Optimal. Leaf size=98 \[ \frac{b^2 \tanh ^{-1}(\tanh (a+b x))^5}{105 x^5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac{\tanh ^{-1}(\tanh (a+b x))^5}{7 x^7 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{b \tanh ^{-1}(\tanh (a+b x))^5}{21 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
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Rubi [A] time = 0.0531644, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2171, 2167} \[ \frac{b^2 \tanh ^{-1}(\tanh (a+b x))^5}{105 x^5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac{\tanh ^{-1}(\tanh (a+b x))^5}{7 x^7 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{b \tanh ^{-1}(\tanh (a+b x))^5}{21 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
Antiderivative was successfully verified.
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Rule 2171
Rule 2167
Rubi steps
\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^4}{x^8} \, dx &=\frac{\tanh ^{-1}(\tanh (a+b x))^5}{7 x^7 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{(2 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))^4}{x^7} \, dx}{7 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{b \tanh ^{-1}(\tanh (a+b x))^5}{21 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac{\tanh ^{-1}(\tanh (a+b x))^5}{7 x^7 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{b^2 \int \frac{\tanh ^{-1}(\tanh (a+b x))^4}{x^6} \, dx}{21 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}\\ &=\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^5}{105 x^5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3}+\frac{b \tanh ^{-1}(\tanh (a+b x))^5}{21 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac{\tanh ^{-1}(\tanh (a+b x))^5}{7 x^7 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.033165, size = 71, normalized size = 0.72 \[ -\frac{3 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+6 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2+10 b x \tanh ^{-1}(\tanh (a+b x))^3+15 \tanh ^{-1}(\tanh (a+b x))^4+b^4 x^4}{105 x^7} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 74, normalized size = 0.8 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4}}{7\,{x}^{7}}}+{\frac{4\,b}{7} \left ( -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{6\,{x}^{6}}}+{\frac{b}{2} \left ( -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{5\,{x}^{5}}}+{\frac{2\,b}{5} \left ( -{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{4\,{x}^{4}}}-{\frac{b}{12\,{x}^{3}}} \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.79776, size = 97, normalized size = 0.99 \begin{align*} -\frac{1}{105} \,{\left (b{\left (\frac{b^{2}}{x^{3}} + \frac{3 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{x^{4}}\right )} + \frac{6 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x^{5}}\right )} b - \frac{2 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{21 \, x^{6}} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{7 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52279, size = 109, normalized size = 1.11 \begin{align*} -\frac{35 \, b^{4} x^{4} + 105 \, a b^{3} x^{3} + 126 \, a^{2} b^{2} x^{2} + 70 \, a^{3} b x + 15 \, a^{4}}{105 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 11.6407, size = 80, normalized size = 0.82 \begin{align*} - \frac{b^{4}}{105 x^{3}} - \frac{b^{3} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{35 x^{4}} - \frac{2 b^{2} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{35 x^{5}} - \frac{2 b \operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{21 x^{6}} - \frac{\operatorname{atanh}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{7 x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15779, size = 62, normalized size = 0.63 \begin{align*} -\frac{35 \, b^{4} x^{4} + 105 \, a b^{3} x^{3} + 126 \, a^{2} b^{2} x^{2} + 70 \, a^{3} b x + 15 \, a^{4}}{105 \, x^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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