Optimal. Leaf size=981 \[ \text{result too large to display} \]
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Rubi [A] time = 1.25169, antiderivative size = 981, normalized size of antiderivative = 1., number of steps used = 44, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {2530, 1591, 203, 6741, 2528, 2524, 2418, 2394, 2393, 2391} \[ \frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{x+1}\right ) \log \left (x+\sqrt{x+1}\right )-\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{x+1}\right ) \log \left (x+\sqrt{x+1}\right )+\frac{1}{2} i \log \left (\sqrt{x+1}+\sqrt{1-i}\right ) \log \left (x+\sqrt{x+1}\right )-\frac{1}{2} i \log \left (\sqrt{x+1}+\sqrt{1+i}\right ) \log \left (x+\sqrt{x+1}\right )-\frac{1}{2} i \log \left (\sqrt{x+1}+\sqrt{1-i}\right ) \log \left (\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1-2 \sqrt{1-i}-\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{x+1}\right ) \log \left (\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1+2 \sqrt{1-i}-\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{x+1}+\sqrt{1+i}\right ) \log \left (\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1-2 \sqrt{1+i}-\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{x+1}\right ) \log \left (\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1+2 \sqrt{1+i}-\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{x+1}+\sqrt{1-i}\right ) \log \left (\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1-2 \sqrt{1-i}+\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{x+1}\right ) \log \left (\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1+2 \sqrt{1-i}+\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{x+1}+\sqrt{1+i}\right ) \log \left (\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1-2 \sqrt{1+i}+\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{x+1}\right ) \log \left (\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1+2 \sqrt{1+i}+\sqrt{5}}\right )-\frac{1}{2} i \text{PolyLog}\left (2,\frac{2 \left (\sqrt{1-i}-\sqrt{x+1}\right )}{1+2 \sqrt{1-i}-\sqrt{5}}\right )-\frac{1}{2} i \text{PolyLog}\left (2,\frac{2 \left (\sqrt{1-i}-\sqrt{x+1}\right )}{1+2 \sqrt{1-i}+\sqrt{5}}\right )+\frac{1}{2} i \text{PolyLog}\left (2,\frac{2 \left (\sqrt{1+i}-\sqrt{x+1}\right )}{1+2 \sqrt{1+i}-\sqrt{5}}\right )+\frac{1}{2} i \text{PolyLog}\left (2,\frac{2 \left (\sqrt{1+i}-\sqrt{x+1}\right )}{1+2 \sqrt{1+i}+\sqrt{5}}\right )-\frac{1}{2} i \text{PolyLog}\left (2,-\frac{2 \left (\sqrt{x+1}+\sqrt{1-i}\right )}{1-2 \sqrt{1-i}-\sqrt{5}}\right )-\frac{1}{2} i \text{PolyLog}\left (2,-\frac{2 \left (\sqrt{x+1}+\sqrt{1-i}\right )}{1-2 \sqrt{1-i}+\sqrt{5}}\right )+\frac{1}{2} i \text{PolyLog}\left (2,-\frac{2 \left (\sqrt{x+1}+\sqrt{1+i}\right )}{1-2 \sqrt{1+i}-\sqrt{5}}\right )+\frac{1}{2} i \text{PolyLog}\left (2,-\frac{2 \left (\sqrt{x+1}+\sqrt{1+i}\right )}{1-2 \sqrt{1+i}+\sqrt{5}}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2530
Rule 1591
Rule 203
Rule 6741
Rule 2528
Rule 2524
Rule 2418
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (x+\sqrt{1+x}\right )}{1+x^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x \log \left (-1+x+x^2\right )}{1+\left (-1+x^2\right )^2} \, dx,x,\sqrt{1+x}\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x \log \left (-1+x+x^2\right )}{2-2 x^2+x^4} \, dx,x,\sqrt{1+x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{i x \log \left (-1+x+x^2\right )}{(2+2 i)-2 x^2}+\frac{i x \log \left (-1+x+x^2\right )}{(-2+2 i)+2 x^2}\right ) \, dx,x,\sqrt{1+x}\right )\\ &=2 i \operatorname{Subst}\left (\int \frac{x \log \left (-1+x+x^2\right )}{(2+2 i)-2 x^2} \, dx,x,\sqrt{1+x}\right )+2 i \operatorname{Subst}\left (\int \frac{x \log \left (-1+x+x^2\right )}{(-2+2 i)+2 x^2} \, dx,x,\sqrt{1+x}\right )\\ &=2 i \operatorname{Subst}\left (\int \left (-\frac{\log \left (-1+x+x^2\right )}{4 \left (\sqrt{1-i}-x\right )}+\frac{\log \left (-1+x+x^2\right )}{4 \left (\sqrt{1-i}+x\right )}\right ) \, dx,x,\sqrt{1+x}\right )+2 i \operatorname{Subst}\left (\int \left (\frac{\log \left (-1+x+x^2\right )}{4 \left (\sqrt{1+i}-x\right )}-\frac{\log \left (-1+x+x^2\right )}{4 \left (\sqrt{1+i}+x\right )}\right ) \, dx,x,\sqrt{1+x}\right )\\ &=-\left (\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (-1+x+x^2\right )}{\sqrt{1-i}-x} \, dx,x,\sqrt{1+x}\right )\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (-1+x+x^2\right )}{\sqrt{1+i}-x} \, dx,x,\sqrt{1+x}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (-1+x+x^2\right )}{\sqrt{1-i}+x} \, dx,x,\sqrt{1+x}\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (-1+x+x^2\right )}{\sqrt{1+i}+x} \, dx,x,\sqrt{1+x}\right )\\ &=\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )+\frac{1}{2} i \log \left (\sqrt{1-i}+\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1+i}+\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{(1+2 x) \log \left (\sqrt{1-i}-x\right )}{-1+x+x^2} \, dx,x,\sqrt{1+x}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{(1+2 x) \log \left (\sqrt{1+i}-x\right )}{-1+x+x^2} \, dx,x,\sqrt{1+x}\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{(1+2 x) \log \left (\sqrt{1-i}+x\right )}{-1+x+x^2} \, dx,x,\sqrt{1+x}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{(1+2 x) \log \left (\sqrt{1+i}+x\right )}{-1+x+x^2} \, dx,x,\sqrt{1+x}\right )\\ &=\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )+\frac{1}{2} i \log \left (\sqrt{1-i}+\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1+i}+\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \left (\frac{2 \log \left (\sqrt{1-i}-x\right )}{1-\sqrt{5}+2 x}+\frac{2 \log \left (\sqrt{1-i}-x\right )}{1+\sqrt{5}+2 x}\right ) \, dx,x,\sqrt{1+x}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \left (\frac{2 \log \left (\sqrt{1+i}-x\right )}{1-\sqrt{5}+2 x}+\frac{2 \log \left (\sqrt{1+i}-x\right )}{1+\sqrt{5}+2 x}\right ) \, dx,x,\sqrt{1+x}\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \left (\frac{2 \log \left (\sqrt{1-i}+x\right )}{1-\sqrt{5}+2 x}+\frac{2 \log \left (\sqrt{1-i}+x\right )}{1+\sqrt{5}+2 x}\right ) \, dx,x,\sqrt{1+x}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \left (\frac{2 \log \left (\sqrt{1+i}+x\right )}{1-\sqrt{5}+2 x}+\frac{2 \log \left (\sqrt{1+i}+x\right )}{1+\sqrt{5}+2 x}\right ) \, dx,x,\sqrt{1+x}\right )\\ &=\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )+\frac{1}{2} i \log \left (\sqrt{1-i}+\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1+i}+\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-i \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{1-i}-x\right )}{1-\sqrt{5}+2 x} \, dx,x,\sqrt{1+x}\right )-i \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{1-i}-x\right )}{1+\sqrt{5}+2 x} \, dx,x,\sqrt{1+x}\right )+i \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{1+i}-x\right )}{1-\sqrt{5}+2 x} \, dx,x,\sqrt{1+x}\right )+i \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{1+i}-x\right )}{1+\sqrt{5}+2 x} \, dx,x,\sqrt{1+x}\right )-i \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{1-i}+x\right )}{1-\sqrt{5}+2 x} \, dx,x,\sqrt{1+x}\right )-i \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{1-i}+x\right )}{1+\sqrt{5}+2 x} \, dx,x,\sqrt{1+x}\right )+i \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{1+i}+x\right )}{1-\sqrt{5}+2 x} \, dx,x,\sqrt{1+x}\right )+i \operatorname{Subst}\left (\int \frac{\log \left (\sqrt{1+i}+x\right )}{1+\sqrt{5}+2 x} \, dx,x,\sqrt{1+x}\right )\\ &=\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )+\frac{1}{2} i \log \left (\sqrt{1-i}+\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1+i}+\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}+\sqrt{1+x}\right ) \log \left (\frac{1-\sqrt{5}+2 \sqrt{1+x}}{1-2 \sqrt{1-i}-\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{1+x}\right ) \log \left (\frac{1-\sqrt{5}+2 \sqrt{1+x}}{1+2 \sqrt{1-i}-\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}+\sqrt{1+x}\right ) \log \left (\frac{1-\sqrt{5}+2 \sqrt{1+x}}{1-2 \sqrt{1+i}-\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{1+x}\right ) \log \left (\frac{1-\sqrt{5}+2 \sqrt{1+x}}{1+2 \sqrt{1+i}-\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}+\sqrt{1+x}\right ) \log \left (\frac{1+\sqrt{5}+2 \sqrt{1+x}}{1-2 \sqrt{1-i}+\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{1+x}\right ) \log \left (\frac{1+\sqrt{5}+2 \sqrt{1+x}}{1+2 \sqrt{1-i}+\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}+\sqrt{1+x}\right ) \log \left (\frac{1+\sqrt{5}+2 \sqrt{1+x}}{1-2 \sqrt{1+i}+\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{1+x}\right ) \log \left (\frac{1+\sqrt{5}+2 \sqrt{1+x}}{1+2 \sqrt{1+i}+\sqrt{5}}\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (\frac{-1-\sqrt{5}-2 x}{-1-2 \sqrt{1-i}-\sqrt{5}}\right )}{\sqrt{1-i}-x} \, dx,x,\sqrt{1+x}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (\frac{-1-\sqrt{5}-2 x}{-1-2 \sqrt{1+i}-\sqrt{5}}\right )}{\sqrt{1+i}-x} \, dx,x,\sqrt{1+x}\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (\frac{-1+\sqrt{5}-2 x}{-1-2 \sqrt{1-i}+\sqrt{5}}\right )}{\sqrt{1-i}-x} \, dx,x,\sqrt{1+x}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (\frac{-1+\sqrt{5}-2 x}{-1-2 \sqrt{1+i}+\sqrt{5}}\right )}{\sqrt{1+i}-x} \, dx,x,\sqrt{1+x}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (\frac{1-\sqrt{5}+2 x}{1-2 \sqrt{1-i}-\sqrt{5}}\right )}{\sqrt{1-i}+x} \, dx,x,\sqrt{1+x}\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (\frac{1-\sqrt{5}+2 x}{1-2 \sqrt{1+i}-\sqrt{5}}\right )}{\sqrt{1+i}+x} \, dx,x,\sqrt{1+x}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (\frac{1+\sqrt{5}+2 x}{1-2 \sqrt{1-i}+\sqrt{5}}\right )}{\sqrt{1-i}+x} \, dx,x,\sqrt{1+x}\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (\frac{1+\sqrt{5}+2 x}{1-2 \sqrt{1+i}+\sqrt{5}}\right )}{\sqrt{1+i}+x} \, dx,x,\sqrt{1+x}\right )\\ &=\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )+\frac{1}{2} i \log \left (\sqrt{1-i}+\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1+i}+\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}+\sqrt{1+x}\right ) \log \left (\frac{1-\sqrt{5}+2 \sqrt{1+x}}{1-2 \sqrt{1-i}-\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{1+x}\right ) \log \left (\frac{1-\sqrt{5}+2 \sqrt{1+x}}{1+2 \sqrt{1-i}-\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}+\sqrt{1+x}\right ) \log \left (\frac{1-\sqrt{5}+2 \sqrt{1+x}}{1-2 \sqrt{1+i}-\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{1+x}\right ) \log \left (\frac{1-\sqrt{5}+2 \sqrt{1+x}}{1+2 \sqrt{1+i}-\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}+\sqrt{1+x}\right ) \log \left (\frac{1+\sqrt{5}+2 \sqrt{1+x}}{1-2 \sqrt{1-i}+\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{1+x}\right ) \log \left (\frac{1+\sqrt{5}+2 \sqrt{1+x}}{1+2 \sqrt{1-i}+\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}+\sqrt{1+x}\right ) \log \left (\frac{1+\sqrt{5}+2 \sqrt{1+x}}{1-2 \sqrt{1+i}+\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{1+x}\right ) \log \left (\frac{1+\sqrt{5}+2 \sqrt{1+x}}{1+2 \sqrt{1+i}+\sqrt{5}}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{-1-2 \sqrt{1-i}-\sqrt{5}}\right )}{x} \, dx,x,\sqrt{1-i}-\sqrt{1+x}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{1-2 \sqrt{1-i}-\sqrt{5}}\right )}{x} \, dx,x,\sqrt{1-i}+\sqrt{1+x}\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{-1-2 \sqrt{1+i}-\sqrt{5}}\right )}{x} \, dx,x,\sqrt{1+i}-\sqrt{1+x}\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{1-2 \sqrt{1+i}-\sqrt{5}}\right )}{x} \, dx,x,\sqrt{1+i}+\sqrt{1+x}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{-1-2 \sqrt{1-i}+\sqrt{5}}\right )}{x} \, dx,x,\sqrt{1-i}-\sqrt{1+x}\right )+\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{1-2 \sqrt{1-i}+\sqrt{5}}\right )}{x} \, dx,x,\sqrt{1-i}+\sqrt{1+x}\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{-1-2 \sqrt{1+i}+\sqrt{5}}\right )}{x} \, dx,x,\sqrt{1+i}-\sqrt{1+x}\right )-\frac{1}{2} i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 x}{1-2 \sqrt{1+i}+\sqrt{5}}\right )}{x} \, dx,x,\sqrt{1+i}+\sqrt{1+x}\right )\\ &=\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )+\frac{1}{2} i \log \left (\sqrt{1-i}+\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1+i}+\sqrt{1+x}\right ) \log \left (x+\sqrt{1+x}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}+\sqrt{1+x}\right ) \log \left (\frac{1-\sqrt{5}+2 \sqrt{1+x}}{1-2 \sqrt{1-i}-\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{1+x}\right ) \log \left (\frac{1-\sqrt{5}+2 \sqrt{1+x}}{1+2 \sqrt{1-i}-\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}+\sqrt{1+x}\right ) \log \left (\frac{1-\sqrt{5}+2 \sqrt{1+x}}{1-2 \sqrt{1+i}-\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{1+x}\right ) \log \left (\frac{1-\sqrt{5}+2 \sqrt{1+x}}{1+2 \sqrt{1+i}-\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}+\sqrt{1+x}\right ) \log \left (\frac{1+\sqrt{5}+2 \sqrt{1+x}}{1-2 \sqrt{1-i}+\sqrt{5}}\right )-\frac{1}{2} i \log \left (\sqrt{1-i}-\sqrt{1+x}\right ) \log \left (\frac{1+\sqrt{5}+2 \sqrt{1+x}}{1+2 \sqrt{1-i}+\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}+\sqrt{1+x}\right ) \log \left (\frac{1+\sqrt{5}+2 \sqrt{1+x}}{1-2 \sqrt{1+i}+\sqrt{5}}\right )+\frac{1}{2} i \log \left (\sqrt{1+i}-\sqrt{1+x}\right ) \log \left (\frac{1+\sqrt{5}+2 \sqrt{1+x}}{1+2 \sqrt{1+i}+\sqrt{5}}\right )-\frac{1}{2} i \text{Li}_2\left (\frac{2 \left (\sqrt{1-i}-\sqrt{1+x}\right )}{1+2 \sqrt{1-i}-\sqrt{5}}\right )-\frac{1}{2} i \text{Li}_2\left (\frac{2 \left (\sqrt{1-i}-\sqrt{1+x}\right )}{1+2 \sqrt{1-i}+\sqrt{5}}\right )+\frac{1}{2} i \text{Li}_2\left (\frac{2 \left (\sqrt{1+i}-\sqrt{1+x}\right )}{1+2 \sqrt{1+i}-\sqrt{5}}\right )+\frac{1}{2} i \text{Li}_2\left (\frac{2 \left (\sqrt{1+i}-\sqrt{1+x}\right )}{1+2 \sqrt{1+i}+\sqrt{5}}\right )-\frac{1}{2} i \text{Li}_2\left (-\frac{2 \left (\sqrt{1-i}+\sqrt{1+x}\right )}{1-2 \sqrt{1-i}-\sqrt{5}}\right )-\frac{1}{2} i \text{Li}_2\left (-\frac{2 \left (\sqrt{1-i}+\sqrt{1+x}\right )}{1-2 \sqrt{1-i}+\sqrt{5}}\right )+\frac{1}{2} i \text{Li}_2\left (-\frac{2 \left (\sqrt{1+i}+\sqrt{1+x}\right )}{1-2 \sqrt{1+i}-\sqrt{5}}\right )+\frac{1}{2} i \text{Li}_2\left (-\frac{2 \left (\sqrt{1+i}+\sqrt{1+x}\right )}{1-2 \sqrt{1+i}+\sqrt{5}}\right )\\ \end{align*}
Mathematica [A] time = 0.481055, size = 868, normalized size = 0.88 \[ \frac{1}{2} i \left (2 i \tan ^{-1}(x) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )+\log \left (\frac{2 \left (\sqrt{1-i}-\sqrt{x+1}\right )}{1+2 \sqrt{1-i}-\sqrt{5}}\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )-\log \left (\frac{2 \left (\sqrt{1+i}-\sqrt{x+1}\right )}{1+2 \sqrt{1+i}-\sqrt{5}}\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )+\log \left (\frac{2 \left (\sqrt{x+1}+\sqrt{1-i}\right )}{-1+2 \sqrt{1-i}+\sqrt{5}}\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )-\log \left (\frac{2 \left (\sqrt{x+1}+\sqrt{1+i}\right )}{-1+2 \sqrt{1+i}+\sqrt{5}}\right ) \log \left (\sqrt{x+1}-\frac{\sqrt{5}}{2}+\frac{1}{2}\right )+2 i \tan ^{-1}(x) \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )+\log \left (\frac{2 \left (\sqrt{1-i}-\sqrt{x+1}\right )}{1+2 \sqrt{1-i}+\sqrt{5}}\right ) \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )-\log \left (\frac{2 \left (\sqrt{1+i}-\sqrt{x+1}\right )}{1+2 \sqrt{1+i}+\sqrt{5}}\right ) \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )+\log \left (\frac{2 \left (\sqrt{x+1}+\sqrt{1-i}\right )}{-1+2 \sqrt{1-i}-\sqrt{5}}\right ) \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )-\log \left (\frac{2 \left (\sqrt{x+1}+\sqrt{1+i}\right )}{-1+2 \sqrt{1+i}-\sqrt{5}}\right ) \log \left (\sqrt{x+1}+\frac{1}{2} \left (1+\sqrt{5}\right )\right )-2 i \tan ^{-1}(x) \log \left (x+\sqrt{x+1}\right )+\text{PolyLog}\left (2,\frac{-2 \sqrt{x+1}+\sqrt{5}-1}{-1+2 \sqrt{1-i}+\sqrt{5}}\right )-\text{PolyLog}\left (2,\frac{-2 \sqrt{x+1}+\sqrt{5}-1}{-1+2 \sqrt{1+i}+\sqrt{5}}\right )+\text{PolyLog}\left (2,\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1+2 \sqrt{1-i}-\sqrt{5}}\right )-\text{PolyLog}\left (2,\frac{2 \sqrt{x+1}-\sqrt{5}+1}{1+2 \sqrt{1+i}-\sqrt{5}}\right )+\text{PolyLog}\left (2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1-2 \sqrt{1-i}+\sqrt{5}}\right )+\text{PolyLog}\left (2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1+2 \sqrt{1-i}+\sqrt{5}}\right )-\text{PolyLog}\left (2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1-2 \sqrt{1+i}+\sqrt{5}}\right )-\text{PolyLog}\left (2,\frac{2 \sqrt{x+1}+\sqrt{5}+1}{1+2 \sqrt{1+i}+\sqrt{5}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 698, normalized size = 0.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (x + \sqrt{x + 1}\right )}{x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left (x + \sqrt{x + 1}\right )}{x^{2} + 1}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (x + \sqrt{x + 1} \right )}}{x^{2} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (x + \sqrt{x + 1}\right )}{x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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