Optimal. Leaf size=45 \[ \frac {\left (a+b x^n\right ) \cot ^{-1}\left (a+b x^n\right )}{b n}+\frac {\log \left (1+\left (a+b x^n\right )^2\right )}{2 b n} \]
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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, 5148,
4931, 266} \begin {gather*} \frac {\log \left (\left (a+b x^n\right )^2+1\right )}{2 b n}+\frac {\left (a+b x^n\right ) \cot ^{-1}\left (a+b x^n\right )}{b n} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 4931
Rule 5148
Rule 6847
Rubi steps
\begin {align*} \int x^{-1+n} \cot ^{-1}\left (a+b x^n\right ) \, dx &=\frac {\text {Subst}\left (\int \cot ^{-1}(a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac {\text {Subst}\left (\int \cot ^{-1}(x) \, dx,x,a+b x^n\right )}{b n}\\ &=\frac {\left (a+b x^n\right ) \cot ^{-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 ) \cot ^{-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 ) \cot ^{-1}\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 = 0.12, size = 149, normalized size = 3.31
method | result | size |
risch | \(\frac {i x^{n} \ln \left (1+i \left (a +b \,x^{n}\right )\right )}{2 n}-\frac {i x^{n} \ln \left (1-i \left (a +b \,x^{n}\right )\right )}{2 n}+\frac {\pi \,x^{n}}{2 n}+\frac {i \ln \left (x^{n}-\frac {i-a}{b}\right ) a}{2 b n}-\frac {i \ln \left (\frac {i+a}{b}+x^{n}\right ) a}{2 b n}+\frac {\ln \left (x^{n}-\frac {i-a}{b}\right )}{2 b n}+\frac {\ln \left (\frac {i+a}{b}+x^{n}\right )}{2 b n}\) | \(149\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 38, normalized size = 0.84 \begin {gather*} \frac {2 \, {\left (b x^{n} + a\right )} \operatorname {arccot}\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 = 1.94, size = 56, normalized size = 1.24 \begin {gather*} \frac {2 \, b x^{n} \operatorname {arccot}\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 = 0.43, size = 60, normalized size = 1.33 \begin {gather*} \frac {b {\left (\frac {2 \, {\left (b x^{n} + a\right )} \arctan \left (\frac {1}{b x^{n} + a}\right )}{b^{2}} + \frac {\log \left (\frac {1}{{\left (b x^{n} + a\right )}^{2}} + 1\right )}{b^{2}} - \frac {\log \left (\frac {1}{{\left (b x^{n} + a\right )}^{2}}\right )}{b^{2}}\right )}}{2 \, n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.38, size = 58, normalized size = 1.29 \begin {gather*} \frac {\frac {\ln \left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n+1\right )}{2}+a\,\mathrm {acot}\left (a+b\,x^n\right )}{b\,n}+\frac {x^n\,\mathrm {acot}\left (a+b\,x^n\right )}{n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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