Optimal. Leaf size=23 \[ \frac {1}{7} \log \left (5+\frac {1}{x}\right )-\frac {1}{7} \log \left (3+\frac {2}{x}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1366, 630, 31}
\begin {gather*} \frac {1}{7} \log \left (\frac {1}{x}+5\right )-\frac {1}{7} \log \left (\frac {2}{x}+3\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 630
Rule 1366
Rubi steps
\begin {align*} \int \frac {1}{\left (15+\frac {2}{x^2}+\frac {13}{x}\right ) x^2} \, dx &=-\text {Subst}\left (\int \frac {1}{15+13 x+2 x^2} \, dx,x,\frac {1}{x}\right )\\ &=-\left (\frac {2}{7} \text {Subst}\left (\int \frac {1}{3+2 x} \, dx,x,\frac {1}{x}\right )\right )+\frac {2}{7} \text {Subst}\left (\int \frac {1}{10+2 x} \, dx,x,\frac {1}{x}\right )\\ &=\frac {1}{7} \log \left (5+\frac {1}{x}\right )-\frac {1}{7} \log \left (3+\frac {2}{x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 21, normalized size = 0.91 \begin {gather*} -\frac {1}{7} \log (2+3 x)+\frac {1}{7} \log (1+5 x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 18, normalized size = 0.78
method | result | size |
default | \(\frac {\ln \left (1+5 x \right )}{7}-\frac {\ln \left (2+3 x \right )}{7}\) | \(18\) |
norman | \(\frac {\ln \left (1+5 x \right )}{7}-\frac {\ln \left (2+3 x \right )}{7}\) | \(18\) |
risch | \(\frac {\ln \left (1+5 x \right )}{7}-\frac {\ln \left (2+3 x \right )}{7}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{7} \, \log \left (5 \, x + 1\right ) - \frac {1}{7} \, \log \left (3 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{7} \, \log \left (5 \, x + 1\right ) - \frac {1}{7} \, \log \left (3 \, x + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 15, normalized size = 0.65 \begin {gather*} \frac {\log {\left (x + \frac {1}{5} \right )}}{7} - \frac {\log {\left (x + \frac {2}{3} \right )}}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.77, size = 19, normalized size = 0.83 \begin {gather*} \frac {1}{7} \, \log \left ({\left | 5 \, x + 1 \right |}\right ) - \frac {1}{7} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.37, size = 8, normalized size = 0.35 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {30\,x}{7}+\frac {13}{7}\right )}{7} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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