Optimal. Leaf size=55 \[ -\frac {b^2 \tanh ^{-1}(\tanh (a+b x))}{x}-\frac {b \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac {\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}+b^3 \log (x) \]
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Rubi [A]
time = 0.03, 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 = {2199, 29}
\begin {gather*} -\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) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2199
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 \begin {gather*} \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 (11+6 \log (x))}{6 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.93, size = 52, normalized size = 0.95
method | result | size |
default | \(-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{3 x^{3}}+b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{2 x^{2}}+b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{x}+b \ln \left (x \right )\right )\right )\) | \(52\) |
risch | \(\text {Expression too large to display}\) | \(7816\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.37, size = 52, normalized size = 0.95 \begin {gather*} {\left (b^{2} \log \left (x\right ) - \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 37, normalized size = 0.67 \begin {gather*} \frac {6 \, b^{3} x^{3} \log \left (x\right ) - 18 \, a b^{2} x^{2} - 9 \, a^{2} b x - 2 \, a^{3}}{6 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.29, size = 51, normalized size = 0.93 \begin {gather*} b^{3} \log {\left (x \right )} - \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 35, normalized size = 0.64 \begin {gather*} 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}} \end {gather*}
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 \begin {gather*} b^3\,\ln \left (x\right )-\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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