Optimal. Leaf size=79 \[ \frac {x^m \left (\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )^{-m} \tanh ^{-1}(\tanh (a+b x))^{n+1} \, _2F_1\left (-m,n+1;n+2;-\frac {\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b (n+1)} \]
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Rubi [A] time = 0.04, 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 = {2173} \[ \frac {x^m \left (\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )^{-m} \tanh ^{-1}(\tanh (a+b x))^{n+1} \, _2F_1\left (-m,n+1;n+2;-\frac {\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 2173
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 = 0.14, size = 71, normalized size = 0.90 \[ \frac {x^{m+1} \tanh ^{-1}(\tanh (a+b x))^n \left (\frac {b x}{\tanh ^{-1}(\tanh (a+b x))-b x}+1\right )^{-n} \, _2F_1\left (m+1,-n;m+2;-\frac {b x}{\tanh ^{-1}(\tanh (a+b x))-b x}\right )}{m+1} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{n}, x\right ) \]
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 \[ \int x^{m} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.58, 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 \[ \int x^{m} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{n}\,{d x} \]
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 \[ \int x^m\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^n \,d x \]
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 \[ \int x^{m} \operatorname {atanh}^{n}{\left (\tanh {\left (a + b x \right )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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