Optimal. Leaf size=35 \[ -\frac {1}{4} i \text {PolyLog}\left (2,-\frac {i}{1+x}\right )+\frac {1}{4} i \text {PolyLog}\left (2,\frac {i}{1+x}\right ) \]
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Rubi [A]
time = 0.03, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5152, 12, 4941,
2438} \begin {gather*} \frac {1}{4} i \text {Li}_2\left (\frac {i}{x+1}\right )-\frac {1}{4} i \text {Li}_2\left (-\frac {i}{x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2438
Rule 4941
Rule 5152
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(1+x)}{2+2 x} \, dx &=\text {Subst}\left (\int \frac {\cot ^{-1}(x)}{2 x} \, dx,x,1+x\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {\cot ^{-1}(x)}{x} \, dx,x,1+x\right )\\ &=\frac {1}{4} i \text {Subst}\left (\int \frac {\log \left (1-\frac {i}{x}\right )}{x} \, dx,x,1+x\right )-\frac {1}{4} i \text {Subst}\left (\int \frac {\log \left (1+\frac {i}{x}\right )}{x} \, dx,x,1+x\right )\\ &=-\frac {1}{4} i \text {Li}_2\left (-\frac {i}{1+x}\right )+\frac {1}{4} i \text {Li}_2\left (\frac {i}{1+x}\right )\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 35, normalized size = 1.00 \begin {gather*} -\frac {1}{4} i \text {PolyLog}\left (2,-\frac {i}{1+x}\right )+\frac {1}{4} i \text {PolyLog}\left (2,\frac {i}{1+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 67 vs. \(2 (27 ) = 54\).
time = 0.11, size = 68, normalized size = 1.94
method | result | size |
risch | \(\frac {\pi \ln \left (-i x -i\right )}{4}+\frac {i \dilog \left (-i x -i+1\right )}{4}-\frac {i \dilog \left (i x +i+1\right )}{4}\) | \(37\) |
derivativedivides | \(\frac {\ln \left (1+x \right ) \mathrm {arccot}\left (1+x \right )}{2}-\frac {i \ln \left (1+x \right ) \ln \left (1+i \left (1+x \right )\right )}{4}+\frac {i \ln \left (1+x \right ) \ln \left (1-i \left (1+x \right )\right )}{4}-\frac {i \dilog \left (1+i \left (1+x \right )\right )}{4}+\frac {i \dilog \left (1-i \left (1+x \right )\right )}{4}\) | \(68\) |
default | \(\frac {\ln \left (1+x \right ) \mathrm {arccot}\left (1+x \right )}{2}-\frac {i \ln \left (1+x \right ) \ln \left (1+i \left (1+x \right )\right )}{4}+\frac {i \ln \left (1+x \right ) \ln \left (1-i \left (1+x \right )\right )}{4}-\frac {i \dilog \left (1+i \left (1+x \right )\right )}{4}+\frac {i \dilog \left (1-i \left (1+x \right )\right )}{4}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 64 vs. \(2 (21) = 42\).
time = 0.50, size = 64, normalized size = 1.83 \begin {gather*} \frac {1}{4} \, \arctan \left (x + 1, 0\right ) \log \left (x^{2} + 2 \, x + 2\right ) + \frac {1}{2} \, \operatorname {arccot}\left (x + 1\right ) \log \left (x + 1\right ) + \frac {1}{2} \, \arctan \left (x + 1\right ) \log \left (x + 1\right ) - \frac {1}{2} \, \arctan \left (x + 1\right ) \log \left ({\left | x + 1 \right |}\right ) + \frac {1}{4} i \, {\rm Li}_2\left (i \, x + i + 1\right ) - \frac {1}{4} i \, {\rm Li}_2\left (-i \, x - i + 1\right ) \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*} \frac {\int \frac {\operatorname {acot}{\left (x + 1 \right )}}{x + 1}\, dx}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 26, normalized size = 0.74 \begin {gather*} -\frac {1}{4} \, {\left (x + 1\right )}^{2} \arctan \left (\frac {1}{x + 1}\right ) - \frac {1}{4} \, x - \frac {1}{4} \, \arctan \left (\frac {1}{x + 1}\right ) - \frac {1}{4} \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 = -0.03 \begin {gather*} \int \frac {\mathrm {acot}\left (x+1\right )}{2\,x+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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