Optimal. Leaf size=27 \[ -\frac {b n x^{1+n}}{1+n}+x \log \left (e^{a+b x^n}\right ) \]
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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 = {2628, 12, 30}
\begin {gather*} x \log \left (e^{a+b x^n}\right )-\frac {b n x^{n+1}}{n+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 30
Rule 2628
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 = 0.02, size = 25, normalized size = 0.93 \begin {gather*} x \left (-\frac {b n x^n}{1+n}+\log \left (e^{a+b x^n}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 27, normalized size = 1.00
method | result | size |
risch | \(x \ln \left ({\mathrm e}^{a +b \,x^{n}}\right )-\frac {b n x \,x^{n}}{1+n}\) | \(26\) |
default | \(-\frac {b n \,x^{1+n}}{1+n}+x \ln \left ({\mathrm e}^{a +b \,x^{n}}\right )\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 16, normalized size = 0.59 \begin {gather*} a x + \frac {b x^{n + 1}}{n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 20, normalized size = 0.74 \begin {gather*} \frac {b x x^{n} + {\left (a n + a\right )} x}{n + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 65 vs.
\(2 (22) = 44\).
time = 0.47, size = 65, normalized size = 2.41 \begin {gather*} \begin {cases} - \frac {b n x x^{n}}{n + 1} + \frac {n x \log {\left (e^{a} e^{b x^{n}} \right )}}{n + 1} + \frac {x \log {\left (e^{a} e^{b x^{n}} \right )}}{n + 1} & \text {for}\: n \neq -1 \\b \log {\left (x \right )} + x \log {\left (e^{a} e^{\frac {b}{x}} \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.96, size = 16, normalized size = 0.59 \begin {gather*} a x + \frac {b x^{n + 1}}{n + 1} \end {gather*}
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 \begin {gather*} \left \{\begin {array}{cl} x\,\ln \left ({\mathrm {e}}^{a+\frac {b}{x}}\right )+b\,\ln \left (x\right ) & \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 . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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