log[x]]^2/x^3, x]

Rubi steps

\begin {align*} \int \frac {\coth ^2(a+2 \log (x))}{x^3} \, dx &=\int \frac {\coth ^2(a+2 \log (x))}{x^3} \, dx\\ \end {align*}

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Mathematica [C]  time = 3.25, size = 155, normalized size = 2.58 \[ \frac {64 \left (e^{3 a} x^6+e^a x^2\right )^2 \, _4F_3\left (\frac {1}{2},2,2,2;1,1,\frac {7}{2};e^{2 a} x^4\right )+15 \left (e^{4 a} x^8-17 e^{2 a} x^4-\frac {27 e^{-2 a}}{x^4}-77\right )-\frac {15 \left (e^{8 a} x^{16}+4 e^{6 a} x^{12}-54 e^{4 a} x^8-52 e^{2 a} x^4-27\right ) \tanh ^{-1}\left (\sqrt {e^{2 a} x^4}\right )}{\left (e^{2 a} x^4\right )^{3/2}}}{480 x^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[a + 2*Log[x]]^2/x^3,x]

[Out]

(15*(-77 - 27/(E^(2*a)*x^4) - 17*E^(2*a)*x^4 + E^(4*a)*x^8) - (15*(-27 - 52*E^(2*a)*x^4 - 54*E^(4*a)*x^8 + 4*E
^(6*a)*x^12 + E^(8*a)*x^16)*ArcTanh[Sqrt[E^(2*a)*x^4]])/(E^(2*a)*x^4)^(3/2) + 64*(E^a*x^2 + E^(3*a)*x^6)^2*Hyp
ergeometricPFQ[{1/2, 2, 2, 2}, {1, 1, 7/2}, E^(2*a)*x^4])/(480*x^2)

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fricas [A]  time = 0.40, size = 82, normalized size = 1.37 \[ -\frac {3 \, x^{4} e^{\left (2 \, a\right )} - {\left (x^{6} e^{\left (3 \, a\right )} - x^{2} e^{a}\right )} \log \left (x^{2} e^{a} + 1\right ) + {\left (x^{6} e^{\left (3 \, a\right )} - x^{2} e^{a}\right )} \log \left (x^{2} e^{a} - 1\right ) - 1}{2 \, {\left (x^{6} e^{\left (2 \, a\right )} - x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*log(x))^2/x^3,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.14, size = 57, normalized size = 0.95 \[ \frac {1}{2} \, e^{a} \log \left (x^{2} e^{a} + 1\right ) - \frac {1}{2} \, e^{a} \log \left ({\left | x^{2} e^{a} - 1 \right |}\right ) - \frac {3 \, x^{4} e^{\left (2 \, a\right )} - 1}{2 \, {\left (x^{6} e^{\left (2 \, a\right )} - x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*log(x))^2/x^3,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.09, size = 55, normalized size = 0.92 \[ \frac {-\frac {3 \,{\mathrm e}^{2 a} x^{4}}{2}+\frac {1}{2}}{x^{2} \left (-1+{\mathrm e}^{2 a} x^{4}\right )}-\frac {{\mathrm e}^{a} \ln \left ({\mathrm e}^{a} x^{2}-1\right )}{2}+\frac {{\mathrm e}^{a} \ln \left ({\mathrm e}^{a} x^{2}+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a+2*ln(x))^2/x^3,x)

[Out]

(-3/2*exp(2*a)*x^4+1/2)/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 = 0.33, size = 50, normalized size = 0.83 \[ \frac {1}{2} \, e^{a} \log \left (\frac {1}{x^{2}} + e^{a}\right ) - \frac {1}{2} \, e^{a} \log \left (\frac {1}{x^{2}} - e^{a}\right ) - \frac {1}{2 \, x^{2}} + \frac {e^{\left (2 \, a\right )}}{x^{2} {\left (\frac {1}{x^{4}} - e^{\left (2 \, a\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*log(x))^2/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.23, size = 48, normalized size = 0.80 \[ \mathrm {atanh}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )\,\sqrt {{\mathrm {e}}^{2\,a}}-\frac {\frac {3\,x^4\,{\mathrm {e}}^{2\,a}}{2}-\frac {1}{2}}{x^6\,{\mathrm {e}}^{2\,a}-x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + 2*log(x))^2/x^3,x)

[Out]

atanh(x^2*exp(2*a)^(1/2))*exp(2*a)^(1/2) - ((3*x^4*exp(2*a))/2 - 1/2)/(x^6*exp(2*a) - x^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{2}{\left (a + 2 \log {\relax (x )} \right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*ln(x))**2/x**3,x)

[Out]

Integral(coth(a + 2*log(x))**2/x**3, x)

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