Optimal. Leaf size=56 \[ -\tanh ^{-1}(1+b x)^2 \log \left (-\frac {2}{b x}\right )-\tanh ^{-1}(1+b x) \text {PolyLog}\left (2,1+\frac {2}{b x}\right )+\frac {1}{2} \text {PolyLog}\left (3,1+\frac {2}{b x}\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {6246, 6055,
6095, 6205, 6745} \begin {gather*} \frac {1}{2} \text {Li}_3\left (1+\frac {2}{b x}\right )-\text {Li}_2\left (1+\frac {2}{b x}\right ) \tanh ^{-1}(b x+1)-\log \left (-\frac {2}{b x}\right ) \tanh ^{-1}(b x+1)^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 6055
Rule 6095
Rule 6205
Rule 6246
Rule 6745
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(1+b x)^2}{x} \, dx &=\frac {\text {Subst}\left (\int \frac {\tanh ^{-1}(x)^2}{-\frac {1}{b}+\frac {x}{b}} \, dx,x,1+b x\right )}{b}\\ &=-\tanh ^{-1}(1+b x)^2 \log \left (-\frac {2}{b x}\right )+2 \text {Subst}\left (\int \frac {\tanh ^{-1}(x) \log \left (\frac {2}{1-x}\right )}{1-x^2} \, dx,x,1+b x\right )\\ &=-\tanh ^{-1}(1+b x)^2 \log \left (-\frac {2}{b x}\right )-\tanh ^{-1}(1+b x) \text {Li}_2\left (1+\frac {2}{b x}\right )+\text {Subst}\left (\int \frac {\text {Li}_2\left (1-\frac {2}{1-x}\right )}{1-x^2} \, dx,x,1+b x\right )\\ &=-\tanh ^{-1}(1+b x)^2 \log \left (-\frac {2}{b x}\right )-\tanh ^{-1}(1+b x) \text {Li}_2\left (1+\frac {2}{b x}\right )+\frac {1}{2} \text {Li}_3\left (1+\frac {2}{b x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 75, normalized size = 1.34 \begin {gather*} -\frac {2}{3} \tanh ^{-1}(1+b x)^3-\tanh ^{-1}(1+b x)^2 \log \left (1+e^{-2 \tanh ^{-1}(1+b x)}\right )+\tanh ^{-1}(1+b x) \text {PolyLog}\left (2,-e^{-2 \tanh ^{-1}(1+b x)}\right )+\frac {1}{2} \text {PolyLog}\left (3,-e^{-2 \tanh ^{-1}(1+b x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 10.56, size = 160, normalized size = 2.86
method | result | size |
derivativedivides | \(\ln \left (b x \right ) \arctanh \left (b x +1\right )^{2}-\arctanh \left (b x +1\right ) \polylog \left (2, -\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}\right )+\frac {\polylog \left (3, -\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}\right )}{2}-\left (i \pi \mathrm {csgn}\left (\frac {i}{1+\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i}{1+\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}}\right )^{2}+i \pi +\ln \left (2\right )\right ) \arctanh \left (b x +1\right )^{2}\) | \(160\) |
default | \(\ln \left (b x \right ) \arctanh \left (b x +1\right )^{2}-\arctanh \left (b x +1\right ) \polylog \left (2, -\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}\right )+\frac {\polylog \left (3, -\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}\right )}{2}-\left (i \pi \mathrm {csgn}\left (\frac {i}{1+\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}}\right )^{3}-i \pi \mathrm {csgn}\left (\frac {i}{1+\frac {\left (b x +2\right )^{2}}{-\left (b x +1\right )^{2}+1}}\right )^{2}+i \pi +\ln \left (2\right )\right ) \arctanh \left (b x +1\right )^{2}\) | \(160\) |
risch | \(\text {Expression too large to display}\) | \(2638\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {\operatorname {atanh}^{2}{\left (b x + 1 \right )}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\mathrm {atanh}\left (b\,x+1\right )}^2}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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