Optimal. Leaf size=101 \[ \frac {b^2 n \coth ^{-1}(\tanh (a+b x))^{n-1} \, _2F_1\left (1,n-1;n;-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac {\coth ^{-1}(\tanh (a+b x))^n}{2 x^2}-\frac {b n \coth ^{-1}(\tanh (a+b x))^{n-1}}{2 x} \]
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Rubi [A] time = 0.07, 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 = {2168, 2164} \[ \frac {b^2 n \coth ^{-1}(\tanh (a+b x))^{n-1} \, _2F_1\left (1,n-1;n;-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac {\coth ^{-1}(\tanh (a+b x))^n}{2 x^2}-\frac {b n \coth ^{-1}(\tanh (a+b x))^{n-1}}{2 x} \]
Antiderivative was successfully verified.
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Rule 2164
Rule 2168
Rubi steps
\begin {align*} \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x^3} \, dx &=-\frac {\coth ^{-1}(\tanh (a+b x))^n}{2 x^2}+\frac {1}{2} (b n) \int \frac {\coth ^{-1}(\tanh (a+b x))^{-1+n}}{x^2} \, dx\\ &=-\frac {b n \coth ^{-1}(\tanh (a+b x))^{-1+n}}{2 x}-\frac {\coth ^{-1}(\tanh (a+b x))^n}{2 x^2}-\frac {1}{2} \left (b^2 (1-n) n\right ) \int \frac {\coth ^{-1}(\tanh (a+b x))^{-2+n}}{x} \, dx\\ &=-\frac {b n \coth ^{-1}(\tanh (a+b x))^{-1+n}}{2 x}-\frac {\coth ^{-1}(\tanh (a+b x))^n}{2 x^2}+\frac {b^2 n \coth ^{-1}(\tanh (a+b x))^{-1+n} \, _2F_1\left (1,-1+n;n;-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 67, normalized size = 0.66 \[ \frac {\coth ^{-1}(\tanh (a+b x))^n \left (\frac {\coth ^{-1}(\tanh (a+b x))}{b x}\right )^{-n} \, _2F_1\left (2-n,-n;3-n;1-\frac {\coth ^{-1}(\tanh (a+b x))}{b x}\right )}{(n-2) x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x^{3}}, x\right ) \]
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 \[ \int \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 13.91, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arccoth}\left (\tanh \left (b x +a \right )\right )^{n}}{x^{3}}\, dx \]
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 \[ \int \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^n}{x^3} \,d x \]
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 \[ \int \frac {\operatorname {acoth}^{n}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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