Optimal. Leaf size=80 \[ -\frac{b^3 \tanh ^{-1}(\tanh (a+b x))}{210 x^7}-\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{60 x^8}-\frac{2 b \tanh ^{-1}(\tanh (a+b x))^3}{45 x^9}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{10 x^{10}}-\frac{b^4}{1260 x^6} \]
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Rubi [A] time = 0.0561019, antiderivative size = 80, normalized size of antiderivative = 1., 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 2168
Rule 30
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*}
Mathematica [A] time = 0.0324927, 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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Maple [A] time = 0.04, size = 74, normalized size = 0.9 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4}}{10\,{x}^{10}}}+{\frac{2\,b}{5} \left ( -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{9\,{x}^{9}}}+{\frac{b}{3} \left ( -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{8\,{x}^{8}}}+{\frac{b}{4} \left ( -{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{7\,{x}^{7}}}-{\frac{b}{42\,{x}^{6}}} \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.80116, size = 97, normalized size = 1.21 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46743, size = 116, normalized size = 1.45 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 78.1129, size = 78, normalized size = 0.98 \begin{align*} - \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16586, size = 62, normalized size = 0.78 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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