Optimal. Leaf size=22 \[ \frac {\log (x)}{2}-\frac {\log \left (b x^n+2\right )}{2 n} \]
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Rubi [A] time = 0.01, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 36, 29, 31} \[ \frac {\log (x)}{2}-\frac {\log \left (b x^n+2\right )}{2 n} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \left (2+b x^n\right )} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x (2+b x)} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^n\right )}{2 n}-\frac {b \operatorname {Subst}\left (\int \frac {1}{2+b x} \, dx,x,x^n\right )}{2 n}\\ &=\frac {\log (x)}{2}-\frac {\log \left (2+b x^n\right )}{2 n}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 22, normalized size = 1.00 \[ \frac {n \log (x)-\log \left (b x^n+2\right )}{2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 20, normalized size = 0.91 \[ \frac {n \log \relax (x) - \log \left (b x^{n} + 2\right )}{2 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{n} + 2\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 1.09 \[ \frac {\ln \left (x^{n}\right )}{2 n}-\frac {\ln \left (b \,x^{n}+2\right )}{2 n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 23, normalized size = 1.05 \[ -\frac {\log \left (b x^{n} + 2\right )}{2 \, n} + \frac {\log \left (x^{n}\right )}{2 \, n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 18, normalized size = 0.82 \[ \frac {\ln \relax (x)}{2}-\frac {\ln \left (b\,x^n+2\right )}{2\,n} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 29, normalized size = 1.32 \[ \begin {cases} \frac {\log {\relax (x )}}{2} & \text {for}\: b = 0 \wedge \left (b = 0 \vee n = 0\right ) \\\frac {\log {\relax (x )}}{b + 2} & \text {for}\: n = 0 \\\frac {\log {\relax (x )}}{2} - \frac {\log {\left (x^{n} + \frac {2}{b} \right )}}{2 n} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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