Optimal. Leaf size=31 \[ \frac {\tanh ^{-1}(\tanh (a+b x))^5}{5 x^5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
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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} \[ \frac {\tanh ^{-1}(\tanh (a+b x))^5}{5 x^5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
Antiderivative was successfully verified.
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Rule 2167
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^4}{x^6} \, dx &=\frac {\tanh ^{-1}(\tanh (a+b x))^5}{5 x^5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [B] time = 0.06, size = 66, normalized size = 2.13 \[ -\frac {b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2+b x \tanh ^{-1}(\tanh (a+b x))^3+\tanh ^{-1}(\tanh (a+b x))^4+b^4 x^4}{5 x^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 44, normalized size = 1.42 \[ -\frac {5 \, b^{4} x^{4} + 10 \, a b^{3} x^{3} + 10 \, a^{2} b^{2} x^{2} + 5 \, a^{3} b x + a^{4}}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 44, normalized size = 1.42 \[ -\frac {5 \, b^{4} x^{4} + 10 \, a b^{3} x^{3} + 10 \, a^{2} b^{2} x^{2} + 5 \, a^{3} b x + a^{4}}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.15, size = 74, normalized size = 2.39 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{5 x^{5}}+\frac {4 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{4 x^{4}}+\frac {3 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{3 x^{3}}+\frac {2 b \left (-\frac {b}{2 x}-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{2 x^{2}}\right )}{3}\right )}{4}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.60, size = 70, normalized size = 2.26 \[ -\frac {1}{5} \, {\left (b {\left (\frac {b^{2}}{x} + \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{x^{2}}\right )} + \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x^{3}}\right )} b - \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{5 \, x^{4}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{5 \, x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.20, size = 64, normalized size = 2.06 \[ -\frac {b^4\,x^4+b^3\,x^3\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )+b^2\,x^2\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2+b\,x\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3+{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^4}{5\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 2.10, size = 75, normalized size = 2.42 \[ - \frac {b^{4}}{5 x} - \frac {b^{3} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{5 x^{2}} - \frac {b^{2} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{5 x^{3}} - \frac {b \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{5 x^{4}} - \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{5 x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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