3.157 \(\int \frac {\coth (a+2 \log (x))}{x^3} \, dx\)

Optimal. Leaf size=21 \[ \frac {1}{2 x^2}-e^a \tanh ^{-1}\left (e^a x^2\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [F]  time = 0.02, 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 = {} \[ \int \frac {\coth (a+2 \log (x))}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Coth[a + 2*Log[x]]/x^3,x]

[Out]

Defer[Int][Coth[a + 2*Log[x]]/x^3, x]

Rubi steps

\begin {align*} \int \frac {\coth (a+2 \log (x))}{x^3} \, dx &=\int \frac {\coth (a+2 \log (x))}{x^3} \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.16, size = 27, normalized size = 1.29 \[ \frac {1}{2 x^2}-(\sinh (a)+\cosh (a)) \tanh ^{-1}\left (\frac {\cosh (a)-\sinh (a)}{x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[a + 2*Log[x]]/x^3,x]

[Out]

1/(2*x^2) - ArcTanh[(Cosh[a] - Sinh[a])/x^2]*(Cosh[a] + Sinh[a])

________________________________________________________________________________________

fricas [B]  time = 0.41, size = 38, normalized size = 1.81 \[ -\frac {x^{2} e^{a} \log \left (x^{2} e^{a} + 1\right ) - x^{2} e^{a} \log \left (x^{2} e^{a} - 1\right ) - 1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*log(x))/x^3,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.13, size = 33, normalized size = 1.57 \[ -\frac {1}{2} \, e^{a} \log \left (x^{2} e^{a} + 1\right ) + \frac {1}{2} \, e^{a} \log \left ({\left | x^{2} e^{a} - 1 \right |}\right ) + \frac {1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*log(x))/x^3,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.10, size = 35, normalized size = 1.67 \[ \frac {1}{2 x^{2}}-\frac {{\mathrm e}^{a} \ln \left (-{\mathrm e}^{a} x^{2}-1\right )}{2}+\frac {{\mathrm e}^{a} \ln \left (-{\mathrm e}^{a} x^{2}+1\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a+2*ln(x))/x^3,x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.50, size = 30, normalized size = 1.43 \[ -\frac {1}{2} \, e^{a} \log \left (\frac {1}{x^{2}} + e^{a}\right ) + \frac {1}{2} \, e^{a} \log \left (\frac {1}{x^{2}} - e^{a}\right ) + \frac {1}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*log(x))/x^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 1.21, size = 25, normalized size = 1.19 \[ \frac {1}{2\,x^2}-\mathrm {atanh}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )\,\sqrt {{\mathrm {e}}^{2\,a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(a + 2*log(x))/x^3,x)

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth {\left (a + 2 \log {\relax (x )} \right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(a+2*ln(x))/x**3,x)

[Out]

Integral(coth(a + 2*log(x))/x**3, x)

________________________________________________________________________________________