Optimal. Leaf size=64 \[ \frac{\tanh ^{-1}(\tanh (a+b x))^5}{6 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{b \tanh ^{-1}(\tanh (a+b x))^5}{30 x^5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
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Rubi [A] time = 0.0313652, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2171, 2167} \[ \frac{\tanh ^{-1}(\tanh (a+b x))^5}{6 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{b \tanh ^{-1}(\tanh (a+b x))^5}{30 x^5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]
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^7} \, dx &=\frac{\tanh ^{-1}(\tanh (a+b x))^5}{6 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{b \int \frac{\tanh ^{-1}(\tanh (a+b x))^4}{x^6} \, dx}{6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{b \tanh ^{-1}(\tanh (a+b x))^5}{30 x^5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac{\tanh ^{-1}(\tanh (a+b x))^5}{6 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}
Mathematica [A] time = 0.0313331, size = 71, normalized size = 1.11 \[ -\frac{2 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+3 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2+4 b x \tanh ^{-1}(\tanh (a+b x))^3+5 \tanh ^{-1}(\tanh (a+b x))^4+b^4 x^4}{30 x^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 74, normalized size = 1.2 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4}}{6\,{x}^{6}}}+{\frac{2\,b}{3} \left ( -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{5\,{x}^{5}}}+{\frac{3\,b}{5} \left ( -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{4\,{x}^{4}}}+{\frac{b}{2} \left ( -{\frac{b}{6\,{x}^{2}}}-{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{3\,{x}^{3}}} \right ) } \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.79142, size = 97, normalized size = 1.52 \begin{align*} -\frac{1}{30} \,{\left (b{\left (\frac{b^{2}}{x^{2}} + \frac{2 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{x^{3}}\right )} + \frac{3 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x^{4}}\right )} b - \frac{2 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{15 \, x^{5}} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{6 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45738, size = 104, normalized size = 1.62 \begin{align*} -\frac{15 \, b^{4} x^{4} + 40 \, a b^{3} x^{3} + 45 \, a^{2} b^{2} x^{2} + 24 \, a^{3} b x + 5 \, a^{4}}{30 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.67547, size = 78, normalized size = 1.22 \begin{align*} - \frac{b^{4}}{30 x^{2}} - \frac{b^{3} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{15 x^{3}} - \frac{b^{2} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{10 x^{4}} - \frac{2 b \operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{15 x^{5}} - \frac{\operatorname{atanh}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{6 x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16793, size = 62, normalized size = 0.97 \begin{align*} -\frac{15 \, b^{4} x^{4} + 40 \, a b^{3} x^{3} + 45 \, a^{2} b^{2} x^{2} + 24 \, a^{3} b x + 5 \, a^{4}}{30 \, x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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