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 - E^(2*a)*x^4) - ArcTan[E^(a/2)*x]/(2*E^(a/2)) - ArcTanh[E^(a/2)*x]/(2*E^(a/2))

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Rubi [F]  time = 0.0098684, 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 = {} \[ \int \coth ^2(a+2 \log (x)) \, dx \]

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*}

Mathematica [C]  time = 1.95104, size = 153, normalized size = 2.55 \[ \frac{16}{585} e^{2 a} x^5 \left (e^{2 a} x^4+1\right )^2 \text{HypergeometricPFQ}\left (\left \{\frac{5}{4},2,2,2\right \},\left \{1,1,\frac{17}{4}\right \},e^{2 a} x^4\right )+\frac{e^{-4 a} \left (5 \left (e^{8 a} x^{16}-248 e^{6 a} x^{12}+102 e^{4 a} x^8+1208 e^{2 a} x^4+729\right ) \, _2F_1\left (\frac{1}{4},1;\frac{5}{4};e^{2 a} x^4\right )+681 e^{6 a} x^{12}-1483 e^{4 a} x^8-6769 e^{2 a} x^4-3645\right )}{640 x^7} \]

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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Maple [A]  time = 0.043, size = 86, normalized size = 1.4 \begin{align*} x-{\frac{x}{{{\rm e}^{2\,a}}{x}^{4}-1}}+{\frac{1}{4}\ln \left ( x\sqrt{{{\rm e}^{a}}}-1 \right ){\frac{1}{\sqrt{{{\rm e}^{a}}}}}}-{\frac{1}{4}\ln \left ( x\sqrt{{{\rm e}^{a}}}+1 \right ){\frac{1}{\sqrt{{{\rm e}^{a}}}}}}-{\frac{1}{4}\ln \left ( x\sqrt{-{{\rm e}^{a}}}+1 \right ){\frac{1}{\sqrt{-{{\rm e}^{a}}}}}}+{\frac{1}{4}\ln \left ( x\sqrt{-{{\rm e}^{a}}}-1 \right ){\frac{1}{\sqrt{-{{\rm e}^{a}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x-x/(exp(2*a)*x^4-1)+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)-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 = 1.57043, size = 81, normalized size = 1.35 \begin{align*} -\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} \end{align*}

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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Fricas [B]  time = 2.64083, size = 246, normalized size = 4.1 \begin{align*} \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 )}} \end{align*}

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

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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Giac [A]  time = 1.119, size = 89, normalized size = 1.48 \begin{align*} -\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} \end{align*}

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)