Optimal. Leaf size=20 \[ -\frac {1}{b n \left (a+b \log \left (c x^n\right )\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2302, 30} \[ -\frac {1}{b n \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2302
Rubi steps
\begin {align*} \int \frac {1}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,a+b \log \left (c x^n\right )\right )}{b n}\\ &=-\frac {1}{b n \left (a+b \log \left (c x^n\right )\right )}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 20, normalized size = 1.00 \[ -\frac {1}{b n \left (a+b \log \left (c x^n\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 25, normalized size = 1.25 \[ -\frac {1}{b^{2} n^{2} \log \relax (x) + b^{2} n \log \relax (c) + a b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.25, size = 21, normalized size = 1.05 \[ -\frac {1}{{\left (b n \log \relax (x) + b \log \relax (c) + a\right )} b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 21, normalized size = 1.05 \[ -\frac {1}{\left (b \ln \left (c \,x^{n}\right )+a \right ) b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 20, normalized size = 1.00 \[ -\frac {1}{{\left (b \log \left (c x^{n}\right ) + a\right )} b n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.27, size = 20, normalized size = 1.00 \[ -\frac {1}{n\,\ln \left (c\,x^n\right )\,b^2+a\,n\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 22.74, size = 70, normalized size = 3.50 \[ \begin {cases} \frac {\tilde {\infty } \log {\relax (x )}}{\log {\relax (c )}^{2}} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\tilde {\infty } n \log {\relax (x )} & \text {for}\: a = - b \left (n \log {\relax (x )} + \log {\relax (c )}\right ) \\\frac {\log {\relax (x )}}{a^{2}} & \text {for}\: b = 0 \\\frac {\log {\relax (x )}}{\left (a + b \log {\relax (c )}\right )^{2}} & \text {for}\: n = 0 \\- \frac {1}{a b n + b^{2} n^{2} \log {\relax (x )} + b^{2} n \log {\relax (c )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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