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 = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 12
Rule 30
Rule 2548
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*}
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Mathematica [A] time = 0.03, 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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fricas [A] time = 0.46, size = 20, normalized size = 0.74 \[ \frac {b x x^{n} + {\left (a n + a\right )} x}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 16, normalized size = 0.59 \[ a x + \frac {b x^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 27, normalized size = 1.00 \[ -\frac {b n \,x^{n +1}}{n +1}+x \ln \left ({\mathrm e}^{b \,x^{n}+a}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 16, normalized size = 0.59 \[ a x + \frac {b x^{n + 1}}{n + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.60, size = 49, normalized size = 1.81 \[ \left \{\begin {array}{cl} x\,\ln \left ({\mathrm {e}}^{a+\frac {b}{x}}\right )+b\,\ln \relax (x) & \text {\ if\ \ }n=-1\\ x\,\ln \left ({\mathrm {e}}^{a+b\,x^n}\right )-\frac {b\,n\,x^{n+1}}{n+1} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.55, size = 34, normalized size = 1.26 \[ \begin {cases} \frac {a n x}{n + 1} + \frac {a x}{n + 1} + \frac {b x x^{n}}{n + 1} & \text {for}\: n \neq -1 \\a x + b \log {\relax (x )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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