Optimal. Leaf size=64 \[ \frac {\coth ^{-1}(\tanh (a+b x))^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(n+1) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A] time = 0.02, antiderivative size = 64, 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 = {2164} \[ \frac {\coth ^{-1}(\tanh (a+b x))^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(n+1) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
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Rule 2164
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x} \, dx &=\frac {\coth ^{-1}(\tanh (a+b x))^{1+n} \, _2F_1\left (1,1+n;2+n;-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{(1+n) \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 60, normalized size = 0.94 \[ \frac {\coth ^{-1}(\tanh (a+b x))^n \left (\frac {\coth ^{-1}(\tanh (a+b x))}{b x}\right )^{-n} \, _2F_1\left (-n,-n;1-n;1-\frac {\coth ^{-1}(\tanh (a+b x))}{b x}\right )}{n} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.92, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x}, x\right ) \]
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 \[ \int \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.68, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{n}}{x}\, dx \]
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 \[ \int \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^n}{x} \,d x \]
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 \[ \int \frac {\operatorname {acoth}^{n}{\left (\tanh {\left (a + b x \right )} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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