Optimal. Leaf size=71 \[ \frac {b \tanh ^{-1}(\tanh (a+b x))^n \, _2F_1\left (1,n;n+1;-\frac {\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b x-\tanh ^{-1}(\tanh (a+b x))}-\frac {\tanh ^{-1}(\tanh (a+b x))^n}{x} \]
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Rubi [A] time = 0.04, antiderivative size = 71, 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} \[ \frac {b \tanh ^{-1}(\tanh (a+b x))^n \, _2F_1\left (1,n;n+1;-\frac {\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b x-\tanh ^{-1}(\tanh (a+b x))}-\frac {\tanh ^{-1}(\tanh (a+b x))^n}{x} \]
Antiderivative was successfully verified.
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Rule 2164
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^n}{x^2} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^n}{x}+(b n) \int \frac {\tanh ^{-1}(\tanh (a+b x))^{-1+n}}{x} \, dx\\ &=-\frac {\tanh ^{-1}(\tanh (a+b x))^n}{x}+\frac {b \tanh ^{-1}(\tanh (a+b x))^n \, _2F_1\left (1,n;1+n;-\frac {\tanh ^{-1}(\tanh (a+b x))}{b x-\tanh ^{-1}(\tanh (a+b x))}\right )}{b x-\tanh ^{-1}(\tanh (a+b x))}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 67, normalized size = 0.94 \[ \frac {\tanh ^{-1}(\tanh (a+b x))^n \left (\frac {\tanh ^{-1}(\tanh (a+b x))}{b x}\right )^{-n} \, _2F_1\left (1-n,-n;2-n;1-\frac {\tanh ^{-1}(\tanh (a+b x))}{b x}\right )}{(n-1) x} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{n}}{x^{2}}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{n}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.88, size = 0, normalized size = 0.00 \[ \int \frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{n}}{x^{2}}\, dx \]
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 \[ \int \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{n}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^n}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {atanh}^{n}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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