Optimal. Leaf size=141 \[ \frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x-1}{\sqrt {1-x^2}}\right )+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x+1}{\sqrt {1-x^2}}\right )-\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {2 x^2-1}{\sqrt {3}}\right )+x \tan ^{-1}\left (\sqrt {1-x^2}+x\right )-\frac {1}{4} \tanh ^{-1}\left (x \sqrt {1-x^2}\right )-\frac {1}{8} \log \left (x^4-x^2+1\right )-\frac {1}{2} \sin ^{-1}(x) \]
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Rubi [C] time = 0.75, antiderivative size = 269, normalized size of antiderivative = 1.91, number of steps used = 40, number of rules used = 15, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {5203, 12, 6742, 216, 1114, 634, 618, 204, 628, 1174, 402, 377, 205, 1293, 1107} \[ -\frac {1}{8} \log \left (x^4-x^2+1\right )+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )+\frac {1}{12} \left (-\sqrt {3}+3 i\right ) \tan ^{-1}\left (\frac {x}{\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt {1-x^2}}\right )+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt {1-x^2}}\right )}{\sqrt {3}}-\frac {1}{12} \left (\sqrt {3}+3 i\right ) \tan ^{-1}\left (\frac {\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x}{\sqrt {1-x^2}}\right )+\frac {\tan ^{-1}\left (\frac {\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}+x \tan ^{-1}\left (\sqrt {1-x^2}+x\right )-\frac {1}{2} \sin ^{-1}(x) \]
Warning: Unable to verify antiderivative.
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Rule 12
Rule 204
Rule 205
Rule 216
Rule 377
Rule 402
Rule 618
Rule 628
Rule 634
Rule 1107
Rule 1114
Rule 1174
Rule 1293
Rule 5203
Rule 6742
Rubi steps
\begin {align*} \int \tan ^{-1}\left (x+\sqrt {1-x^2}\right ) \, dx &=x \tan ^{-1}\left (x+\sqrt {1-x^2}\right )-\int \frac {x \left (1-\frac {x}{\sqrt {1-x^2}}\right )}{2 \left (1+x \sqrt {1-x^2}\right )} \, dx\\ &=x \tan ^{-1}\left (x+\sqrt {1-x^2}\right )-\frac {1}{2} \int \frac {x \left (1-\frac {x}{\sqrt {1-x^2}}\right )}{1+x \sqrt {1-x^2}} \, dx\\ &=x \tan ^{-1}\left (x+\sqrt {1-x^2}\right )-\frac {1}{2} \int \left (\frac {x^2}{-x+x^3-\sqrt {1-x^2}}+\frac {x}{1+x \sqrt {1-x^2}}\right ) \, dx\\ &=x \tan ^{-1}\left (x+\sqrt {1-x^2}\right )-\frac {1}{2} \int \frac {x^2}{-x+x^3-\sqrt {1-x^2}} \, dx-\frac {1}{2} \int \frac {x}{1+x \sqrt {1-x^2}} \, dx\\ &=x \tan ^{-1}\left (x+\sqrt {1-x^2}\right )-\frac {1}{2} \int \left (\frac {x}{1-x^2+x^4}-\frac {x^2 \sqrt {1-x^2}}{1-x^2+x^4}\right ) \, dx-\frac {1}{2} \int \left (-\frac {1}{\sqrt {1-x^2}}+\frac {x^3}{1-x^2+x^4}+\frac {\sqrt {1-x^2}}{1-x^2+x^4}-\frac {x^2 \sqrt {1-x^2}}{1-x^2+x^4}\right ) \, dx\\ &=x \tan ^{-1}\left (x+\sqrt {1-x^2}\right )+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx-\frac {1}{2} \int \frac {x}{1-x^2+x^4} \, dx-\frac {1}{2} \int \frac {x^3}{1-x^2+x^4} \, dx-\frac {1}{2} \int \frac {\sqrt {1-x^2}}{1-x^2+x^4} \, dx+2 \left (\frac {1}{2} \int \frac {x^2 \sqrt {1-x^2}}{1-x^2+x^4} \, dx\right )\\ &=\frac {1}{2} \sin ^{-1}(x)+x \tan ^{-1}\left (x+\sqrt {1-x^2}\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^2\right )-\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{1-x+x^2} \, dx,x,x^2\right )+2 \left (-\left (\frac {1}{2} \int \frac {1}{\sqrt {1-x^2}} \, dx\right )+\frac {1}{2} \int \frac {1}{\sqrt {1-x^2} \left (1-x^2+x^4\right )} \, dx\right )+\frac {i \int \frac {\sqrt {1-x^2}}{-1-i \sqrt {3}+2 x^2} \, dx}{\sqrt {3}}-\frac {i \int \frac {\sqrt {1-x^2}}{-1+i \sqrt {3}+2 x^2} \, dx}{\sqrt {3}}\\ &=\frac {1}{2} \sin ^{-1}(x)+x \tan ^{-1}\left (x+\sqrt {1-x^2}\right )-\frac {1}{8} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,x^2\right )-\frac {1}{8} \operatorname {Subst}\left (\int \frac {-1+2 x}{1-x+x^2} \, dx,x,x^2\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^2\right )+2 \left (-\frac {1}{2} \sin ^{-1}(x)-\frac {i \int \frac {1}{\sqrt {1-x^2} \left (-1-i \sqrt {3}+2 x^2\right )} \, dx}{\sqrt {3}}+\frac {i \int \frac {1}{\sqrt {1-x^2} \left (-1+i \sqrt {3}+2 x^2\right )} \, dx}{\sqrt {3}}\right )+\frac {1}{6} \left (3-i \sqrt {3}\right ) \int \frac {1}{\sqrt {1-x^2} \left (-1+i \sqrt {3}+2 x^2\right )} \, dx+\frac {1}{6} \left (3+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1-x^2} \left (-1-i \sqrt {3}+2 x^2\right )} \, dx\\ &=\frac {1}{2} \sin ^{-1}(x)+\frac {\tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )}{2 \sqrt {3}}+x \tan ^{-1}\left (x+\sqrt {1-x^2}\right )-\frac {1}{8} \log \left (1-x^2+x^4\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 x^2\right )+2 \left (-\frac {1}{2} \sin ^{-1}(x)+\frac {i \operatorname {Subst}\left (\int \frac {1}{-1+i \sqrt {3}-\left (-1-i \sqrt {3}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}-\frac {i \operatorname {Subst}\left (\int \frac {1}{-1-i \sqrt {3}-\left (-1+i \sqrt {3}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}\right )+\frac {1}{6} \left (3-i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{-1+i \sqrt {3}-\left (-1-i \sqrt {3}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )+\frac {1}{6} \left (3+i \sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-i \sqrt {3}-\left (-1+i \sqrt {3}\right ) x^2} \, dx,x,\frac {x}{\sqrt {1-x^2}}\right )\\ &=\frac {1}{2} \sin ^{-1}(x)+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1-2 x^2}{\sqrt {3}}\right )+\frac {1}{12} \left (3 i-\sqrt {3}\right ) \tan ^{-1}\left (\frac {x}{\sqrt {-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt {1-x^2}}\right )-\frac {1}{12} \left (3 i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\sqrt {-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt {1-x^2}}\right )+2 \left (-\frac {1}{2} \sin ^{-1}(x)+\frac {\tan ^{-1}\left (\frac {x}{\sqrt {-\frac {i-\sqrt {3}}{i+\sqrt {3}}} \sqrt {1-x^2}}\right )}{2 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-\frac {i-\sqrt {3}}{i+\sqrt {3}}} x}{\sqrt {1-x^2}}\right )}{2 \sqrt {3}}\right )+x \tan ^{-1}\left (x+\sqrt {1-x^2}\right )-\frac {1}{8} \log \left (1-x^2+x^4\right )\\ \end {align*}
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Mathematica [C] time = 3.80, size = 1822, normalized size = 12.92 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.49, size = 191, normalized size = 1.35 \[ x \arctan \left (x + \sqrt {-x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) - \frac {1}{8} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {-x^{2} + 1} x + \sqrt {3}}{3 \, {\left (2 \, x^{2} - 1\right )}}\right ) - \frac {1}{8} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {-x^{2} + 1} x - \sqrt {3}}{3 \, {\left (2 \, x^{2} - 1\right )}}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{2} + 1} x}{x^{2} - 1}\right ) - \frac {1}{8} \, \log \left (x^{4} - x^{2} + 1\right ) - \frac {1}{16} \, \log \left (-x^{4} + x^{2} + 2 \, \sqrt {-x^{2} + 1} x + 1\right ) + \frac {1}{16} \, \log \left (-x^{4} + x^{2} - 2 \, \sqrt {-x^{2} + 1} x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.47, size = 364, normalized size = 2.58 \[ x \arctan \left (x + \sqrt {-x^{2} + 1}\right ) - \frac {1}{4} \, \pi \mathrm {sgn}\relax (x) + \frac {1}{8} \, \sqrt {3} {\left (\pi \mathrm {sgn}\relax (x) + 2 \, \arctan \left (-\frac {\sqrt {3} x {\left (\frac {\sqrt {-x^{2} + 1} - 1}{x} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} + \frac {1}{8} \, \sqrt {3} {\left (\pi \mathrm {sgn}\relax (x) + 2 \, \arctan \left (\frac {\sqrt {3} x {\left (\frac {\sqrt {-x^{2} + 1} - 1}{x} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} - \frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) - \frac {1}{2} \, \arctan \left (-\frac {x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right ) - \frac {1}{8} \, \log \left (x^{4} - x^{2} + 1\right ) + \frac {1}{8} \, \log \left ({\left (\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}\right )}^{2} + \frac {2 \, x}{\sqrt {-x^{2} + 1} - 1} - \frac {2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} + 4\right ) - \frac {1}{8} \, \log \left ({\left (\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}\right )}^{2} - \frac {2 \, x}{\sqrt {-x^{2} + 1} - 1} + \frac {2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 439, normalized size = 3.11 \[ x \arctan \left (x +\sqrt {-x^{2}+1}\right )+\arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x}\right )-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right )}{4}+\frac {i \sqrt {3}\, \ln \left (\frac {\left (-1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}+\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}-1\right )}{8}+\frac {\ln \left (\frac {\left (-1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}+\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}-1\right )}{8}+\frac {\ln \left (\frac {\left (-1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}+\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}-1\right )}{8}-\frac {i \sqrt {3}\, \ln \left (\frac {\left (-1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}+\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}-1\right )}{8}+\frac {i \sqrt {3}\, \ln \left (\frac {\left (1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}+\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}-1\right )}{8}-\frac {\ln \left (\frac {\left (1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}+\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}-1\right )}{8}-\frac {i \sqrt {3}\, \ln \left (\frac {\left (1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}+\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}-1\right )}{8}-\frac {\ln \left (\frac {\left (1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}+\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}-1\right )}{8}-\frac {\ln \left (x^{4}-x^{2}+1\right )}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ x \arctan \left (x + \sqrt {x + 1} \sqrt {-x + 1}\right ) - \int \frac {x^{3} + x^{2} e^{\left (\frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (-x + 1\right )\right )} - x}{x^{4} + {\left (x^{2} - 1\right )} e^{\left (\log \left (x + 1\right ) + \log \left (-x + 1\right )\right )} + 2 \, {\left (x^{3} - x\right )} e^{\left (\frac {1}{2} \, \log \left (x + 1\right ) + \frac {1}{2} \, \log \left (-x + 1\right )\right )} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.37, size = 661, normalized size = 4.69 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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