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

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

[Out]

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

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Rubi [F]  time = 0.01, 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 = {} \[ \int \coth ^2(a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + 2*Log[x]]^2,x]

[Out]

Defer[Int][Coth[a + 2*Log[x]]^2, x]

Rubi steps

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

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Mathematica [C]  time = 2.24, size = 153, normalized size = 2.55 \[ \frac {16}{585} e^{2 a} x^5 \left (e^{2 a} x^4+1\right )^2 \, _4F_3\left (\frac {5}{4},2,2,2;1,1,\frac {17}{4};e^{2 a} x^4\right )+\frac {e^{-4 a} \left (5 \left (e^{8 a} x^{16}-248 e^{6 a} x^{12}+102 e^{4 a} x^8+1208 e^{2 a} x^4+729\right ) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};e^{2 a} x^4\right )+681 e^{6 a} x^{12}-1483 e^{4 a} x^8-6769 e^{2 a} x^4-3645\right )}{640 x^7} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Coth[a + 2*Log[x]]^2,x]

[Out]

(-3645 - 6769*E^(2*a)*x^4 - 1483*E^(4*a)*x^8 + 681*E^(6*a)*x^12 + 5*(729 + 1208*E^(2*a)*x^4 + 102*E^(4*a)*x^8
- 248*E^(6*a)*x^12 + E^(8*a)*x^16)*Hypergeometric2F1[1/4, 1, 5/4, E^(2*a)*x^4])/(640*E^(4*a)*x^7) + (16*E^(2*a
)*x^5*(1 + E^(2*a)*x^4)^2*HypergeometricPFQ[{5/4, 2, 2, 2}, {1, 1, 17/4}, E^(2*a)*x^4])/585

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fricas [B]  time = 0.41, size = 97, normalized size = 1.62 \[ \frac {4 \, x^{5} e^{\left (3 \, a\right )} - 2 \, {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} + {\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {x^{2} e^{a} - 2 \, x e^{\left (\frac {1}{2} \, a\right )} + 1}{x^{2} e^{a} - 1}\right ) - 8 \, x e^{a}}{4 \, {\left (x^{4} e^{\left (3 \, a\right )} - e^{a}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*x^5*e^(3*a) - 2*(x^4*e^(2*a) - 1)*arctan(x*e^(1/2*a))*e^(1/2*a) + (x^4*e^(2*a) - 1)*e^(1/2*a)*log((x^2*
e^a - 2*x*e^(1/2*a) + 1)/(x^2*e^a - 1)) - 8*x*e^a)/(x^4*e^(3*a) - e^a)

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giac [A]  time = 0.14, size = 66, normalized size = 1.10 \[ -\frac {1}{2} \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {1}{2} \, a\right )} + \frac {1}{4} \, e^{\left (-\frac {1}{2} \, a\right )} \log \left (\frac {{\left | 2 \, x e^{a} - 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}{{\left | 2 \, x e^{a} + 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}\right ) + x - \frac {x}{x^{4} e^{\left (2 \, a\right )} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maple [A]  time = 0.13, size = 86, normalized size = 1.43 \[ x -\frac {x}{-1+{\mathrm e}^{2 a} x^{4}}+\frac {\ln \left (\sqrt {{\mathrm e}^{a}}\, x -1\right )}{4 \sqrt {{\mathrm e}^{a}}}-\frac {\ln \left (\sqrt {{\mathrm e}^{a}}\, x +1\right )}{4 \sqrt {{\mathrm e}^{a}}}-\frac {\ln \left (x \sqrt {-{\mathrm e}^{a}}+1\right )}{4 \sqrt {-{\mathrm e}^{a}}}+\frac {\ln \left (x \sqrt {-{\mathrm e}^{a}}-1\right )}{4 \sqrt {-{\mathrm e}^{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.42, size = 60, normalized size = 1.00 \[ -\frac {1}{2} \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {1}{2} \, a\right )} + \frac {1}{4} \, e^{\left (-\frac {1}{2} \, a\right )} \log \left (\frac {x e^{a} - e^{\left (\frac {1}{2} \, a\right )}}{x e^{a} + e^{\left (\frac {1}{2} \, a\right )}}\right ) + x - \frac {x}{x^{4} e^{\left (2 \, a\right )} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.21, size = 54, normalized size = 0.90 \[ x-\frac {\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{2\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}}-\frac {x}{x^4\,{\mathrm {e}}^{2\,a}-1}+\frac {\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + 2*log(x))^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(a + 2*log(x))**2, x)

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