Optimal. Leaf size=80 \[ -\frac {b^3 \tanh ^{-1}(\tanh (a+b x))}{70 x^5}-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{28 x^6}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{8 x^8}-\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{14 x^7}-\frac {b^4}{280 x^4} \]
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Rubi [A] time = 0.08, antiderivative size = 132, normalized size of antiderivative = 1.65, number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2171, 2167} \[ \frac {b^2 \tanh ^{-1}(\tanh (a+b x))^5}{56 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac {b^3 \tanh ^{-1}(\tanh (a+b x))^5}{280 x^5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4}+\frac {\tanh ^{-1}(\tanh (a+b x))^5}{8 x^8 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac {3 b \tanh ^{-1}(\tanh (a+b x))^5}{56 x^7 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
Antiderivative was successfully verified.
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Rule 2167
Rule 2171
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^4}{x^9} \, dx &=\frac {\tanh ^{-1}(\tanh (a+b x))^5}{8 x^8 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac {(3 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))^4}{x^8} \, dx}{8 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac {3 b \tanh ^{-1}(\tanh (a+b x))^5}{56 x^7 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac {\tanh ^{-1}(\tanh (a+b x))^5}{8 x^8 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac {\left (3 b^2\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))^4}{x^7} \, dx}{28 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}\\ &=\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^5}{56 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac {3 b \tanh ^{-1}(\tanh (a+b x))^5}{56 x^7 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac {\tanh ^{-1}(\tanh (a+b x))^5}{8 x^8 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac {b^3 \int \frac {\tanh ^{-1}(\tanh (a+b x))^4}{x^6} \, dx}{56 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}\\ &=\frac {b^3 \tanh ^{-1}(\tanh (a+b x))^5}{280 x^5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^4}+\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^5}{56 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac {3 b \tanh ^{-1}(\tanh (a+b x))^5}{56 x^7 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac {\tanh ^{-1}(\tanh (a+b x))^5}{8 x^8 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 71, normalized size = 0.89 \[ -\frac {4 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+10 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2+20 b x \tanh ^{-1}(\tanh (a+b x))^3+35 \tanh ^{-1}(\tanh (a+b x))^4+b^4 x^4}{280 x^8} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 46, normalized size = 0.58 \[ -\frac {70 \, b^{4} x^{4} + 224 \, a b^{3} x^{3} + 280 \, a^{2} b^{2} x^{2} + 160 \, a^{3} b x + 35 \, a^{4}}{280 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 46, normalized size = 0.58 \[ -\frac {70 \, b^{4} x^{4} + 224 \, a b^{3} x^{3} + 280 \, a^{2} b^{2} x^{2} + 160 \, a^{3} b x + 35 \, a^{4}}{280 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 74, normalized size = 0.92 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{8 x^{8}}+\frac {b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{7 x^{7}}+\frac {3 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{6 x^{6}}+\frac {b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{5 x^{5}}-\frac {b}{20 x^{4}}\right )}{3}\right )}{7}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 72, normalized size = 0.90 \[ -\frac {1}{280} \, {\left (b {\left (\frac {b^{2}}{x^{4}} + \frac {4 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{x^{5}}\right )} + \frac {10 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x^{6}}\right )} b - \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{14 \, x^{7}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{8 \, x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.07, size = 70, normalized size = 0.88 \[ -\frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^4}{8\,x^8}-\frac {b^4}{280\,x^4}-\frac {b^2\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{28\,x^6}-\frac {b^3\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{70\,x^5}-\frac {b\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{14\,x^7} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 8.25, size = 76, normalized size = 0.95 \[ - \frac {b^{4}}{280 x^{4}} - \frac {b^{3} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{70 x^{5}} - \frac {b^{2} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{28 x^{6}} - \frac {b \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{14 x^{7}} - \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{8 x^{8}} \]
Verification of antiderivative is not currently implemented for this CAS.
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