Optimal. Leaf size=94 \[ -\frac {x^m}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {m x^{-1+m}}{2 b^2 \tanh ^{-1}(\tanh (a+b x))}-\frac {m x^{-1+m} \, _2F_1\left (1,-1+m;m;\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{2 b^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A]
time = 0.04, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 {m x^{m-1} \, _2F_1\left (1,m-1;m;\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{2 b^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {m x^{m-1}}{2 b^2 \tanh ^{-1}(\tanh (a+b x))}-\frac {x^m}{2 b \tanh ^{-1}(\tanh (a+b x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2195
Rule 2199
Rubi steps
\begin {align*} \int \frac {x^m}{\tanh ^{-1}(\tanh (a+b x))^3} \, dx &=-\frac {x^m}{2 b \tanh ^{-1}(\tanh (a+b x))^2}+\frac {m \int \frac {x^{-1+m}}{\tanh ^{-1}(\tanh (a+b x))^2} \, dx}{2 b}\\ &=-\frac {x^m}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {m x^{-1+m}}{2 b^2 \tanh ^{-1}(\tanh (a+b x))}-\frac {((1-m) m) \int \frac {x^{-2+m}}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{2 b^2}\\ &=-\frac {x^m}{2 b \tanh ^{-1}(\tanh (a+b x))^2}-\frac {m x^{-1+m}}{2 b^2 \tanh ^{-1}(\tanh (a+b x))}-\frac {m x^{-1+m} \, _2F_1\left (1,-1+m;m;\frac {b x}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{2 b^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 51, normalized size = 0.54 \begin {gather*} \frac {x^{1+m} \, _2F_1\left (3,1+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 )^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x^{m}}{\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}\, dx\]
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 {atanh}^{3}{\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 = -0.01 \begin {gather*} \int \frac {x^m}{{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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