Optimal. Leaf size=31 \[ \frac {1}{4} i \text {Li}_2(-i (x+1))-\frac {1}{4} i \text {Li}_2(i (x+1)) \]
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Rubi [A] time = 0.04, 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 = {5043, 12, 4848, 2391} \[ \frac {1}{4} i \text {PolyLog}(2,-i (x+1))-\frac {1}{4} i \text {PolyLog}(2,i (x+1)) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2391
Rule 4848
Rule 5043
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(1+x)}{2+2 x} \, dx &=\operatorname {Subst}\left (\int \frac {\tan ^{-1}(x)}{2 x} \, dx,x,1+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {\tan ^{-1}(x)}{x} \, dx,x,1+x\right )\\ &=\frac {1}{4} i \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,1+x\right )-\frac {1}{4} i \operatorname {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 \[ \frac {1}{4} i \text {Li}_2(-i (x+1))-\frac {1}{4} i \text {Li}_2(i (x+1)) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arctan \left (x + 1\right )}{2 \, {\left (x + 1\right )}}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 68, normalized size = 2.19 \[ \frac {\ln \left (x +1\right ) \arctan \left (x +1\right )}{2}+\frac {i \ln \left (x +1\right ) \ln \left (1+i \left (x +1\right )\right )}{4}-\frac {i \ln \left (x +1\right ) \ln \left (1-i \left (x +1\right )\right )}{4}+\frac {i \dilog \left (1+i \left (x +1\right )\right )}{4}-\frac {i \dilog \left (1-i \left (x +1\right )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 44, normalized size = 1.42 \[ -\frac {1}{4} \, \arctan \left (x + 1, 0\right ) \log \left (x^{2} + 2 \, x + 2\right ) + \frac {1}{2} \, \arctan \left (x + 1\right ) \log \left ({\left | x + 1 \right |}\right ) - \frac {1}{4} i \, {\rm Li}_2\left (i \, x + i + 1\right ) + \frac {1}{4} i \, {\rm Li}_2\left (-i \, x - i + 1\right ) \]
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 \[ -\frac {{\mathrm {Li}}_{\mathrm {2}}\left (1-x\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{4}+\frac {{\mathrm {Li}}_{\mathrm {2}}\left (x\,1{}\mathrm {i}+1+1{}\mathrm {i}\right )\,1{}\mathrm {i}}{4} \]
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 \[ \frac {\int \frac {\operatorname {atan}{\left (x + 1 \right )}}{x + 1}\, dx}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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