Optimal. Leaf size=45 \[ \frac {\left (a+b x^n\right ) \text {ArcTan}\left (a+b x^n\right )}{b n}-\frac {\log \left (1+\left (a+b x^n\right )^2\right )}{2 b n} \]
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Rubi [A]
time = 0.03, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6847, 5147,
4930, 266} \begin {gather*} \frac {\left (a+b x^n\right ) \text {ArcTan}\left (a+b x^n\right )}{b n}-\frac {\log \left (\left (a+b x^n\right )^2+1\right )}{2 b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 4930
Rule 5147
Rule 6847
Rubi steps
\begin {align*} \int x^{-1+n} \tan ^{-1}\left (a+b x^n\right ) \, dx &=\frac {\text {Subst}\left (\int \tan ^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \tan ^{-1}(x) \, dx,x,a+b x^n\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \tan ^{-1}\left (a+b x^n\right )}{b n}-\frac {\text {Subst}\left (\int \frac {x}{1+x^2} \, dx,x,a+b x^n\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \tan ^{-1}\left (a+b x^n\right )}{b n}-\frac {\log \left (1+\left (a+b x^n\right )^2\right )}{2 b n}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 40, normalized size = 0.89 \begin {gather*} -\frac {-2 \left (a+b x^n\right ) \text {ArcTan}\left (a+b x^n\right )+\log \left (1+\left (a+b x^n\right )^2\right )}{2 b n} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.08, size = 140, normalized size = 3.11
method | result | size |
risch | \(-\frac {i x^{n} \ln \left (1+i \left (a +b \,x^{n}\right )\right )}{2 n}+\frac {i x^{n} \ln \left (1-i \left (a +b \,x^{n}\right )\right )}{2 n}-\frac {\ln \left (x^{n}-\frac {i-a}{b}\right )}{2 b n}-\frac {\ln \left (\frac {i+a}{b}+x^{n}\right )}{2 b n}-\frac {i \ln \left (x^{n}-\frac {i-a}{b}\right ) a}{2 b n}+\frac {i \ln \left (\frac {i+a}{b}+x^{n}\right ) a}{2 b n}\) | \(140\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 40, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (b x^{n} + a\right )} \arctan \left (b x^{n} + a\right ) - \log \left ({\left (b x^{n} + a\right )}^{2} + 1\right )}{2 \, b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.34, size = 58, normalized size = 1.29 \begin {gather*} \frac {2 \, b x^{n} \arctan \left (b x^{n} + a\right ) + 2 \, a \arctan \left (b x^{n} + a\right ) - \log \left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2} + 1\right )}{2 \, b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: HeuristicGCDFailed} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 40, normalized size = 0.89 \begin {gather*} \frac {2 \, {\left (b x^{n} + a\right )} \arctan \left (b x^{n} + a\right ) - \log \left ({\left (b x^{n} + a\right )}^{2} + 1\right )}{2 \, b n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.63, size = 58, normalized size = 1.29 \begin {gather*} \frac {x^n\,\mathrm {atan}\left (a+b\,x^n\right )}{n}-\frac {\ln \left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n+1\right )-2\,a\,\mathrm {atan}\left (a+b\,x^n\right )}{2\,b\,n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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