Optimal. Leaf size=31 \[ \frac {1}{4} i \text {PolyLog}(2,-i (1+x))-\frac {1}{4} i \text {PolyLog}(2,i (1+x)) \]
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Rubi [A]
time = 0.03, antiderivative size = 31, 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 = {5151, 12, 4940,
2438} \begin {gather*} \frac {1}{4} i \text {Li}_2(-i (x+1))-\frac {1}{4} i \text {Li}_2(i (x+1)) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2438
Rule 4940
Rule 5151
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(1+x)}{2+2 x} \, dx &=\text {Subst}\left (\int \frac {\tan ^{-1}(x)}{2 x} \, dx,x,1+x\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {\tan ^{-1}(x)}{x} \, dx,x,1+x\right )\\ &=\frac {1}{4} i \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,1+x\right )-\frac {1}{4} i \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,1+x\right )\\ &=\frac {1}{4} i \text {Li}_2(-i (1+x))-\frac {1}{4} i \text {Li}_2(i (1+x))\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 31, normalized size = 1.00 \begin {gather*} \frac {1}{4} i \text {PolyLog}(2,-i (1+x))-\frac {1}{4} i \text {PolyLog}(2,i (1+x)) \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. \(2 (23 ) = 46\).
time = 0.06, size = 68, normalized size = 2.19
method | result | size |
risch | \(-\frac {i \dilog \left (-i x -i+1\right )}{4}+\frac {i \dilog \left (i x +i+1\right )}{4}\) | \(26\) |
derivativedivides | \(\frac {\ln \left (1+x \right ) \arctan \left (1+x \right )}{2}+\frac {i \ln \left (1+x \right ) \ln \left (1+i \left (1+x \right )\right )}{4}-\frac {i \ln \left (1+x \right ) \ln \left (1-i \left (1+x \right )\right )}{4}+\frac {i \dilog \left (1+i \left (1+x \right )\right )}{4}-\frac {i \dilog \left (1-i \left (1+x \right )\right )}{4}\) | \(68\) |
default | \(\frac {\ln \left (1+x \right ) \arctan \left (1+x \right )}{2}+\frac {i \ln \left (1+x \right ) \ln \left (1+i \left (1+x \right )\right )}{4}-\frac {i \ln \left (1+x \right ) \ln \left (1-i \left (1+x \right )\right )}{4}+\frac {i \dilog \left (1+i \left (1+x \right )\right )}{4}-\frac {i \dilog \left (1-i \left (1+x \right )\right )}{4}\) | \(68\) |
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. 44 vs. \(2 (17) = 34\).
time = 0.51, size = 44, normalized size = 1.42 \begin {gather*} -\frac {1}{4} \, \arctan \left (x + 1, 0\right ) \log \left (x^{2} + 2 \, x + 2\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 {atan}{\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 [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 [B]
time = 0.08, size = 25, normalized size = 0.81 \begin {gather*} -\frac {{\mathrm {Li}}_{\mathrm {2}}\left (1-x\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{4}+\frac {{\mathrm {Li}}_{\mathrm {2}}\left (x\,1{}\mathrm {i}+1+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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