Optimal. Leaf size=35 \[ \frac {p}{2 x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {2504, 2436,
2332} \begin {gather*} \frac {p}{2 x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2332
Rule 2436
Rule 2504
Rubi steps
\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{x^3} \, dx &=-\left (\frac {1}{2} \text {Subst}\left (\int \log \left (c (a+b x)^p\right ) \, dx,x,\frac {1}{x^2}\right )\right )\\ &=-\frac {\text {Subst}\left (\int \log \left (c x^p\right ) \, dx,x,a+\frac {b}{x^2}\right )}{2 b}\\ &=\frac {p}{2 x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{2 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 34, normalized size = 0.97 \begin {gather*} \frac {1}{2} \left (\frac {p}{x^2}-\frac {\left (a+\frac {b}{x^2}\right ) \log \left (c \left (a+\frac {b}{x^2}\right )^p\right )}{b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 37, normalized size = 1.06
method | result | size |
derivativedivides | \(-\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) \left (a +\frac {b}{x^{2}}\right )-\left (a +\frac {b}{x^{2}}\right ) p}{2 b}\) | \(37\) |
default | \(-\frac {\ln \left (c \left (a +\frac {b}{x^{2}}\right )^{p}\right ) \left (a +\frac {b}{x^{2}}\right )-\left (a +\frac {b}{x^{2}}\right ) p}{2 b}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 54, normalized size = 1.54 \begin {gather*} -\frac {1}{2} \, b p {\left (\frac {a \log \left (a x^{2} + b\right )}{b^{2}} - \frac {a \log \left (x^{2}\right )}{b^{2}} - \frac {1}{b x^{2}}\right )} - \frac {\log \left ({\left (a + \frac {b}{x^{2}}\right )}^{p} c\right )}{2 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 41, normalized size = 1.17 \begin {gather*} \frac {b p - b \log \left (c\right ) - {\left (a p x^{2} + b p\right )} \log \left (\frac {a x^{2} + b}{x^{2}}\right )}{2 \, b x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.05, size = 53, normalized size = 1.51 \begin {gather*} \begin {cases} - \frac {a \log {\left (c \left (a + \frac {b}{x^{2}}\right )^{p} \right )}}{2 b} + \frac {p}{2 x^{2}} - \frac {\log {\left (c \left (a + \frac {b}{x^{2}}\right )^{p} \right )}}{2 x^{2}} & \text {for}\: b \neq 0 \\- \frac {\log {\left (a^{p} c \right )}}{2 x^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.28, size = 57, normalized size = 1.63 \begin {gather*} -\frac {p {\left (\frac {{\left (a x^{2} + b\right )} \log \left (\frac {a x^{2} + b}{x^{2}}\right )}{x^{2}} - \frac {a x^{2} + b}{x^{2}}\right )} + \frac {{\left (a x^{2} + b\right )} \log \left (c\right )}{x^{2}}}{2 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 47, normalized size = 1.34 \begin {gather*} \frac {p}{2\,x^2}-\frac {\ln \left (c\,{\left (a+\frac {b}{x^2}\right )}^p\right )}{2\,x^2}-\frac {a\,p\,\ln \left (a\,x^2+b\right )}{2\,b}+\frac {a\,p\,\ln \left (x\right )}{b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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