Optimal. Leaf size=15 \[ \log (x)+\frac {1}{3} \log \left (b+c x^3\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {457, 78}
\begin {gather*} \frac {1}{3} \log \left (b+c x^3\right )+\log (x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {b+2 c x^3}{x \left (b+c x^3\right )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {b+2 c x}{x (b+c x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{x}+\frac {c}{b+c x}\right ) \, dx,x,x^3\right )\\ &=\log (x)+\frac {1}{3} \log \left (b+c x^3\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} \log (x)+\frac {1}{3} \log \left (b+c x^3\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.32, size = 14, normalized size = 0.93
method | result | size |
default | \(\ln \left (x \right )+\frac {\ln \left (c \,x^{3}+b \right )}{3}\) | \(14\) |
norman | \(\ln \left (x \right )+\frac {\ln \left (c \,x^{3}+b \right )}{3}\) | \(14\) |
risch | \(\ln \left (x \right )+\frac {\ln \left (c \,x^{3}+b \right )}{3}\) | \(14\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.31, size = 17, normalized size = 1.13 \begin {gather*} \frac {1}{3} \, \log \left (c x^{3} + b\right ) + \frac {1}{3} \, \log \left (x^{3}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.14, size = 13, normalized size = 0.87 \begin {gather*} \frac {1}{3} \, \log \left (c x^{3} + b\right ) + \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.08, size = 12, normalized size = 0.80 \begin {gather*} \log {\left (x \right )} + \frac {\log {\left (\frac {b}{c} + x^{3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.57, size = 15, normalized size = 1.00 \begin {gather*} \frac {1}{3} \, \log \left ({\left | c x^{3} + b \right |}\right ) + \log \left ({\left | x \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 4.63, size = 13, normalized size = 0.87 \begin {gather*} \frac {\ln \left (c\,x^3+b\right )}{3}+\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________