Optimal. Leaf size=14 \[ -\frac {1}{x}+b \log (1+b x) \]
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Rubi [A]
time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {46}
\begin {gather*} b \log (b x+1)-\frac {1}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rubi steps
\begin {align*} \int \left (\frac {b}{x}+\frac {1}{x^2 (1+b x)}\right ) \, dx &=b \log (x)+\int \frac {1}{x^2 (1+b x)} \, dx\\ &=b \log (x)+\int \left (\frac {1}{x^2}-\frac {b}{x}+\frac {b^2}{1+b x}\right ) \, dx\\ &=-\frac {1}{x}+b \log (1+b x)\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 14, normalized size = 1.00 \begin {gather*} -\frac {1}{x}+b \log (1+b x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 15, normalized size = 1.07
method | result | size |
default | \(-\frac {1}{x}+b \ln \left (b x +1\right )\) | \(15\) |
norman | \(-\frac {1}{x}+b \ln \left (b x +1\right )\) | \(15\) |
risch | \(-\frac {1}{x}+b \ln \left (-b x -1\right )\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 14, normalized size = 1.00 \begin {gather*} b \log \left (b x + 1\right ) - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 15, normalized size = 1.07 \begin {gather*} \frac {b x \log \left (b x + 1\right ) - 1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 10, normalized size = 0.71 \begin {gather*} b \log {\left (b x + 1 \right )} - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 15, normalized size = 1.07 \begin {gather*} b \log \left ({\left | b x + 1 \right |}\right ) - \frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 20, normalized size = 1.43 \begin {gather*} b\,\ln \left (x\right )+2\,b\,\mathrm {atanh}\left (2\,b\,x+1\right )-\frac {1}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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