Integrand size = 13, antiderivative size = 101 \[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x^3} \, dx=-\frac {b n \coth ^{-1}(\tanh (a+b x))^{-1+n}}{2 x}-\frac {\coth ^{-1}(\tanh (a+b x))^n}{2 x^2}+\frac {b^2 n \coth ^{-1}(\tanh (a+b x))^{-1+n} \operatorname {Hypergeometric2F1}\left (1,-1+n,n,-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )} \]
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Time = 0.05 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2199, 2195} \[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x^3} \, dx=\frac {b^2 n \coth ^{-1}(\tanh (a+b x))^{n-1} \operatorname {Hypergeometric2F1}\left (1,n-1,n,-\frac {\coth ^{-1}(\tanh (a+b x))}{b x-\coth ^{-1}(\tanh (a+b x))}\right )}{2 \left (b x-\coth ^{-1}(\tanh (a+b x))\right )}-\frac {\coth ^{-1}(\tanh (a+b x))^n}{2 x^2}-\frac {b n \coth ^{-1}(\tanh (a+b x))^{n-1}}{2 x} \]
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Rule 2195
Rule 2199
Rubi steps \begin{align*} \text {integral}& = -\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} \operatorname {Hypergeometric2F1}\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*}
Time = 0.03 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.66 \[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x^3} \, dx=\frac {\coth ^{-1}(\tanh (a+b x))^n \left (\frac {\coth ^{-1}(\tanh (a+b x))}{b x}\right )^{-n} \operatorname {Hypergeometric2F1}\left (2-n,-n,3-n,1-\frac {\coth ^{-1}(\tanh (a+b x))}{b x}\right )}{(-2+n) x^2} \]
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\[\int \frac {\operatorname {arccoth}\left (\tanh \left (b x +a \right )\right )^{n}}{x^{3}}d x\]
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\[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x^3} \, dx=\int { \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x^{3}} \,d x } \]
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\[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x^3} \, dx=\int \frac {\operatorname {acoth}^{n}{\left (\tanh {\left (a + b x \right )} \right )}}{x^{3}}\, dx \]
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\[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x^3} \, dx=\int { \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x^{3}} \,d x } \]
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\[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x^3} \, dx=\int { \frac {\operatorname {arcoth}\left (\tanh \left (b x + a\right )\right )^{n}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\coth ^{-1}(\tanh (a+b x))^n}{x^3} \, dx=\int \frac {{\mathrm {acoth}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^n}{x^3} \,d x \]
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