Optimal. Leaf size=555 \[ 6 \operatorname {PolyLog}\left (2,-\frac {2 \sqrt {x+1}}{1+\sqrt {5}}\right )-\left (3+\sqrt {5}\right ) \operatorname {PolyLog}\left (2,-\frac {2 \sqrt {x+1}-\sqrt {5}+1}{2 \sqrt {5}}\right )-\left (3-\sqrt {5}\right ) \operatorname {PolyLog}\left (2,\frac {2 \sqrt {x+1}+\sqrt {5}+1}{2 \sqrt {5}}\right )-6 \operatorname {PolyLog}\left (2,\frac {2 \sqrt {x+1}}{1-\sqrt {5}}+1\right )-\frac {\log ^2\left (x+\sqrt {x+1}\right )}{x+1}-\frac {1}{2} \left (3+\sqrt {5}\right ) \log ^2\left (2 \sqrt {x+1}-\sqrt {5}+1\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \log ^2\left (2 \sqrt {x+1}+\sqrt {5}+1\right )-6 \log \left (\sqrt {x+1}\right ) \log \left (x+\sqrt {x+1}\right )+\left (3+\sqrt {5}\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right ) \log \left (x+\sqrt {x+1}\right )+\left (3-\sqrt {5}\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right ) \log \left (x+\sqrt {x+1}\right )+\frac {2 \log \left (x+\sqrt {x+1}\right )}{\sqrt {x+1}}+\log (x+1)+6 \log \left (\frac {1}{2} \left (\sqrt {5}-1\right )\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right )-\left (1+\sqrt {5}\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right )-\left (3-\sqrt {5}\right ) \log \left (-\frac {2 \sqrt {x+1}-\sqrt {5}+1}{2 \sqrt {5}}\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )-\left (1-\sqrt {5}\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )-\left (3+\sqrt {5}\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right ) \log \left (\frac {2 \sqrt {x+1}+\sqrt {5}+1}{2 \sqrt {5}}\right )+6 \log \left (\sqrt {x+1}\right ) \log \left (\frac {2 \sqrt {x+1}}{1+\sqrt {5}}+1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.71, antiderivative size = 555, normalized size of antiderivative = 1.00, number of steps used = 35, number of rules used = 16, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.889, Rules used = {2525, 2528, 800, 632, 31, 2524, 2357, 2316, 2315, 2317, 2391, 2418, 2390, 2301, 2394, 2393} \[ 6 \text {PolyLog}\left (2,-\frac {2 \sqrt {x+1}}{1+\sqrt {5}}\right )-\left (3+\sqrt {5}\right ) \text {PolyLog}\left (2,-\frac {2 \sqrt {x+1}-\sqrt {5}+1}{2 \sqrt {5}}\right )-\left (3-\sqrt {5}\right ) \text {PolyLog}\left (2,\frac {2 \sqrt {x+1}+\sqrt {5}+1}{2 \sqrt {5}}\right )-6 \text {PolyLog}\left (2,\frac {2 \sqrt {x+1}}{1-\sqrt {5}}+1\right )-\frac {\log ^2\left (x+\sqrt {x+1}\right )}{x+1}-\frac {1}{2} \left (3+\sqrt {5}\right ) \log ^2\left (2 \sqrt {x+1}-\sqrt {5}+1\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \log ^2\left (2 \sqrt {x+1}+\sqrt {5}+1\right )-6 \log \left (\sqrt {x+1}\right ) \log \left (x+\sqrt {x+1}\right )+\left (3+\sqrt {5}\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right ) \log \left (x+\sqrt {x+1}\right )+\left (3-\sqrt {5}\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right ) \log \left (x+\sqrt {x+1}\right )+\frac {2 \log \left (x+\sqrt {x+1}\right )}{\sqrt {x+1}}+\log (x+1)+6 \log \left (\frac {1}{2} \left (\sqrt {5}-1\right )\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right )-\left (1+\sqrt {5}\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right )-\left (3-\sqrt {5}\right ) \log \left (-\frac {2 \sqrt {x+1}-\sqrt {5}+1}{2 \sqrt {5}}\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )-\left (1-\sqrt {5}\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )-\left (3+\sqrt {5}\right ) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right ) \log \left (\frac {2 \sqrt {x+1}+\sqrt {5}+1}{2 \sqrt {5}}\right )+6 \log \left (\sqrt {x+1}\right ) \log \left (\frac {2 \sqrt {x+1}}{1+\sqrt {5}}+1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 632
Rule 800
Rule 2301
Rule 2315
Rule 2316
Rule 2317
Rule 2357
Rule 2390
Rule 2391
Rule 2393
Rule 2394
Rule 2418
Rule 2524
Rule 2525
Rule 2528
Rubi steps
\begin {align*} \int \frac {\log ^2\left (x+\sqrt {1+x}\right )}{(1+x)^2} \, dx &=2 \operatorname {Subst}\left (\int \frac {\log ^2\left (-1+x+x^2\right )}{x^3} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}+2 \operatorname {Subst}\left (\int \frac {(1+2 x) \log \left (-1+x+x^2\right )}{x^2 \left (-1+x+x^2\right )} \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}+2 \operatorname {Subst}\left (\int \left (-\frac {\log \left (-1+x+x^2\right )}{x^2}-\frac {3 \log \left (-1+x+x^2\right )}{x}+\frac {(4+3 x) \log \left (-1+x+x^2\right )}{-1+x+x^2}\right ) \, dx,x,\sqrt {1+x}\right )\\ &=-\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}-2 \operatorname {Subst}\left (\int \frac {\log \left (-1+x+x^2\right )}{x^2} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \frac {(4+3 x) \log \left (-1+x+x^2\right )}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right )-6 \operatorname {Subst}\left (\int \frac {\log \left (-1+x+x^2\right )}{x} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {2 \log \left (x+\sqrt {1+x}\right )}{\sqrt {1+x}}-6 \log \left (\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}-2 \operatorname {Subst}\left (\int \frac {1+2 x}{x \left (-1+x+x^2\right )} \, dx,x,\sqrt {1+x}\right )+2 \operatorname {Subst}\left (\int \left (\frac {\left (3+\sqrt {5}\right ) \log \left (-1+x+x^2\right )}{1-\sqrt {5}+2 x}+\frac {\left (3-\sqrt {5}\right ) \log \left (-1+x+x^2\right )}{1+\sqrt {5}+2 x}\right ) \, dx,x,\sqrt {1+x}\right )+6 \operatorname {Subst}\left (\int \frac {(1+2 x) \log (x)}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=\frac {2 \log \left (x+\sqrt {1+x}\right )}{\sqrt {1+x}}-6 \log \left (\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}-2 \operatorname {Subst}\left (\int \left (-\frac {1}{x}+\frac {3+x}{-1+x+x^2}\right ) \, dx,x,\sqrt {1+x}\right )+6 \operatorname {Subst}\left (\int \left (\frac {2 \log (x)}{1-\sqrt {5}+2 x}+\frac {2 \log (x)}{1+\sqrt {5}+2 x}\right ) \, dx,x,\sqrt {1+x}\right )+\left (2 \left (3-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (-1+x+x^2\right )}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )+\left (2 \left (3+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (-1+x+x^2\right )}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )\\ &=\log (1+x)+\frac {2 \log \left (x+\sqrt {1+x}\right )}{\sqrt {1+x}}-6 \log \left (\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}+\left (3+\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+\left (3-\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-2 \operatorname {Subst}\left (\int \frac {3+x}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right )+12 \operatorname {Subst}\left (\int \frac {\log (x)}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )+12 \operatorname {Subst}\left (\int \frac {\log (x)}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )+\left (-3-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {(1+2 x) \log \left (1-\sqrt {5}+2 x\right )}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right )+\left (-3+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {(1+2 x) \log \left (1+\sqrt {5}+2 x\right )}{-1+x+x^2} \, dx,x,\sqrt {1+x}\right )\\ &=\log (1+x)+\frac {2 \log \left (x+\sqrt {1+x}\right )}{\sqrt {1+x}}-6 \log \left (\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}+6 \log \left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+\left (3+\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+\left (3-\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )+6 \log \left (\sqrt {1+x}\right ) \log \left (1+\frac {2 \sqrt {1+x}}{1+\sqrt {5}}\right )-6 \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{1+\sqrt {5}}\right )}{x} \, dx,x,\sqrt {1+x}\right )+12 \operatorname {Subst}\left (\int \frac {\log \left (-\frac {2 x}{1-\sqrt {5}}\right )}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )+\left (-3-\sqrt {5}\right ) \operatorname {Subst}\left (\int \left (\frac {2 \log \left (1-\sqrt {5}+2 x\right )}{1-\sqrt {5}+2 x}+\frac {2 \log \left (1-\sqrt {5}+2 x\right )}{1+\sqrt {5}+2 x}\right ) \, dx,x,\sqrt {1+x}\right )-\left (1-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2}+\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+x}\right )+\left (-3+\sqrt {5}\right ) \operatorname {Subst}\left (\int \left (\frac {2 \log \left (1+\sqrt {5}+2 x\right )}{1-\sqrt {5}+2 x}+\frac {2 \log \left (1+\sqrt {5}+2 x\right )}{1+\sqrt {5}+2 x}\right ) \, dx,x,\sqrt {1+x}\right )-\left (1+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2}-\frac {\sqrt {5}}{2}+x} \, dx,x,\sqrt {1+x}\right )\\ &=\log (1+x)+\frac {2 \log \left (x+\sqrt {1+x}\right )}{\sqrt {1+x}}-6 \log \left (\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}-\left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+6 \log \left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+\left (3+\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )-\left (1-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )+\left (3-\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )+6 \log \left (\sqrt {1+x}\right ) \log \left (1+\frac {2 \sqrt {1+x}}{1+\sqrt {5}}\right )+6 \text {Li}_2\left (-\frac {2 \sqrt {1+x}}{1+\sqrt {5}}\right )-6 \text {Li}_2\left (1+\frac {2 \sqrt {1+x}}{1-\sqrt {5}}\right )-\left (2 \left (3-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\sqrt {5}+2 x\right )}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )-\left (2 \left (3-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\sqrt {5}+2 x\right )}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )-\left (2 \left (3+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\sqrt {5}+2 x\right )}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )-\left (2 \left (3+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (1-\sqrt {5}+2 x\right )}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )\\ &=\log (1+x)+\frac {2 \log \left (x+\sqrt {1+x}\right )}{\sqrt {1+x}}-6 \log \left (\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}-\left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+6 \log \left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+\left (3+\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )-\left (1-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )+\left (3-\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\left (3-\sqrt {5}\right ) \log \left (-\frac {1-\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right )+6 \log \left (\sqrt {1+x}\right ) \log \left (1+\frac {2 \sqrt {1+x}}{1+\sqrt {5}}\right )+6 \text {Li}_2\left (-\frac {2 \sqrt {1+x}}{1+\sqrt {5}}\right )-6 \text {Li}_2\left (1+\frac {2 \sqrt {1+x}}{1-\sqrt {5}}\right )-\left (3-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1+\sqrt {5}+2 \sqrt {1+x}\right )+\left (2 \left (3-\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2 \left (1-\sqrt {5}+2 x\right )}{2 \left (1-\sqrt {5}\right )-2 \left (1+\sqrt {5}\right )}\right )}{1+\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )-\left (3+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,1-\sqrt {5}+2 \sqrt {1+x}\right )+\left (2 \left (3+\sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {\log \left (\frac {2 \left (1+\sqrt {5}+2 x\right )}{-2 \left (1-\sqrt {5}\right )+2 \left (1+\sqrt {5}\right )}\right )}{1-\sqrt {5}+2 x} \, dx,x,\sqrt {1+x}\right )\\ &=\log (1+x)+\frac {2 \log \left (x+\sqrt {1+x}\right )}{\sqrt {1+x}}-6 \log \left (\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}-\left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+6 \log \left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+\left (3+\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )-\frac {1}{2} \left (3+\sqrt {5}\right ) \log ^2\left (1-\sqrt {5}+2 \sqrt {1+x}\right )-\left (1-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )+\left (3-\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\left (3-\sqrt {5}\right ) \log \left (-\frac {1-\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \log ^2\left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right )+6 \log \left (\sqrt {1+x}\right ) \log \left (1+\frac {2 \sqrt {1+x}}{1+\sqrt {5}}\right )+6 \text {Li}_2\left (-\frac {2 \sqrt {1+x}}{1+\sqrt {5}}\right )-6 \text {Li}_2\left (1+\frac {2 \sqrt {1+x}}{1-\sqrt {5}}\right )+\left (3-\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{2 \left (1-\sqrt {5}\right )-2 \left (1+\sqrt {5}\right )}\right )}{x} \, dx,x,1+\sqrt {5}+2 \sqrt {1+x}\right )+\left (3+\sqrt {5}\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {2 x}{-2 \left (1-\sqrt {5}\right )+2 \left (1+\sqrt {5}\right )}\right )}{x} \, dx,x,1-\sqrt {5}+2 \sqrt {1+x}\right )\\ &=\log (1+x)+\frac {2 \log \left (x+\sqrt {1+x}\right )}{\sqrt {1+x}}-6 \log \left (\sqrt {1+x}\right ) \log \left (x+\sqrt {1+x}\right )-\frac {\log ^2\left (x+\sqrt {1+x}\right )}{1+x}-\left (1+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+6 \log \left (\frac {1}{2} \left (-1+\sqrt {5}\right )\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )+\left (3+\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right )-\frac {1}{2} \left (3+\sqrt {5}\right ) \log ^2\left (1-\sqrt {5}+2 \sqrt {1+x}\right )-\left (1-\sqrt {5}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )+\left (3-\sqrt {5}\right ) \log \left (x+\sqrt {1+x}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\left (3-\sqrt {5}\right ) \log \left (-\frac {1-\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right ) \log \left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\frac {1}{2} \left (3-\sqrt {5}\right ) \log ^2\left (1+\sqrt {5}+2 \sqrt {1+x}\right )-\left (3+\sqrt {5}\right ) \log \left (1-\sqrt {5}+2 \sqrt {1+x}\right ) \log \left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right )+6 \log \left (\sqrt {1+x}\right ) \log \left (1+\frac {2 \sqrt {1+x}}{1+\sqrt {5}}\right )+6 \text {Li}_2\left (-\frac {2 \sqrt {1+x}}{1+\sqrt {5}}\right )-\left (3+\sqrt {5}\right ) \text {Li}_2\left (-\frac {1-\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right )-\left (3-\sqrt {5}\right ) \text {Li}_2\left (\frac {1+\sqrt {5}+2 \sqrt {1+x}}{2 \sqrt {5}}\right )-6 \text {Li}_2\left (1+\frac {2 \sqrt {1+x}}{1-\sqrt {5}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.58, size = 1076, normalized size = 1.94 \[ \frac {1}{2} \left (\sqrt {5} \log ^2\left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )+3 \log ^2\left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )-12 \log \left (\frac {2 \sqrt {x+1}}{-1+\sqrt {5}}\right ) \log \left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )+6 \log (x+1) \log \left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )-2 \sqrt {5} \log \left (-2 \sqrt {x+1}+\sqrt {5}-1\right ) \log \left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )-6 \log \left (-2 \sqrt {x+1}+\sqrt {5}-1\right ) \log \left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )+2 \sqrt {5} \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right ) \log \left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )-6 \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right ) \log \left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )-2 \sqrt {5} \log \left (\frac {2 \sqrt {x+1}+\sqrt {5}+1}{2 \sqrt {5}}\right ) \log \left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )+6 \log \left (\frac {2 \sqrt {x+1}+\sqrt {5}+1}{2 \sqrt {5}}\right ) \log \left (\sqrt {x+1}-\frac {\sqrt {5}}{2}+\frac {1}{2}\right )-\sqrt {5} \log ^2\left (\sqrt {x+1}+\frac {1}{2} \left (1+\sqrt {5}\right )\right )+3 \log ^2\left (\sqrt {x+1}+\frac {1}{2} \left (1+\sqrt {5}\right )\right )-\frac {2 \log ^2\left (x+\sqrt {x+1}\right )}{x+1}-6 \log \left (\frac {1}{2} \left (1+\sqrt {5}\right )\right ) \log (x+1)+2 \log (x+1)+6 \log (x+1) \log \left (\sqrt {x+1}+\frac {1}{2} \left (1+\sqrt {5}\right )\right )-2 \sqrt {5} \log \left (-2 \sqrt {x+1}+\sqrt {5}-1\right ) \log \left (\sqrt {x+1}+\frac {1}{2} \left (1+\sqrt {5}\right )\right )-6 \log \left (-2 \sqrt {x+1}+\sqrt {5}-1\right ) \log \left (\sqrt {x+1}+\frac {1}{2} \left (1+\sqrt {5}\right )\right )-6 \log (x+1) \log \left (x+\sqrt {x+1}\right )+2 \sqrt {5} \log \left (-2 \sqrt {x+1}+\sqrt {5}-1\right ) \log \left (x+\sqrt {x+1}\right )+6 \log \left (-2 \sqrt {x+1}+\sqrt {5}-1\right ) \log \left (x+\sqrt {x+1}\right )+\frac {4 \log \left (x+\sqrt {x+1}\right )}{\sqrt {x+1}}+\sqrt {5} \log (5) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right )+3 \log (5) \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right )-2 \sqrt {5} \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right )-2 \log \left (2 \sqrt {x+1}-\sqrt {5}+1\right )+2 \sqrt {5} \log \left (\sqrt {x+1}+\frac {1}{2} \left (1+\sqrt {5}\right )\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )-6 \log \left (\sqrt {x+1}+\frac {1}{2} \left (1+\sqrt {5}\right )\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )-2 \sqrt {5} \log \left (x+\sqrt {x+1}\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )+6 \log \left (x+\sqrt {x+1}\right ) \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )+2 \sqrt {5} \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )-2 \log \left (2 \sqrt {x+1}+\sqrt {5}+1\right )+12 \operatorname {PolyLog}\left (2,-\frac {2 \sqrt {x+1}}{1+\sqrt {5}}\right )-4 \sqrt {5} \operatorname {PolyLog}\left (2,\frac {-2 \sqrt {x+1}+\sqrt {5}-1}{2 \sqrt {5}}\right )-12 \operatorname {PolyLog}\left (2,\frac {-2 \sqrt {x+1}+\sqrt {5}-1}{-1+\sqrt {5}}\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (x + \sqrt {x + 1}\right )^{2}}{x^{2} + 2 \, x + 1}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (x + \sqrt {x + 1}\right )^{2}}{{\left (x + 1\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (x +\sqrt {x +1}\right )^{2}}{\left (x +1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {\log \left (x + \sqrt {x + 1}\right )^{2}}{x + 1} + \int \frac {{\left (2 \, x + \sqrt {x + 1} + 2\right )} \log \left (x + \sqrt {x + 1}\right )}{x^{3} + 2 \, x^{2} + {\left (x^{2} + 2 \, x + 1\right )} \sqrt {x + 1} + x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\ln \left (x+\sqrt {x+1}\right )}^2}{{\left (x+1\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (x + \sqrt {x + 1} \right )}^{2}}{\left (x + 1\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________