Optimal. Leaf size=31 \[ \frac {\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2198}
\begin {gather*} \frac {\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2198
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(\tanh (a+b x))^2}{x^4} \, dx &=\frac {\tanh ^{-1}(\tanh (a+b x))^3}{3 x^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 34, normalized size = 1.10 \begin {gather*} -\frac {b^2 x^2+b x \tanh ^{-1}(\tanh (a+b x))+\tanh ^{-1}(\tanh (a+b x))^2}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 38, normalized size = 1.23
method | result | size |
default | \(-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )^{2}}{3 x^{3}}+\frac {2 b \left (-\frac {b}{2 x}-\frac {\arctanh \left (\tanh \left (b x +a \right )\right )}{2 x^{2}}\right )}{3}\) | \(38\) |
risch | \(\text {Expression too large to display}\) | \(1978\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 36, normalized size = 1.16 \begin {gather*} -\frac {b^{2}}{3 \, x} - \frac {b \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{3 \, x^{2}} - \frac {\operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 22, normalized size = 0.71 \begin {gather*} -\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.27, size = 37, normalized size = 1.19 \begin {gather*} - \frac {b^{2}}{3 x} - \frac {b \operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{3 x^{2}} - \frac {\operatorname {atanh}^{2}{\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 = 22, normalized size = 0.71 \begin {gather*} -\frac {3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.94, size = 32, normalized size = 1.03 \begin {gather*} -\frac {b^2\,x^2+b\,x\,\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )+{\mathrm {atanh}\left (\mathrm {tanh}\left (a+b\,x\right )\right )}^2}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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