Optimal. Leaf size=53 \[ -\frac {x^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{(m+1) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A] time = 0.03, antiderivative size = 53, 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 = {2164} \[ -\frac {x^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{(m+1) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
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Rule 2164
Rubi steps
\begin {align*} \int \frac {x^m}{\tanh ^{-1}(\tanh (a+b x))} \, dx &=-\frac {x^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{(1+m) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 51, normalized size = 0.96 \[ \frac {x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {b x}{\tanh ^{-1}(\tanh (a+b x))-b x}\right )}{(m+1) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}, x\right ) \]
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 \[ \int \frac {x^{m}}{\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.04, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\arctanh \left (\tanh \left (b x +a \right )\right )}\, dx \]
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 \[ \int \frac {x^{m}}{\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^m}{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )} \,d x \]
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 \[ \int \frac {x^{m}}{\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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