Optimal. Leaf size=21 \[ \frac {\text {Li}_2(x+1)}{4}-\frac {\text {Li}_2(-x-1)}{4} \]
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Rubi [A] time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6107, 12, 5912} \[ \frac {1}{4} \text {PolyLog}(2,x+1)-\frac {1}{4} \text {PolyLog}(2,-x-1) \]
Antiderivative was successfully verified.
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Rule 12
Rule 5912
Rule 6107
Rubi steps
\begin {align*} \int \frac {\tanh ^{-1}(1+x)}{2+2 x} \, dx &=\operatorname {Subst}\left (\int \frac {\tanh ^{-1}(x)}{2 x} \, dx,x,1+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\tanh ^{-1}(x)}{x} \, dx,x,1+x\right )\\ &=-\frac {1}{4} \text {Li}_2(-1-x)+\frac {\text {Li}_2(1+x)}{4}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 31, normalized size = 1.48 \[ \frac {1}{4} \text {Li}_2\left (\frac {1}{2} (2 x+2)\right )-\frac {1}{4} \text {Li}_2\left (\frac {1}{2} (-2 x-2)\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {artanh}\left (x + 1\right )}{2 \, {\left (x + 1\right )}}, x\right ) \]
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 \[ \int \frac {\operatorname {artanh}\left (x + 1\right )}{2 \, {\left (x + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 34, normalized size = 1.62 \[ \frac {\ln \left (1+x \right ) \arctanh \left (1+x \right )}{2}-\frac {\dilog \left (1+x \right )}{4}-\frac {\dilog \left (x +2\right )}{4}-\frac {\ln \left (1+x \right ) \ln \left (x +2\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 58, normalized size = 2.76 \[ -\frac {1}{4} \, {\left (\log \left (x + 2\right ) - \log \relax (x)\right )} \log \left (x + 1\right ) + \frac {1}{2} \, \operatorname {artanh}\left (x + 1\right ) \log \left (x + 1\right ) - \frac {1}{4} \, \log \left (x + 1\right ) \log \relax (x) + \frac {1}{4} \, \log \left (x + 2\right ) \log \left (-x - 1\right ) - \frac {1}{4} \, {\rm Li}_2\left (-x\right ) + \frac {1}{4} \, {\rm Li}_2\left (x + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\mathrm {atanh}\left (x+1\right )}{2\,x+2} \,d x \]
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 \[ \frac {\int \frac {\operatorname {atanh}{\left (x + 1 \right )}}{x + 1}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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