Optimal. Leaf size=101 \[ -\frac {b n \tanh ^{-1}(\tanh (a+b x))^{-1+n}}{2 x}-\frac {\tanh ^{-1}(\tanh (a+b x))^n}{2 x^2}+\frac {b^2 n \tanh ^{-1}(\tanh (a+b x))^{-1+n} \, _2F_1\left (1,-1+n;n;-\frac {\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A]
time = 0.05, antiderivative size = 101, 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 {b^2 n \tanh ^{-1}(\tanh (a+b x))^{n-1} \, _2F_1\left (1,n-1;n;-\frac {\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}-\frac {\tanh ^{-1}(\tanh (a+b x))^n}{2 x^2}-\frac {b n \tanh ^{-1}(\tanh (a+b x))^{n-1}}{2 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 2195
Rule 2199
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^n}{x^3} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^n}{2 x^2}+\frac {1}{2} (b n) \int \frac {\tanh ^{-1}(\tanh (a+b x))^{-1+n}}{x^2} \, dx\\ &=-\frac {b n \tanh ^{-1}(\tanh (a+b x))^{-1+n}}{2 x}-\frac {\tanh ^{-1}(\tanh (a+b x))^n}{2 x^2}-\frac {1}{2} \left (b^2 (1-n) n\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))^{-2+n}}{x} \, dx\\ &=-\frac {b n \tanh ^{-1}(\tanh (a+b x))^{-1+n}}{2 x}-\frac {\tanh ^{-1}(\tanh (a+b x))^n}{2 x^2}+\frac {b^2 n \tanh ^{-1}(\tanh (a+b x))^{-1+n} \, _2F_1\left (1,-1+n;n;-\frac {\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 67, normalized size = 0.66 \begin {gather*} \frac {\tanh ^{-1}(\tanh (a+b x))^n \left (\frac {\tanh ^{-1}(\tanh (a+b x))}{b x}\right )^{-n} \, _2F_1\left (2-n,-n;3-n;1-\frac {\tanh ^{-1}(\tanh (a+b x))}{b x}\right )}{(-2+n) x^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{n}}{x^{3}}\, dx\]
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 {\operatorname {atanh}^{n}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{3}}\, 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 {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^n}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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