Optimal. Leaf size=56 \[ \frac {x^2}{2 b}+\frac {x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3} \]
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Rubi [A]
time = 0.02, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2190, 2189,
2188, 29} \begin {gather*} \frac {\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}+\frac {x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {x^2}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2188
Rule 2189
Rule 2190
Rubi steps
\begin {align*} \int \frac {x^2}{\tanh ^{-1}(\tanh (a+b x))} \, dx &=\frac {x^2}{2 b}-\frac {\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac {x}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b}\\ &=\frac {x^2}{2 b}+\frac {x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2 \int \frac {1}{\tanh ^{-1}(\tanh (a+b x))} \, dx}{b^2}\\ &=\frac {x^2}{2 b}+\frac {x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2 \text {Subst}\left (\int \frac {1}{x} \, dx,x,\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}\\ &=\frac {x^2}{2 b}+\frac {x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 55, normalized size = 0.98 \begin {gather*} \frac {x^2}{2 b}-\frac {x \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )}{b^2}+\frac {\left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )^2 \log \left (\tanh ^{-1}(\tanh (a+b x))\right )}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(110\) vs.
\(2(54)=108\).
time = 0.65, size = 111, normalized size = 1.98
method | result | size |
default | \(\frac {x^{2}}{2 b}-\frac {x a}{b^{2}}-\frac {x \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{b^{2}}+\frac {\ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right ) a^{2}}{b^{3}}+\frac {2 \ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right ) a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )}{b^{3}}+\frac {\ln \left (\arctanh \left (\tanh \left (b x +a \right )\right )\right ) \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}}{b^{3}}\) | \(111\) |
risch | \(\text {Expression too large to display}\) | \(14523\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 29, normalized size = 0.52 \begin {gather*} \frac {a^{2} \log \left (b x + a\right )}{b^{3}} + \frac {b x^{2} - 2 \, a x}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 29, normalized size = 0.52 \begin {gather*} \frac {b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.38, size = 30, normalized size = 0.54 \begin {gather*} \frac {a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac {b x^{2} - 2 \, a x}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.27, size = 234, normalized size = 4.18 \begin {gather*} \frac {x^2}{2\,b}+\frac {x\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{2\,b^2}+\frac {\ln \left (\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\right )\,\left ({\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2-4\,a\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )+4\,a^2\right )}{4\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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