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

Optimal. Leaf size=41 \[ e^{a/2} \tan ^{-1}\left (e^{a/2} x\right )-e^{a/2} \tanh ^{-1}\left (e^{a/2} x\right )+\frac{1}{x} \]

[Out]

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

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Rubi [F]  time = 0.0222044, 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 = {} \[ \int \frac{\coth (a+2 \log (x))}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + 2*Log[x]]/x^2,x]

[Out]

Defer[Int][Coth[a + 2*Log[x]]/x^2, x]

Rubi steps

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

Mathematica [C]  time = 0.162025, size = 62, normalized size = 1.51 \[ \frac{x (\sinh (a)+\cosh (a))^2 \text{RootSum}\left [-\text{$\#$1}^4 \sinh (a)+\text{$\#$1}^4 \cosh (a)-\sinh (a)-\cosh (a)\& ,\frac{\log \left (\frac{1}{x}-\text{$\#$1}\right )+\log (x)}{\text{$\#$1}^3}\& \right ]+2}{2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + 2*Log[x]]/x^2,x]

[Out]

(2 + x*RootSum[-Cosh[a] - Sinh[a] + Cosh[a]*#1^4 - Sinh[a]*#1^4 & , (Log[x] + Log[x^(-1) - #1])/#1^3 & ]*(Cosh
[a] + Sinh[a])^2)/(2*x)

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Maple [B]  time = 0.042, size = 93, normalized size = 2.3 \begin{align*}{x}^{-1}+{\frac{1}{2}\sqrt{{{\rm e}^{a}}}\ln \left ( -{{\rm e}^{2\,a}}x+ \left ({{\rm e}^{a}} \right ) ^{{\frac{3}{2}}} \right ) }-{\frac{1}{2}\sqrt{{{\rm e}^{a}}}\ln \left ( -{{\rm e}^{2\,a}}x- \left ({{\rm e}^{a}} \right ) ^{{\frac{3}{2}}} \right ) }+{\frac{1}{2}\sqrt{-{{\rm e}^{a}}}\ln \left ( -{{\rm e}^{2\,a}}x+ \left ( -{{\rm e}^{a}} \right ) ^{{\frac{3}{2}}} \right ) }-{\frac{1}{2}\sqrt{-{{\rm e}^{a}}}\ln \left ( -{{\rm e}^{2\,a}}x- \left ( -{{\rm e}^{a}} \right ) ^{{\frac{3}{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.73337, size = 63, normalized size = 1.54 \begin{align*} -\arctan \left (\frac{e^{\left (-\frac{1}{2} \, a\right )}}{x}\right ) e^{\left (\frac{1}{2} \, a\right )} + \frac{1}{2} \, 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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.6559, size = 149, normalized size = 3.63 \begin{align*} \frac{2 \, x \arctan \left (x e^{\left (\frac{1}{2} \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} + x 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 ) + 2}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(a + 2*log(x))/x**2, x)

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Giac [A]  time = 1.15757, size = 70, normalized size = 1.71 \begin{align*} \arctan \left (x e^{\left (\frac{1}{2} \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} + \frac{1}{2} \, 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{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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