Optimal. Leaf size=79 \[ \frac {x^m \left (\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )^{-m} \tanh ^{-1}(\tanh (a+b x))^{1+n} \, _2F_1\left (-m,1+n;2+n;-\frac {\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b (1+n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2204}
\begin {gather*} \frac {x^m \left (\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )^{-m} \tanh ^{-1}(\tanh (a+b x))^{n+1} \, _2F_1\left (-m,n+1;n+2;-\frac {\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2204
Rubi steps
\begin {align*} \int x^m \tanh ^{-1}(\tanh (a+b x))^n \, dx &=\frac {x^m \left (\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )^{-m} \tanh ^{-1}(\tanh (a+b x))^{1+n} \, _2F_1\left (-m,1+n;2+n;-\frac {\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 71, normalized size = 0.90 \begin {gather*} \frac {x^{1+m} \tanh ^{-1}(\tanh (a+b x))^n \left (1+\frac {b x}{-b x+\tanh ^{-1}(\tanh (a+b x))}\right )^{-n} \, _2F_1\left (1+m,-n;2+m;-\frac {b x}{-b x+\tanh ^{-1}(\tanh (a+b x))}\right )}{1+m} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int x^{m} \arctanh \left (\tanh \left (b x +a \right )\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{m} \operatorname {atanh}^{n}{\left (\tanh {\left (a + b x \right )} \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^m\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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