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(exp(1/2*a)*x)/exp(1/2*a)-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 = {} \[ \int \coth (a+2 \log (x)) \, dx \]

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*}

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Mathematica [C]  time = 0.19, 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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fricas [B]  time = 0.41, size = 58, normalized size = 1.45 \[ -\frac {1}{2} \, {\left (2 \, \arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (\frac {1}{2} \, a\right )} - 2 \, x e^{a} - e^{\left (\frac {1}{2} \, a\right )} \log \left (\frac {x^{2} e^{a} - 2 \, x e^{\left (\frac {1}{2} \, a\right )} + 1}{x^{2} e^{a} - 1}\right )\right )} e^{\left (-a\right )} \]

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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giac [A]  time = 0.14, size = 51, normalized size = 1.28 \[ -\arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {1}{2} \, a\right )} + \frac {1}{2} \, e^{\left (-\frac {1}{2} \, a\right )} \log \left (\frac {{\left | 2 \, x e^{a} - 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}{{\left | 2 \, x e^{a} + 2 \, e^{\left (\frac {1}{2} \, a\right )} \right |}}\right ) + x \]

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

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maple [B]  time = 0.14, size = 71, normalized size = 1.78 \[ x +\frac {\ln \left (\sqrt {{\mathrm e}^{a}}\, x -1\right )}{2 \sqrt {{\mathrm e}^{a}}}-\frac {\ln \left (\sqrt {{\mathrm e}^{a}}\, x +1\right )}{2 \sqrt {{\mathrm e}^{a}}}-\frac {\ln \left (x \sqrt {-{\mathrm e}^{a}}+1\right )}{2 \sqrt {-{\mathrm e}^{a}}}+\frac {\ln \left (x \sqrt {-{\mathrm e}^{a}}-1\right )}{2 \sqrt {-{\mathrm e}^{a}}} \]

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(exp(a)^(1/2)*x-1)-1/2/exp(a)^(1/2)*ln(exp(a)^(1/2)*x+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 = 0.60, size = 45, normalized size = 1.12 \[ -\arctan \left (x e^{\left (\frac {1}{2} \, a\right )}\right ) e^{\left (-\frac {1}{2} \, a\right )} + \frac {1}{2} \, e^{\left (-\frac {1}{2} \, a\right )} \log \left (\frac {x e^{a} - e^{\left (\frac {1}{2} \, a\right )}}{x e^{a} + e^{\left (\frac {1}{2} \, a\right )}}\right ) + x \]

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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mupad [B]  time = 1.19, size = 36, normalized size = 0.90 \[ x-\frac {\mathrm {atan}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}}-\frac {\mathrm {atanh}\left (x\,{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}\right )}{{\left ({\mathrm {e}}^{2\,a}\right )}^{1/4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + 2*log(x)),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(coth(a + 2*log(x)), x)

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