Optimal. Leaf size=15 \[ \frac {\log \left (a+b x^n\right )}{b n} \]
[Out]
________________________________________________________________________________________
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 (a+b x^n\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{-1+n}}{a+b x^n} \, dx &=\frac {\log \left (a+b x^n\right )}{b n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.02, size = 18, normalized size = 1.20 \begin {gather*} \frac {\log \left (a n+b n x^n\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.25, size = 18, normalized size = 1.20
method | result | size |
norman | \(\frac {\ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}{b n}\) | \(18\) |
risch | \(\frac {\ln \left (x^{n}+\frac {a}{b}\right )}{b n}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.30, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log \left (b x^{n} + a\right )}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.41, size = 15, normalized size = 1.00 \begin {gather*} \frac {\log \left (b x^{n} + a\right )}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs.
\(2 (10) = 20\).
time = 0.88, size = 27, normalized size = 1.80 \begin {gather*} \begin {cases} \frac {\log {\left (x \right )}}{a} & \text {for}\: b = 0 \wedge n = 0 \\\frac {x^{n}}{a n} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{a + b} & \text {for}\: n = 0 \\\frac {\log {\left (\frac {a}{b} + x^{n} \right )}}{b n} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.20, size = 16, normalized size = 1.07 \begin {gather*} \frac {\log \left ({\left | b x^{n} + a \right |}\right )}{b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.22, size = 15, normalized size = 1.00 \begin {gather*} \frac {\ln \left (a+b\,x^n\right )}{b\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________