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

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

[Out]

-1/(2*x^2*(1 - E^(2*a)*x^4)) + (3*E^(2*a)*x^2)/(2*(1 - E^(2*a)*x^4)) + E^a*ArcTanh[E^a*x^2]

________________________________________________________________________________________

Rubi [F]  time = 0.0460271, 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 = {} \[ \int \frac{\coth ^2(a+2 \log (x))}{x^3} \, dx \]

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*}

Mathematica [C]  time = 2.86299, size = 155, normalized size = 2.58 \[ \frac{64 \left (e^{3 a} x^6+e^a x^2\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},2,2,2\right \},\left \{1,1,\frac{7}{2}\right \},e^{2 a} x^4\right )+15 \left (e^{4 a} x^8-17 e^{2 a} x^4-\frac{27 e^{-2 a}}{x^4}-77\right )-\frac{15 \left (e^{8 a} x^{16}+4 e^{6 a} x^{12}-54 e^{4 a} x^8-52 e^{2 a} x^4-27\right ) \tanh ^{-1}\left (\sqrt{e^{2 a} x^4}\right )}{\left (e^{2 a} x^4\right )^{3/2}}}{480 x^2} \]

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)

________________________________________________________________________________________

Maple [A]  time = 0.023, size = 55, normalized size = 0.9 \begin{align*}{\frac{1}{{x}^{2} \left ({{\rm e}^{2\,a}}{x}^{4}-1 \right ) } \left ( -{\frac{3\,{{\rm e}^{2\,a}}{x}^{4}}{2}}+{\frac{1}{2}} \right ) }+{\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(coth(a+2*ln(x))^2/x^3,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.07843, size = 68, normalized size = 1.13 \begin{align*} \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 )}} \end{align*}

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)))

________________________________________________________________________________________

Fricas [A]  time = 2.58052, size = 178, normalized size = 2.97 \begin{align*} -\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 )}} \end{align*}

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)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth ^{2}{\left (a + 2 \log{\left (x \right )} \right )}}{x^{3}}\, dx \end{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 1.15239, size = 77, normalized size = 1.28 \begin{align*} \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 )}} \end{align*}

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)