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 = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.00, 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 2167
Rule 2171
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.40, size = 46, normalized size = 0.72 \[ -\frac {15 \, b^{4} x^{4} + 40 \, a b^{3} x^{3} + 45 \, a^{2} b^{2} x^{2} + 24 \, a^{3} b x + 5 \, a^{4}}{30 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 46, normalized size = 0.72 \[ -\frac {15 \, b^{4} x^{4} + 40 \, a b^{3} x^{3} + 45 \, a^{2} b^{2} x^{2} + 24 \, a^{3} b x + 5 \, a^{4}}{30 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.15, size = 74, normalized size = 1.16 \[ -\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{6 x^{6}}+\frac {2 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{5 x^{5}}+\frac {3 b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{4 x^{4}}+\frac {b \left (-\frac {b}{6 x^{2}}-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{3 x^{3}}\right )}{2}\right )}{5}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.60, size = 72, normalized size = 1.12 \[ -\frac {1}{30} \, {\left (b {\left (\frac {b^{2}}{x^{2}} + \frac {2 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{x^{3}}\right )} + \frac {3 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x^{4}}\right )} b - \frac {2 \, b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{15 \, x^{5}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{6 \, x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.04, size = 70, normalized size = 1.09 \[ -\frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^4}{6\,x^6}-\frac {b^4}{30\,x^2}-\frac {b^2\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{10\,x^4}-\frac {b^3\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{15\,x^3}-\frac {2\,b\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{15\,x^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.29, size = 78, normalized size = 1.22 \[ - \frac {b^{4}}{30 x^{2}} - \frac {b^{3} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{15 x^{3}} - \frac {b^{2} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{10 x^{4}} - \frac {2 b \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{15 x^{5}} - \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{6 x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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