Optimal. Leaf size=27 \[ x \log \left (e^{a+b x^n}\right )-\frac{b n x^{n+1}}{n+1} \]
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Rubi [A] time = 0.0085458, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2548, 12, 30} \[ x \log \left (e^{a+b x^n}\right )-\frac{b n x^{n+1}}{n+1} \]
Antiderivative was successfully verified.
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Rule 2548
Rule 12
Rule 30
Rubi steps
\begin{align*} \int \log \left (e^{a+b x^n}\right ) \, dx &=x \log \left (e^{a+b x^n}\right )-\int b n x^n \, dx\\ &=x \log \left (e^{a+b x^n}\right )-(b n) \int x^n \, dx\\ &=-\frac{b n x^{1+n}}{1+n}+x \log \left (e^{a+b x^n}\right )\\ \end{align*}
Mathematica [A] time = 0.0237789, size = 25, normalized size = 0.93 \[ x \left (\log \left (e^{a+b x^n}\right )-\frac{b n x^n}{n+1}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 27, normalized size = 1. \begin{align*} -{\frac{bn{x}^{1+n}}{1+n}}+x\ln \left ({{\rm e}^{a+b{x}^{n}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.08307, size = 45, normalized size = 1.67 \begin{align*} \frac{b x x^{n} +{\left (a n + a\right )} x}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.81085, size = 34, normalized size = 1.26 \begin{align*} \begin{cases} \frac{a n x}{n + 1} + \frac{a x}{n + 1} + \frac{b x x^{n}}{n + 1} & \text{for}\: n \neq -1 \\a x + b \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37261, size = 22, normalized size = 0.81 \begin{align*} a x + \frac{b x^{n + 1}}{n + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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