Optimal. Leaf size=14 \[ \log (x)-\frac{1}{2} \coth (a+2 \log (x)) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0239965, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3473, 8} \[ \log (x)-\frac{1}{2} \coth (a+2 \log (x)) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3473
Rule 8
Rubi steps
\begin{align*} \int \frac{\coth ^2(a+2 \log (x))}{x} \, dx &=\operatorname{Subst}\left (\int \coth ^2(a+2 x) \, dx,x,\log (x)\right )\\ &=-\frac{1}{2} \coth (a+2 \log (x))+\operatorname{Subst}(\int 1 \, dx,x,\log (x))\\ &=-\frac{1}{2} \coth (a+2 \log (x))+\log (x)\\ \end{align*}
Mathematica [C] time = 0.0539936, size = 28, normalized size = 2. \[ -\frac{1}{2} \coth (a+2 \log (x)) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\tanh ^2(a+2 \log (x))\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.003, size = 35, normalized size = 2.5 \begin{align*} -{\frac{{\rm coth} \left (a+2\,\ln \left ( x \right ) \right )}{2}}-{\frac{\ln \left ({\rm coth} \left (a+2\,\ln \left ( x \right ) \right )-1 \right ) }{4}}+{\frac{\ln \left ({\rm coth} \left (a+2\,\ln \left ( x \right ) \right )+1 \right ) }{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.03175, size = 26, normalized size = 1.86 \begin{align*} \frac{1}{2} \, a + \frac{1}{e^{\left (-2 \, a - 4 \, \log \left (x\right )\right )} - 1} + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.52145, size = 68, normalized size = 4.86 \begin{align*} \frac{{\left (x^{4} e^{\left (2 \, a\right )} - 1\right )} \log \left (x\right ) - 1}{x^{4} e^{\left (2 \, a\right )} - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 17.2266, size = 32, normalized size = 2.29 \begin{align*} \begin{cases} \tilde{\infty } \log{\left (x \right )} & \text{for}\: a = \log{\left (- \frac{1}{x^{2}} \right )} \vee a = \log{\left (\frac{1}{x^{2}} \right )} \\\log{\left (x \right )} - \frac{1}{2 \tanh{\left (a + 2 \log{\left (x \right )} \right )}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.11646, size = 28, normalized size = 2. \begin{align*} -\frac{1}{x^{4} e^{\left (2 \, a\right )} - 1} + \frac{1}{4} \, \log \left (x^{4}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]