Optimal. Leaf size=14 \[ a x+b \log ^2\left (c x^n\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2301} \[ a x+b \log ^2\left (c x^n\right ) \]
Antiderivative was successfully verified.
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Rule 2301
Rubi steps
\begin {align*} \int \left (a+\frac {2 b n \log \left (c x^n\right )}{x}\right ) \, dx &=a x+(2 b n) \int \frac {\log \left (c x^n\right )}{x} \, dx\\ &=a x+b \log ^2\left (c x^n\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 14, normalized size = 1.00 \[ a x+b \log ^2\left (c x^n\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 21, normalized size = 1.50 \[ b n^{2} \log \relax (x)^{2} + 2 \, b n \log \relax (c) \log \relax (x) + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 20, normalized size = 1.43 \[ {\left (n \log \relax (x)^{2} + 2 \, \log \relax (c) \log \relax (x)\right )} b n + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 15, normalized size = 1.07 \[ b \ln \left (c \,x^{n}\right )^{2}+a x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 14, normalized size = 1.00 \[ b \log \left (c x^{n}\right )^{2} + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.26, size = 14, normalized size = 1.00 \[ b\,{\ln \left (c\,x^n\right )}^2+a\,x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.72, size = 60, normalized size = 4.29 \[ a x + 2 b n \left (\begin {cases} \frac {\log {\left (c x^{n} \right )}^{2}}{2 n} & \text {for}\: \left |{c x^{n}}\right | < 1 \\\frac {\log {\left (\frac {x^{- n}}{c} \right )}^{2}}{2 n} & \text {for}\: \frac {1}{\left |{c x^{n}}\right |} < 1 \\\frac {{G_{3, 3}^{3, 0}\left (\begin {matrix} & 1, 1, 1 \\0, 0, 0 & \end {matrix} \middle | {c x^{n}} \right )}}{n} + \frac {{G_{3, 3}^{0, 3}\left (\begin {matrix} 1, 1, 1 & \\ & 0, 0, 0 \end {matrix} \middle | {c x^{n}} \right )}}{n} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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