Optimal. Leaf size=80 \[ -\frac {b^3 \tanh ^{-1}(\tanh (a+b x))}{210 x^7}-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{60 x^8}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{10 x^{10}}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{45 x^9}-\frac {b^4}{1260 x^6} \]
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Rubi [A] time = 0.06, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2168, 30} \[ -\frac {b^3 \tanh ^{-1}(\tanh (a+b x))}{210 x^7}-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{60 x^8}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{45 x^9}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{10 x^{10}}-\frac {b^4}{1260 x^6} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^4}{x^{11}} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{10 x^{10}}+\frac {1}{5} (2 b) \int \frac {\tanh ^{-1}(\tanh (a+b x))^3}{x^{10}} \, dx\\ &=-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{45 x^9}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{10 x^{10}}+\frac {1}{15} \left (2 b^2\right ) \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x^9} \, dx\\ &=-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{60 x^8}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{45 x^9}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{10 x^{10}}+\frac {1}{30} b^3 \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x^8} \, dx\\ &=-\frac {b^3 \tanh ^{-1}(\tanh (a+b x))}{210 x^7}-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{60 x^8}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{45 x^9}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{10 x^{10}}+\frac {1}{210} b^4 \int \frac {1}{x^7} \, dx\\ &=-\frac {b^4}{1260 x^6}-\frac {b^3 \tanh ^{-1}(\tanh (a+b x))}{210 x^7}-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{60 x^8}-\frac {2 b \tanh ^{-1}(\tanh (a+b x))^3}{45 x^9}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{10 x^{10}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 71, normalized size = 0.89 \[ -\frac {6 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+21 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2+56 b x \tanh ^{-1}(\tanh (a+b x))^3+126 \tanh ^{-1}(\tanh (a+b x))^4+b^4 x^4}{1260 x^{10}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 46, normalized size = 0.58 \[ -\frac {210 \, b^{4} x^{4} + 720 \, a b^{3} x^{3} + 945 \, a^{2} b^{2} x^{2} + 560 \, a^{3} b x + 126 \, a^{4}}{1260 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 46, normalized size = 0.58 \[ -\frac {210 \, b^{4} x^{4} + 720 \, a b^{3} x^{3} + 945 \, a^{2} b^{2} x^{2} + 560 \, a^{3} b x + 126 \, a^{4}}{1260 \, x^{10}} \]
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 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{10 x^{10}}+\frac {2 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{9 x^{9}}+\frac {b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{8 x^{8}}+\frac {b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{7 x^{7}}-\frac {b}{42 x^{6}}\right )}{4}\right )}{3}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 72, normalized size = 0.90 \[ -\frac {1}{1260} \, {\left (b {\left (\frac {b^{2}}{x^{6}} + \frac {6 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{x^{7}}\right )} + \frac {21 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x^{8}}\right )} b - \frac {2 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{45 \, x^{9}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{10 \, x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 70, normalized size = 0.88 \[ -\frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^4}{10\,x^{10}}-\frac {b^4}{1260\,x^6}-\frac {b^2\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{60\,x^8}-\frac {b^3\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{210\,x^7}-\frac {2\,b\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{45\,x^9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 19.49, size = 78, normalized size = 0.98 \[ - \frac {b^{4}}{1260 x^{6}} - \frac {b^{3} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{210 x^{7}} - \frac {b^{2} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{60 x^{8}} - \frac {2 b \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{45 x^{9}} - \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{10 x^{10}} \]
Verification of antiderivative is not currently implemented for this CAS.
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