Optimal. Leaf size=80 \[ -\frac{b^3 \tanh ^{-1}(\tanh (a+b x))}{126 x^6}-\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}-\frac{b^4}{630 x^5} \]
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Rubi [A] time = 0.0554779, 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} \[ -\frac{b^3 \tanh ^{-1}(\tanh (a+b x))}{126 x^6}-\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}-\frac{b^4}{630 x^5} \]
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^{10}} \, dx &=-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}+\frac{1}{9} (4 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))^3}{x^9} \, dx\\ &=-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}+\frac{1}{6} b^2 \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^8} \, dx\\ &=-\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}+\frac{1}{21} b^3 \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x^7} \, dx\\ &=-\frac{b^3 \tanh ^{-1}(\tanh (a+b x))}{126 x^6}-\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}+\frac{1}{126} b^4 \int \frac{1}{x^6} \, dx\\ &=-\frac{b^4}{630 x^5}-\frac{b^3 \tanh ^{-1}(\tanh (a+b x))}{126 x^6}-\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}\\ \end{align*}
Mathematica [A] time = 0.0599034, size = 71, normalized size = 0.89 \[ -\frac{5 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+15 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2+35 b x \tanh ^{-1}(\tanh (a+b x))^3+70 \tanh ^{-1}(\tanh (a+b x))^4+b^4 x^4}{630 x^9} \]
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}}{9\,{x}^{9}}}+{\frac{4\,b}{9} \left ( -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{8\,{x}^{8}}}+{\frac{3\,b}{8} \left ( -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{7\,{x}^{7}}}+{\frac{2\,b}{7} \left ( -{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{6\,{x}^{6}}}-{\frac{b}{30\,{x}^{5}}} \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.78264, size = 97, normalized size = 1.21 \begin{align*} -\frac{1}{630} \,{\left (b{\left (\frac{b^{2}}{x^{5}} + \frac{5 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{x^{6}}\right )} + \frac{15 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x^{7}}\right )} b - \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{18 \, x^{8}} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{9 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46187, size = 112, normalized size = 1.4 \begin{align*} -\frac{126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 25.8709, size = 76, normalized size = 0.95 \begin{align*} - \frac{b^{4}}{630 x^{5}} - \frac{b^{3} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{126 x^{6}} - \frac{b^{2} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{42 x^{7}} - \frac{b \operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{18 x^{8}} - \frac{\operatorname{atanh}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{9 x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12085, size = 62, normalized size = 0.78 \begin{align*} -\frac{126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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