Optimal. Leaf size=55 \[ -\frac {b^2 \tanh ^{-1}(\tanh (a+b x))}{x}-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}-\frac {b \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}+b^3 \log (x) \]
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Rubi [A] time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2168, 29} \[ -\frac {b^2 \tanh ^{-1}(\tanh (a+b x))}{x}-\frac {b \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}+b^3 \log (x) \]
Antiderivative was successfully verified.
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Rule 29
Rule 2168
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^3}{x^4} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}+b \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x^3} \, dx\\ &=-\frac {b \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}+b^2 \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x^2} \, dx\\ &=-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))}{x}-\frac {b \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}+b^3 \int \frac {1}{x} \, dx\\ &=-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))}{x}-\frac {b \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}+b^3 \log (x)\\ \end {align*}
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Mathematica [A] time = 0.02, size = 60, normalized size = 1.09 \[ \frac {-6 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))-3 b x \tanh ^{-1}(\tanh (a+b x))^2-2 \tanh ^{-1}(\tanh (a+b x))^3+b^3 x^3 (6 \log (x)+11)}{6 x^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 37, normalized size = 0.67 \[ \frac {6 \, b^{3} x^{3} \log \relax (x) - 18 \, a b^{2} x^{2} - 9 \, a^{2} b x - 2 \, a^{3}}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 35, normalized size = 0.64 \[ b^{3} \log \left ({\left | x \right |}\right ) - \frac {18 \, a b^{2} x^{2} + 9 \, a^{2} b x + 2 \, a^{3}}{6 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 52, normalized size = 0.95 \[ -\frac {b^{2} \arctanh \left (\tanh \left (b x +a \right )\right )}{x}-\frac {b \arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{2 x^{2}}-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{3 x^{3}}+b^{3} \ln \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 52, normalized size = 0.95 \[ {\left (b^{2} \log \relax (x) - \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{x}\right )} b - \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{2 \, x^{2}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.00, size = 51, normalized size = 0.93 \[ b^3\,\ln \relax (x)-\frac {b^2\,x^2\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )+\frac {b\,x\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{2}+\frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^3}{3}}{x^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.84, size = 51, normalized size = 0.93 \[ b^{3} \log {\relax (x )} - \frac {b^{2} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{x} - \frac {b \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{2 x^{2}} - \frac {\operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{3 x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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