Optimal. Leaf size=65 \[ -\frac {x^m}{b \coth ^{-1}(\tanh (a+b x))}-\frac {x^m \, _2F_1\left (1,m;1+m;\frac {b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A]
time = 0.03, 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 = {2199, 2195}
\begin {gather*} -\frac {x^m \, _2F_1\left (1,m;m+1;\frac {b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac {x^m}{b \coth ^{-1}(\tanh (a+b x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 2195
Rule 2199
Rubi steps
\begin {align*} \int \frac {x^m}{\coth ^{-1}(\tanh (a+b x))^2} \, dx &=-\frac {x^m}{b \coth ^{-1}(\tanh (a+b x))}+\frac {m \int \frac {x^{-1+m}}{\coth ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=-\frac {x^m}{b \coth ^{-1}(\tanh (a+b x))}-\frac {x^m \, _2F_1\left (1,m;1+m;\frac {b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{b \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [A]
time = 0.37, size = 51, normalized size = 0.78 \begin {gather*} \frac {x^{1+m} \, _2F_1\left (2,1+m;2+m;-\frac {b x}{-b x+\coth ^{-1}(\tanh (a+b x))}\right )}{(1+m) \left (-b x+\coth ^{-1}(\tanh (a+b x))\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{2}}\, dx\]
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 \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 \frac {x^{m}}{\operatorname {acoth}^{2}{\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 = -0.02 \begin {gather*} \int \frac {x^m}{{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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