Optimal. Leaf size=31 \[ \frac {\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A] time = 0.01, antiderivative size = 31, 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 = {2167} \[ \frac {\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
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Rule 2167
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^3}{x^5} \, dx &=\frac {\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 50, normalized size = 1.61 \[ -\frac {b^2 x^2 \tanh ^{-1}(\tanh (a+b x))+b x \tanh ^{-1}(\tanh (a+b x))^2+\tanh ^{-1}(\tanh (a+b x))^3+b^3 x^3}{4 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 33, normalized size = 1.06 \[ -\frac {4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 33, normalized size = 1.06 \[ -\frac {4 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 4 \, a^{2} b x + a^{3}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 56, normalized size = 1.81 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{4 x^{4}}+\frac {3 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{3 x^{3}}+\frac {2 b \left (-\frac {b}{2 x}-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{2 x^{2}}\right )}{3}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.53, size = 53, normalized size = 1.71 \[ -\frac {1}{4} \, b {\left (\frac {b^{2}}{x} + \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{x^{2}}\right )} - \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{4 \, x^{3}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 48, normalized size = 1.55 \[ -\frac {b^3\,x^3+b^2\,x^2\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )+b\,x\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2+{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{4\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.31, size = 56, normalized size = 1.81 \[ - \frac {b^{3}}{4 x} - \frac {b^{2} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{4 x^{2}} - \frac {b \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{4 x^{3}} - \frac {\operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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