Optimal. Leaf size=28 \[ \frac {1}{2 \left (b x^2+1\right )}-\frac {1}{2} \log \left (b x^2+1\right )+\log (x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, 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 = {266, 44} \[ \frac {1}{2 \left (b x^2+1\right )}-\frac {1}{2} \log \left (b x^2+1\right )+\log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \left (1+b x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (1+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{x}-\frac {b}{(1+b x)^2}-\frac {b}{1+b x}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{2 \left (1+b x^2\right )}+\log (x)-\frac {1}{2} \log \left (1+b x^2\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 25, normalized size = 0.89 \[ \frac {1}{2 b x^2+2}-\frac {1}{2} \log \left (b x^2+1\right )+\log (x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.72, size = 40, normalized size = 1.43 \[ -\frac {{\left (b x^{2} + 1\right )} \log \left (b x^{2} + 1\right ) - 2 \, {\left (b x^{2} + 1\right )} \log \relax (x) - 1}{2 \, {\left (b x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.63, size = 36, normalized size = 1.29 \[ \frac {b x^{2} + 2}{2 \, {\left (b x^{2} + 1\right )}} + \frac {1}{2} \, \log \left (x^{2}\right ) - \frac {1}{2} \, \log \left ({\left | b x^{2} + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 25, normalized size = 0.89 \[ \ln \relax (x )-\frac {\ln \left (b \,x^{2}+1\right )}{2}+\frac {1}{2 b \,x^{2}+2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.35, size = 28, normalized size = 1.00 \[ \frac {1}{2 \, {\left (b x^{2} + 1\right )}} - \frac {1}{2} \, \log \left (b x^{2} + 1\right ) + \frac {1}{2} \, \log \left (x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.03, size = 24, normalized size = 0.86 \[ \ln \relax (x)-\frac {\ln \left (\frac {3\,b\,x^2}{2}+\frac {3}{2}\right )}{2}+\frac {1}{2\,\left (b\,x^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.20, size = 22, normalized size = 0.79 \[ \log {\relax (x )} - \frac {\log {\left (x^{2} + \frac {1}{b} \right )}}{2} + \frac {1}{2 b x^{2} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________