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

Optimal. Leaf size=40 \[ -e^{-a/2} \tan ^{-1}\left (e^{a/2} x\right )-e^{-a/2} \tanh ^{-1}\left (e^{a/2} x\right )+x \]

[Out]

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

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Rubi [F]  time = 0.0071696, 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 = {} \[ \int \coth (a+2 \log (x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + 2*Log[x]],x]

[Out]

Defer[Int][Coth[a + 2*Log[x]], x]

Rubi steps

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

Mathematica [C]  time = 0.178869, size = 58, normalized size = 1.45 \[ \frac{1}{2} (\sinh (2 a)-\cosh (2 a)) \text{RootSum}\left [\text{$\#$1}^4 \sinh (a)+\text{$\#$1}^4 \cosh (a)+\sinh (a)-\cosh (a)\& ,\frac{\log (x)-\log (x-\text{$\#$1})}{\text{$\#$1}^3}\& \right ]+x \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + 2*Log[x]],x]

[Out]

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

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Maple [B]  time = 0.045, size = 71, normalized size = 1.8 \begin{align*} x-{\frac{1}{2}\ln \left ( x\sqrt{-{{\rm e}^{a}}}+1 \right ){\frac{1}{\sqrt{-{{\rm e}^{a}}}}}}+{\frac{1}{2}\ln \left ( x\sqrt{-{{\rm e}^{a}}}-1 \right ){\frac{1}{\sqrt{-{{\rm e}^{a}}}}}}+{\frac{1}{2}\ln \left ( x\sqrt{{{\rm e}^{a}}}-1 \right ){\frac{1}{\sqrt{{{\rm e}^{a}}}}}}-{\frac{1}{2}\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)),x)

[Out]

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

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Maxima [A]  time = 1.60909, size = 61, normalized size = 1.52 \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{x e^{a} - e^{\left (\frac{1}{2} \, a\right )}}{x e^{a} + e^{\left (\frac{1}{2} \, a\right )}}\right ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.58194, size = 159, normalized size = 3.98 \begin{align*} -\frac{1}{2} \,{\left (2 \, \arctan \left (x e^{\left (\frac{1}{2} \, a\right )}\right ) e^{\left (\frac{1}{2} \, a\right )} - 2 \, x e^{a} - 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 )\right )} e^{\left (-a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth{\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)),x)

[Out]

Integral(coth(a + 2*log(x)), x)

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Giac [A]  time = 1.12146, size = 69, normalized size = 1.72 \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 ) + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*log(x)),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))) +
x