Optimal. Leaf size=53 \[ -\frac {x^{1+m} \, _2F_1\left (1,1+m;2+m;\frac {b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(1+m) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
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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 = {2195}
\begin {gather*} -\frac {x^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {b x}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(m+1) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2195
Rubi steps
\begin {align*} \int \frac {x^m}{\coth ^{-1}(\tanh (a+b x))} \, dx &=-\frac {x^{1+m} \, _2F_1\left (1,1+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 )}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 51, normalized size = 0.96 \begin {gather*} \frac {x^{1+m} \, _2F_1\left (1,1+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 )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )}\, dx\]
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 \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}{\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 )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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