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

Optimal. Leaf size=86 \[ -\frac {1}{x \left (1-e^{2 a} x^4\right )}+\frac {2 e^{2 a} x^3}{1-e^{2 a} x^4}-\frac {1}{2} e^{a/2} \tan ^{-1}\left (e^{a/2} x\right )+\frac {1}{2} e^{a/2} \tanh ^{-1}\left (e^{a/2} x\right ) \]

[Out]

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

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Rubi [F]  time = 0.04, 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 = {} \[ \int \frac {\coth ^2(a+2 \log (x))}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + 2*Log[x]]^2/x^2,x]

[Out]

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

Rubi steps

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

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Mathematica [C]  time = 2.85, size = 153, normalized size = 1.78 \[ \frac {16}{231} e^{2 a} x^3 \left (e^{2 a} x^4+1\right )^2 \, _4F_3\left (\frac {3}{4},2,2,2;1,1,\frac {15}{4};e^{2 a} x^4\right )+\frac {e^{-2 a} \left (\left (-e^{8 a} x^{16}-56 e^{6 a} x^{12}+362 e^{4 a} x^8+632 e^{2 a} x^4+343\right ) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};e^{2 a} x^4\right )+3 e^{6 a} x^{12}-241 e^{4 a} x^8-1163 e^{2 a} x^4-343\right )}{384 x^5} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[a + 2*Log[x]]^2/x^2,x]

[Out]

(-343 - 1163*E^(2*a)*x^4 - 241*E^(4*a)*x^8 + 3*E^(6*a)*x^12 + (343 + 632*E^(2*a)*x^4 + 362*E^(4*a)*x^8 - 56*E^
(6*a)*x^12 - E^(8*a)*x^16)*Hypergeometric2F1[3/4, 1, 7/4, E^(2*a)*x^4])/(384*E^(2*a)*x^5) + (16*E^(2*a)*x^3*(1
+ E^(2*a)*x^4)^2*HypergeometricPFQ[{3/4, 2, 2, 2}, {1, 1, 15/4}, E^(2*a)*x^4])/231

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fricas [A]  time = 0.41, size = 97, normalized size = 1.13 \[ -\frac {8 \, x^{4} e^{\left (2 \, a\right )} + 2 \, {\left (x^{5} e^{\left (2 \, a\right )} - x\right )} \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - {\left (x^{5} e^{\left (2 \, a\right )} - x\right )} e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {x^{2} e^{a} + 2 \, x e^{\left (\frac {1}{2} \, a\right )} + 1}{x^{2} e^{a} - 1}\right ) - 4}{4 \, {\left (x^{5} e^{\left (2 \, a\right )} - x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(8*x^4*e^(2*a) + 2*(x^5*e^(2*a) - x)*arctan(x*e^(1/2*a))*e^(1/2*a) - (x^5*e^(2*a) - x)*e^(1/2*a)*log((x^2
*e^a + 2*x*e^(1/2*a) + 1)/(x^2*e^a - 1)) - 4)/(x^5*e^(2*a) - x)

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giac [A]  time = 0.12, size = 77, normalized size = 0.90 \[ -\frac {1}{2} \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - \frac {1}{4} \, e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {{\left | 2 \, x e^{a} - 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}{{\left | 2 \, x e^{a} + 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}\right ) - \frac {2 \, x^{4} e^{\left (2 \, a\right )} - 1}{x^{5} e^{\left (2 \, a\right )} - x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*arctan(x*e^(1/2*a))*e^(1/2*a) - 1/4*e^(1/2*a)*log(abs(2*x*e^a - 2*e^(1/2*a))/abs(2*x*e^a + 2*e^(1/2*a)))
- (2*x^4*e^(2*a) - 1)/(x^5*e^(2*a) - x)

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maple [C]  time = 0.13, size = 114, normalized size = 1.33 \[ \frac {-2 \,{\mathrm e}^{2 a} x^{4}+1}{x \left (-1+{\mathrm e}^{2 a} x^{4}\right )}+\frac {\sqrt {-{\mathrm e}^{a}}\, \ln \left (-{\mathrm e}^{2 a} x -\left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{4}-\frac {\sqrt {-{\mathrm e}^{a}}\, \ln \left (-{\mathrm e}^{2 a} x +\left (-{\mathrm e}^{a}\right )^{\frac {3}{2}}\right )}{4}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\textit {\_Z}^{2}-{\mathrm e}^{a}\right )}{\sum }\textit {\_R} \ln \left (\left (-5 \textit {\_R}^{4}+4 \,{\mathrm e}^{2 a}\right ) x -\textit {\_R}^{3}\right )\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-2*exp(2*a)*x^4+1)/x/(-1+exp(2*a)*x^4)+1/4*(-exp(a))^(1/2)*ln(-exp(2*a)*x-(-exp(a))^(3/2))-1/4*(-exp(a))^(1/2
)*ln(-exp(2*a)*x+(-exp(a))^(3/2))+1/4*sum(_R*ln((-5*_R^4+4*exp(2*a))*x-_R^3),_R=RootOf(_Z^2-exp(a)))

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maxima [A]  time = 0.42, size = 69, normalized size = 0.80 \[ \frac {1}{2} \, \arctan \left (\frac {e^{\left (-\frac {1}{2} \, a\right )}}{x}\right ) e^{\left (\frac {1}{2} \, a\right )} - \frac {1}{4} \, e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {\frac {1}{x} - e^{\left (\frac {1}{2} \, a\right )}}{\frac {1}{x} + e^{\left (\frac {1}{2} \, a\right )}}\right ) - \frac {1}{x} + \frac {e^{\left (2 \, a\right )}}{x {\left (\frac {1}{x^{4}} - e^{\left (2 \, a\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.21, size = 60, normalized size = 0.70 \[ \frac {{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\,\mathrm {atanh}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{2}-\frac {{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\,\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{2}+\frac {2\,x^4\,{\mathrm {e}}^{2\,a}-1}{x-x^5\,{\mathrm {e}}^{2\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + 2*log(x))^2/x^2,x)

[Out]

(exp(2*a)^(1/4)*atanh(x*exp(2*a)^(1/4)))/2 - (exp(2*a)^(1/4)*atan(x*exp(2*a)^(1/4)))/2 + (2*x^4*exp(2*a) - 1)/
(x - x^5*exp(2*a))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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