Optimal. Leaf size=18 \[ \frac{1}{2} \log \left (1-b x^2\right )-\log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0102575, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {1593, 266, 36, 29, 31} \[ \frac{1}{2} \log \left (1-b x^2\right )-\log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1593
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{-x+b x^3} \, dx &=\int \frac{1}{x \left (-1+b x^2\right )} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x (-1+b x)} \, dx,x,x^2\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )\right )+\frac{1}{2} b \operatorname{Subst}\left (\int \frac{1}{-1+b x} \, dx,x,x^2\right )\\ &=-\log (x)+\frac{1}{2} \log \left (1-b x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0034299, size = 18, normalized size = 1. \[ \frac{1}{2} \log \left (1-b x^2\right )-\log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.003, size = 16, normalized size = 0.9 \begin{align*} -\ln \left ( x \right ) +{\frac{\ln \left ( b{x}^{2}-1 \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.05053, size = 20, normalized size = 1.11 \begin{align*} \frac{1}{2} \, \log \left (b x^{2} - 1\right ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.40847, size = 39, normalized size = 2.17 \begin{align*} \frac{1}{2} \, \log \left (b x^{2} - 1\right ) - \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.122316, size = 12, normalized size = 0.67 \begin{align*} - \log{\left (x \right )} + \frac{\log{\left (x^{2} - \frac{1}{b} \right )}}{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.23484, size = 24, normalized size = 1.33 \begin{align*} -\frac{1}{2} \, \log \left (x^{2}\right ) + \frac{1}{2} \, \log \left ({\left | b x^{2} - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]