Optimal. Leaf size=42 \[ -\frac {\tanh ^{-1}(\tanh (a+b x))^2}{4 x^4}-\frac {b \tanh ^{-1}(\tanh (a+b x))}{6 x^3}-\frac {b^2}{12 x^2} \]
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Rubi [A] time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.52, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2171, 2167} \[ \frac {\tanh ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{12 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
Antiderivative was successfully verified.
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Rule 2167
Rule 2171
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x^5} \, dx &=\frac {\tanh ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac {b \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x^4} \, dx}{4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{12 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac {\tanh ^{-1}(\tanh (a+b x))^3}{4 x^4 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 37, normalized size = 0.88 \[ -\frac {2 b x \tanh ^{-1}(\tanh (a+b x))+3 \tanh ^{-1}(\tanh (a+b x))^2+b^2 x^2}{12 x^4} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 24, normalized size = 0.57 \[ -\frac {6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.40, size = 24, normalized size = 0.57 \[ -\frac {6 \, b^{2} x^{2} + 8 \, a b x + 3 \, a^{2}}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 38, normalized size = 0.90 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{4 x^{4}}+\frac {b \left (-\frac {b}{6 x^{2}}-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{3 x^{3}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 36, normalized size = 0.86 \[ -\frac {b^{2}}{12 \, x^{2}} - \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{6 \, x^{3}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.95, size = 36, normalized size = 0.86 \[ -\frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{4\,x^4}-\frac {b^2}{12\,x^2}-\frac {b\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{6\,x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.27, size = 39, normalized size = 0.93 \[ - \frac {b^{2}}{12 x^{2}} - \frac {b \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{6 x^{3}} - \frac {\operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{4 x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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