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 = 0.05, antiderivative size = 98, normalized size of antiderivative = 1.00, 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 2167
Rule 2171
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.56, size = 46, normalized size = 0.47 \[ -\frac {35 \, b^{4} x^{4} + 105 \, a b^{3} x^{3} + 126 \, a^{2} b^{2} x^{2} + 70 \, a^{3} b x + 15 \, a^{4}}{105 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 46, normalized size = 0.47 \[ -\frac {35 \, b^{4} x^{4} + 105 \, a b^{3} x^{3} + 126 \, a^{2} b^{2} x^{2} + 70 \, a^{3} b x + 15 \, a^{4}}{105 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 74, normalized size = 0.76 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{7 x^{7}}+\frac {4 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{6 x^{6}}+\frac {b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{5 x^{5}}+\frac {2 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{4 x^{4}}-\frac {b}{12 x^{3}}\right )}{5}\right )}{2}\right )}{7} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 72, normalized size = 0.73 \[ -\frac {1}{105} \, {\left (b {\left (\frac {b^{2}}{x^{3}} + \frac {3 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{x^{4}}\right )} + \frac {6 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x^{5}}\right )} b - \frac {2 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{21 \, x^{6}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{7 \, x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.01, size = 70, normalized size = 0.71 \[ -\frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^4}{7\,x^7}-\frac {b^4}{105\,x^3}-\frac {2\,b^2\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{35\,x^5}-\frac {b^3\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{35\,x^4}-\frac {2\,b\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{21\,x^6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.41, size = 80, normalized size = 0.82 \[ - \frac {b^{4}}{105 x^{3}} - \frac {b^{3} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{35 x^{4}} - \frac {2 b^{2} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{35 x^{5}} - \frac {2 b \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{21 x^{6}} - \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{7 x^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
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