Optimal. Leaf size=74 \[ -\frac {b^3 \tanh ^{-1}(\tanh (a+b x))}{x}-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4}+b^4 \log (x) \]
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Rubi [A]
time = 0.04, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps
used = 5, 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^3 \tanh ^{-1}(\tanh (a+b x))}{x}-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4}-\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}+b^4 \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))^4}{x^5} \, dx &=-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4}+b \int \frac {\tanh ^{-1}(\tanh (a+b x))^3}{x^4} \, dx\\ &=-\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4}+b^2 \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x^3} \, dx\\ &=-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4}+b^3 \int \frac {\tanh ^{-1}(\tanh (a+b x))}{x^2} \, dx\\ &=-\frac {b^3 \tanh ^{-1}(\tanh (a+b x))}{x}-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4}+b^4 \int \frac {1}{x} \, dx\\ &=-\frac {b^3 \tanh ^{-1}(\tanh (a+b x))}{x}-\frac {b^2 \tanh ^{-1}(\tanh (a+b x))^2}{2 x^2}-\frac {b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^3}-\frac {\tanh ^{-1}(\tanh (a+b x))^4}{4 x^4}+b^4 \log (x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 78, normalized size = 1.05 \begin {gather*} -\frac {12 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+6 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2+4 b x \tanh ^{-1}(\tanh (a+b x))^3+3 \tanh ^{-1}(\tanh (a+b x))^4-b^4 x^4 (25+12 \log (x))}{12 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.36, size = 69, normalized size = 0.93
method | result | size |
default | \(-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{4}}{4 x^{4}}+b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{3}}{3 x^{3}}+b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{2 x^{2}}+b \left (-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{x}+b \ln \left (x \right )\right )\right )\right )\) | \(69\) |
risch | \(\text {Expression too large to display}\) | \(22627\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.42, size = 72, normalized size = 0.97 \begin {gather*} \frac {1}{2} \, {\left (2 \, {\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}}{x^{2}}\right )} b - \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{3 \, x^{3}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{4 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 48, normalized size = 0.65 \begin {gather*} \frac {12 \, b^{4} x^{4} \log \left (x\right ) - 48 \, a b^{3} x^{3} - 36 \, a^{2} b^{2} x^{2} - 16 \, a^{3} b x - 3 \, a^{4}}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.36, size = 70, normalized size = 0.95 \begin {gather*} b^{4} \log {\left (x \right )} - \frac {b^{3} \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{x} - \frac {b^{2} \operatorname {atanh}^{2}{\left (\tanh {\left (a + b x \right )} \right )}}{2 x^{2}} - \frac {b \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}{3 x^{3}} - \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{4 x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 46, normalized size = 0.62 \begin {gather*} b^{4} \log \left ({\left | x \right |}\right ) - \frac {48 \, a b^{3} x^{3} + 36 \, a^{2} b^{2} x^{2} + 16 \, a^{3} b x + 3 \, a^{4}}{12 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.14, size = 68, normalized size = 0.92 \begin {gather*} b^4\,\ln \left (x\right )-\frac {{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^4}{4\,x^4}-\frac {b^2\,{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{2\,x^2}-\frac {b^3\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}{x}-\frac {b\,{\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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