Optimal. Leaf size=65 \[ -\frac {x^m \, _2F_1\left (1,m;m+1;\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {x^m}{b \tanh ^{-1}(\tanh (a+b x))} \]
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Rubi [A] time = 0.04, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2168, 2164} \[ -\frac {x^m \, _2F_1\left (1,m;m+1;\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {x^m}{b \tanh ^{-1}(\tanh (a+b x))} \]
Antiderivative was successfully verified.
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Rule 2164
Rule 2168
Rubi steps
\begin {align*} \int \frac {x^m}{\tanh ^{-1}(\tanh (a+b x))^2} \, dx &=-\frac {x^m}{b \tanh ^{-1}(\tanh (a+b x))}+\frac {m \int \frac {x^{-1+m}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=-\frac {x^m}{b \tanh ^{-1}(\tanh (a+b x))}-\frac {x^m \, _2F_1\left (1,m;1+m;\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [A] time = 0.55, size = 51, normalized size = 0.78 \[ \frac {x^{m+1} \, _2F_1\left (2,m+1;m+2;-\frac {b x}{\tanh ^{-1}(\tanh (a+b x))-b x}\right )}{(m+1) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{m}}{\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}, x\right ) \]
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 \[ \int \frac {x^{m}}{\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.78, size = 0, normalized size = 0.00 \[ \int \frac {x^{m}}{\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}\, dx \]
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 \[ \int \frac {x^{m}}{\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}\,{d x} \]
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 \[ \int \frac {x^m}{{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2} \,d x \]
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 \[ \int \frac {x^{m}}{\operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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