Optimal. Leaf size=15 \[ \frac {\log \left (2+b x^n\right )}{b n} \]
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Rubi [A]
time = 0.00, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {266}
\begin {gather*} \frac {\log \left (b x^n+2\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rubi steps
\begin {align*} \int \frac {x^{-1+n}}{2+b x^n} \, dx &=\frac {\log \left (2+b x^n\right )}{b n}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.20 \begin {gather*} \frac {\log \left (2 n+b n x^n\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 18, normalized size = 1.20
method | result | size |
norman | \(\frac {\ln \left (2+b \,{\mathrm e}^{n \ln \left (x \right )}\right )}{b n}\) | \(18\) |
risch | \(\frac {\ln \left (x^{n}+\frac {2}{b}\right )}{b n}\) | \(18\) |
meijerg | \(-\frac {i \left (-1\right )^{-\frac {\mathrm {csgn}\left (i b \right )}{2}-\frac {\mathrm {csgn}\left (i x^{n}\right )}{2}+\frac {\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i b \right )}{2}-\frac {-1+n}{n}-\frac {1}{n}} \ln \left (1-\frac {i x^{n} b \left (-1\right )^{\frac {\mathrm {csgn}\left (i b \right )}{2}+\frac {\mathrm {csgn}\left (i x^{n}\right )}{2}-\frac {\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i b \right )}{2}}}{2}\right )}{b n}\) | \(99\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log \left (b x^{n} + 2\right )}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.40, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log \left (b x^{n} + 2\right )}{b n} \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. 27 vs.
\(2 (10) = 20\).
time = 0.70, size = 27, normalized size = 1.80 \begin {gather*} \begin {cases} \frac {\log {\left (x \right )}}{2} & \text {for}\: b = 0 \wedge n = 0 \\\frac {x^{n}}{2 n} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{b + 2} & \text {for}\: n = 0 \\\frac {\log {\left (x^{n} + \frac {2}{b} \right )}}{b n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.30, size = 16, normalized size = 1.07 \begin {gather*} \frac {\log \left ({\left | b x^{n} + 2 \right |}\right )}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.21, size = 15, normalized size = 1.00 \begin {gather*} \frac {\ln \left (b\,x^n+2\right )}{b\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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