Optimal. Leaf size=15 \[ -\frac {\log \left (a+\frac {b}{x^2}\right )}{2 b} \]
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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {266}
\begin {gather*} -\frac {\log \left (a+\frac {b}{x^2}\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right ) x^3} \, dx &=-\frac {\log \left (a+\frac {b}{x^2}\right )}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 22, normalized size = 1.47 \begin {gather*} \frac {\log (x)}{b}-\frac {\log \left (b+a x^2\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 21, normalized size = 1.40
method | result | size |
derivativedivides | \(-\frac {\ln \left (\frac {b}{x^{2}}+a \right )}{2 b}\) | \(14\) |
default | \(-\frac {\ln \left (a \,x^{2}+b \right )}{2 b}+\frac {\ln \left (x \right )}{b}\) | \(21\) |
norman | \(-\frac {\ln \left (a \,x^{2}+b \right )}{2 b}+\frac {\ln \left (x \right )}{b}\) | \(21\) |
risch | \(-\frac {\ln \left (a \,x^{2}+b \right )}{2 b}+\frac {\ln \left (x \right )}{b}\) | \(21\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 13, normalized size = 0.87 \begin {gather*} -\frac {\log \left (a + \frac {b}{x^{2}}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 18, normalized size = 1.20 \begin {gather*} -\frac {\log \left (a x^{2} + b\right ) - 2 \, \log \left (x\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log {\left (x \right )}}{b} - \frac {\log {\left (x^{2} + \frac {b}{a} \right )}}{2 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.06, size = 24, normalized size = 1.60 \begin {gather*} \frac {\log \left (x^{2}\right )}{2 \, b} - \frac {\log \left ({\left | a x^{2} + b \right |}\right )}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.07, size = 18, normalized size = 1.20 \begin {gather*} -\frac {\ln \left (a\,x^2+b\right )-2\,\ln \left (x\right )}{2\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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