Optimal. Leaf size=21 \[ \frac {\log \left (\frac {x^3}{a+b x^3}\right )}{3 a} \]
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Rubi [A]
time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.05, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {272, 36, 29, 31}
\begin {gather*} \frac {\log (x)}{a}-\frac {\log \left (a+b x^3\right )}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 272
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x (a+b x)} \, dx,x,x^3\right )\\ &=\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,x^3\right )}{3 a}-\frac {b \text {Subst}\left (\int \frac {1}{a+b x} \, dx,x,x^3\right )}{3 a}\\ &=\frac {\log (x)}{a}-\frac {\log \left (a+b x^3\right )}{3 a}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.05 \begin {gather*} \frac {\log (x)}{a}-\frac {\log \left (a+b x^3\right )}{3 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 21, normalized size = 1.00
method | result | size |
default | \(-\frac {\ln \left (b \,x^{3}+a \right )}{3 a}+\frac {\ln \left (x \right )}{a}\) | \(21\) |
norman | \(-\frac {\ln \left (b \,x^{3}+a \right )}{3 a}+\frac {\ln \left (x \right )}{a}\) | \(21\) |
risch | \(-\frac {\ln \left (b \,x^{3}+a \right )}{3 a}+\frac {\ln \left (x \right )}{a}\) | \(21\) |
parallelrisch | \(\frac {3 \ln \left (x \right )-\ln \left (b \,x^{3}+a \right )}{3 a}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 23, normalized size = 1.10 \begin {gather*} -\frac {\log \left (b x^{3} + a\right )}{3 \, a} + \frac {\log \left (x^{3}\right )}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.60, size = 18, normalized size = 0.86 \begin {gather*} -\frac {\log \left (b x^{3} + a\right ) - 3 \, \log \left (x\right )}{3 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.10, size = 15, normalized size = 0.71 \begin {gather*} \frac {\log {\left (x \right )}}{a} - \frac {\log {\left (\frac {a}{b} + x^{3} \right )}}{3 a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 22, normalized size = 1.05 \begin {gather*} -\frac {\log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a} + \frac {\log \left ({\left | x \right |}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 18, normalized size = 0.86 \begin {gather*} -\frac {\ln \left (b\,x^3+a\right )-3\,\ln \left (x\right )}{3\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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