Optimal. Leaf size=15 \[ -\frac{1}{3 x^3}-\frac{1}{x}+\tanh ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0097331, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {1584, 325, 206} \[ -\frac{1}{3 x^3}-\frac{1}{x}+\tanh ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1584
Rule 325
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (x-x^3\right )} \, dx &=\int \frac{1}{x^4 \left (1-x^2\right )} \, dx\\ &=-\frac{1}{3 x^3}+\int \frac{1}{x^2 \left (1-x^2\right )} \, dx\\ &=-\frac{1}{3 x^3}-\frac{1}{x}+\int \frac{1}{1-x^2} \, dx\\ &=-\frac{1}{3 x^3}-\frac{1}{x}+\tanh ^{-1}(x)\\ \end{align*}
Mathematica [B] time = 0.0036898, size = 31, normalized size = 2.07 \[ -\frac{1}{3 x^3}-\frac{1}{x}-\frac{1}{2} \log (1-x)+\frac{1}{2} \log (x+1) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 24, normalized size = 1.6 \begin{align*} -{\frac{1}{3\,{x}^{3}}}-{x}^{-1}+{\frac{\ln \left ( 1+x \right ) }{2}}-{\frac{\ln \left ( -1+x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.09703, size = 34, normalized size = 2.27 \begin{align*} -\frac{3 \, x^{2} + 1}{3 \, x^{3}} + \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.40575, size = 80, normalized size = 5.33 \begin{align*} \frac{3 \, x^{3} \log \left (x + 1\right ) - 3 \, x^{3} \log \left (x - 1\right ) - 6 \, x^{2} - 2}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.108128, size = 24, normalized size = 1.6 \begin{align*} - \frac{\log{\left (x - 1 \right )}}{2} + \frac{\log{\left (x + 1 \right )}}{2} - \frac{3 x^{2} + 1}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.23086, size = 36, normalized size = 2.4 \begin{align*} -\frac{3 \, x^{2} + 1}{3 \, x^{3}} + \frac{1}{2} \, \log \left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \, \log \left ({\left | x - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]