Integrand size = 14, antiderivative size = 141 \[ \int \arctan \left (x+\sqrt {1-x^2}\right ) \, dx=-\frac {\arcsin (x)}{2}+\frac {1}{4} \sqrt {3} \arctan \left (\frac {-1+\sqrt {3} x}{\sqrt {1-x^2}}\right )+\frac {1}{4} \sqrt {3} \arctan \left (\frac {1+\sqrt {3} x}{\sqrt {1-x^2}}\right )-\frac {1}{4} \sqrt {3} \arctan \left (\frac {-1+2 x^2}{\sqrt {3}}\right )+x \arctan \left (x+\sqrt {1-x^2}\right )-\frac {1}{4} \text {arctanh}\left (x \sqrt {1-x^2}\right )-\frac {1}{8} \log \left (1-x^2+x^4\right ) \] Output:
-1/2*arcsin(x)+x*arctan(x+(-x^2+1)^(1/2))-1/4*arctanh(x*(-x^2+1)^(1/2))-1/ 8*ln(x^4-x^2+1)-1/4*arctan(1/3*(2*x^2-1)*3^(1/2))*3^(1/2)+1/4*arctan((-1+x *3^(1/2))/(-x^2+1)^(1/2))*3^(1/2)+1/4*arctan((1+x*3^(1/2))/(-x^2+1)^(1/2)) *3^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.13 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.60 \[ \int \arctan \left (x+\sqrt {1-x^2}\right ) \, dx=-\arctan \left (\frac {x}{-1+\sqrt {1-x^2}}\right )+x \arctan \left (x+\sqrt {1-x^2}\right )-\log (x)+\frac {1}{2} \log \left (-1+\sqrt {1-x^2}\right )-\frac {1}{2} \text {RootSum}\left [1-2 \text {$\#$1}+2 \text {$\#$1}^2+2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {-\log (x)+\log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right )-\log (x) \text {$\#$1}+\log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}-3 \log (x) \text {$\#$1}^2+3 \log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^2-\log (x) \text {$\#$1}^3+\log \left (-1+\sqrt {1-x^2}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-1+2 \text {$\#$1}+3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[ArcTan[x + Sqrt[1 - x^2]],x]
Output:
-ArcTan[x/(-1 + Sqrt[1 - x^2])] + x*ArcTan[x + Sqrt[1 - x^2]] - Log[x] + L og[-1 + Sqrt[1 - x^2]]/2 - RootSum[1 - 2*#1 + 2*#1^2 + 2*#1^3 + #1^4 & , ( -Log[x] + Log[-1 + Sqrt[1 - x^2] - x*#1] - Log[x]*#1 + Log[-1 + Sqrt[1 - x ^2] - x*#1]*#1 - 3*Log[x]*#1^2 + 3*Log[-1 + Sqrt[1 - x^2] - x*#1]*#1^2 - L og[x]*#1^3 + Log[-1 + Sqrt[1 - x^2] - x*#1]*#1^3)/(-1 + 2*#1 + 3*#1^2 + 2* #1^3) & ]/2
Result contains complex when optimal does not.
Time = 0.92 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.94, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5726, 27, 7293, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \arctan \left (\sqrt {1-x^2}+x\right ) \, dx\) |
| \(\Big \downarrow \) 5726 |
| \(\displaystyle x \arctan \left (\sqrt {1-x^2}+x\right )-\int \frac {x \left (1-\frac {x}{\sqrt {1-x^2}}\right )}{2 \left (\sqrt {1-x^2} x+1\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle x \arctan \left (\sqrt {1-x^2}+x\right )-\frac {1}{2} \int \frac {x \left (1-\frac {x}{\sqrt {1-x^2}}\right )}{\sqrt {1-x^2} x+1}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle x \arctan \left (\sqrt {1-x^2}+x\right )-\frac {1}{2} \int \left (\frac {x^2}{x^3-x-\sqrt {1-x^2}}+\frac {x}{\sqrt {1-x^2} x+1}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle x \arctan \left (\sqrt {1-x^2}+x\right )+\frac {1}{2} \left (-\arcsin (x)+\frac {1}{2} \sqrt {3} \arctan \left (\frac {1-2 x^2}{\sqrt {3}}\right )+\frac {1}{6} \left (-\sqrt {3}+3 i\right ) \arctan \left (\frac {x}{\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt {1-x^2}}\right )+\frac {2 \arctan \left (\frac {x}{\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} \sqrt {1-x^2}}\right )}{\sqrt {3}}-\frac {1}{6} \left (\sqrt {3}+3 i\right ) \arctan \left (\frac {\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x}{\sqrt {1-x^2}}\right )+\frac {2 \arctan \left (\frac {\sqrt {-\frac {-\sqrt {3}+i}{\sqrt {3}+i}} x}{\sqrt {1-x^2}}\right )}{\sqrt {3}}-\frac {1}{4} \log \left (x^4-x^2+1\right )\right )\) |
Input:
Int[ArcTan[x + Sqrt[1 - x^2]],x]
Output:
x*ArcTan[x + Sqrt[1 - x^2]] + (-ArcSin[x] + (Sqrt[3]*ArcTan[(1 - 2*x^2)/Sq rt[3]])/2 + (2*ArcTan[x/(Sqrt[-((I - Sqrt[3])/(I + Sqrt[3]))]*Sqrt[1 - x^2 ])])/Sqrt[3] + ((3*I - Sqrt[3])*ArcTan[x/(Sqrt[-((I - Sqrt[3])/(I + Sqrt[3 ]))]*Sqrt[1 - x^2])])/6 + (2*ArcTan[(Sqrt[-((I - Sqrt[3])/(I + Sqrt[3]))]* x)/Sqrt[1 - x^2]])/Sqrt[3] - ((3*I + Sqrt[3])*ArcTan[(Sqrt[-((I - Sqrt[3]) /(I + Sqrt[3]))]*x)/Sqrt[1 - x^2]])/6 - Log[1 - x^2 + x^4]/4)/2
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[ArcTan[u_], x_Symbol] :> Simp[x*ArcTan[u], x] - Int[SimplifyIntegrand[x *(D[u, x]/(1 + u^2)), x], x] /; InverseFunctionFreeQ[u, x]
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 471, normalized size of antiderivative = 3.34
| method | result | size |
| default | \(x \arctan \left (x +\sqrt {-x^{2}+1}\right )-\frac {\ln \left (x^{4}-x^{2}+1\right )}{8}-\frac {\arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{4}-\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{4}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{2}-\frac {\left (\frac {1}{4}-\frac {i \sqrt {3}}{12}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{2}+\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{4}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{2}+\frac {\left (\frac {1}{4}-\frac {i \sqrt {3}}{12}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{2}-\frac {i \sqrt {3}\, \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{12}+\frac {i \sqrt {3}\, \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{12}-\frac {i \sqrt {3}\, \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{12}+\frac {i \sqrt {3}\, \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{12}+\arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x}\right )\) | \(471\) |
| parts | \(x \arctan \left (x +\sqrt {-x^{2}+1}\right )-\frac {\ln \left (x^{4}-x^{2}+1\right )}{8}-\frac {\arctan \left (\frac {\left (2 x^{2}-1\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{4}-\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{4}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{2}-\frac {\left (\frac {1}{4}-\frac {i \sqrt {3}}{12}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{2}+\frac {\left (\frac {i \sqrt {3}}{12}+\frac {1}{4}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{2}+\frac {\left (\frac {1}{4}-\frac {i \sqrt {3}}{12}\right ) \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{2}-\frac {i \sqrt {3}\, \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{12}+\frac {i \sqrt {3}\, \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{12}-\frac {i \sqrt {3}\, \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1+i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{12}+\frac {i \sqrt {3}\, \ln \left (\frac {\left (\sqrt {-x^{2}+1}-1\right )^{2}}{x^{2}}+\frac {\left (-1-i \sqrt {3}\right ) \left (\sqrt {-x^{2}+1}-1\right )}{x}-1\right )}{12}+\arctan \left (\frac {\sqrt {-x^{2}+1}-1}{x}\right )\) | \(471\) |
Input:
int(arctan(x+(-x^2+1)^(1/2)),x,method=_RETURNVERBOSE)
Output:
x*arctan(x+(-x^2+1)^(1/2))-1/8*ln(x^4-x^2+1)-1/4*arctan(1/3*(2*x^2-1)*3^(1 /2))*3^(1/2)-1/2*(1/12*I*3^(1/2)+1/4)*ln(((-x^2+1)^(1/2)-1)^2/x^2+(1+I*3^( 1/2))*((-x^2+1)^(1/2)-1)/x-1)-1/2*(1/4-1/12*I*3^(1/2))*ln(((-x^2+1)^(1/2)- 1)^2/x^2+(1-I*3^(1/2))*((-x^2+1)^(1/2)-1)/x-1)+1/2*(1/12*I*3^(1/2)+1/4)*ln (((-x^2+1)^(1/2)-1)^2/x^2+(-1-I*3^(1/2))*((-x^2+1)^(1/2)-1)/x-1)+1/2*(1/4- 1/12*I*3^(1/2))*ln(((-x^2+1)^(1/2)-1)^2/x^2+(-1+I*3^(1/2))*((-x^2+1)^(1/2) -1)/x-1)-1/12*I*3^(1/2)*ln(((-x^2+1)^(1/2)-1)^2/x^2+(1+I*3^(1/2))*((-x^2+1 )^(1/2)-1)/x-1)+1/12*I*3^(1/2)*ln(((-x^2+1)^(1/2)-1)^2/x^2+(1-I*3^(1/2))*( (-x^2+1)^(1/2)-1)/x-1)-1/12*I*3^(1/2)*ln(((-x^2+1)^(1/2)-1)^2/x^2+(-1+I*3^ (1/2))*((-x^2+1)^(1/2)-1)/x-1)+1/12*I*3^(1/2)*ln(((-x^2+1)^(1/2)-1)^2/x^2+ (-1-I*3^(1/2))*((-x^2+1)^(1/2)-1)/x-1)+arctan(((-x^2+1)^(1/2)-1)/x)
Time = 0.10 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.35 \[ \int \arctan \left (x+\sqrt {1-x^2}\right ) \, dx=x \arctan \left (x + \sqrt {-x^{2} + 1}\right ) - \frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) - \frac {1}{8} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {-x^{2} + 1} x + \sqrt {3}}{3 \, {\left (2 \, x^{2} - 1\right )}}\right ) - \frac {1}{8} \, \sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {-x^{2} + 1} x - \sqrt {3}}{3 \, {\left (2 \, x^{2} - 1\right )}}\right ) + \frac {1}{2} \, \arctan \left (\frac {\sqrt {-x^{2} + 1} x}{x^{2} - 1}\right ) - \frac {1}{8} \, \log \left (x^{4} - x^{2} + 1\right ) - \frac {1}{16} \, \log \left (-x^{4} + x^{2} + 2 \, \sqrt {-x^{2} + 1} x + 1\right ) + \frac {1}{16} \, \log \left (-x^{4} + x^{2} - 2 \, \sqrt {-x^{2} + 1} x + 1\right ) \] Input:
integrate(arctan(x+(-x^2+1)^(1/2)),x, algorithm="fricas")
Output:
x*arctan(x + sqrt(-x^2 + 1)) - 1/4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - 1)) - 1/8*sqrt(3)*arctan(1/3*(4*sqrt(3)*sqrt(-x^2 + 1)*x + sqrt(3))/(2*x^2 - 1)) - 1/8*sqrt(3)*arctan(1/3*(4*sqrt(3)*sqrt(-x^2 + 1)*x - sqrt(3))/(2*x^2 - 1)) + 1/2*arctan(sqrt(-x^2 + 1)*x/(x^2 - 1)) - 1/8*log(x^4 - x^2 + 1) - 1/16*log(-x^4 + x^2 + 2*sqrt(-x^2 + 1)*x + 1) + 1/16*log(-x^4 + x^2 - 2*s qrt(-x^2 + 1)*x + 1)
\[ \int \arctan \left (x+\sqrt {1-x^2}\right ) \, dx=\int \operatorname {atan}{\left (x + \sqrt {1 - x^{2}} \right )}\, dx \] Input:
integrate(atan(x+(-x**2+1)**(1/2)),x)
Output:
Integral(atan(x + sqrt(1 - x**2)), x)
\[ \int \arctan \left (x+\sqrt {1-x^2}\right ) \, dx=\int { \arctan \left (x + \sqrt {-x^{2} + 1}\right ) \,d x } \] Input:
integrate(arctan(x+(-x^2+1)^(1/2)),x, algorithm="maxima")
Output:
x*arctan(x + sqrt(x + 1)*sqrt(-x + 1)) - integrate((x^3 + x^2*e^(1/2*log(x + 1) + 1/2*log(-x + 1)) - x)/(x^4 + (x^2 - 1)*e^(log(x + 1) + log(-x + 1) ) + 2*(x^3 - x)*e^(1/2*log(x + 1) + 1/2*log(-x + 1)) - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (110) = 220\).
Time = 0.15 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.58 \[ \int \arctan \left (x+\sqrt {1-x^2}\right ) \, dx=x \arctan \left (x + \sqrt {-x^{2} + 1}\right ) - \frac {1}{4} \, \pi \mathrm {sgn}\left (x\right ) + \frac {1}{8} \, \sqrt {3} {\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (-\frac {\sqrt {3} x {\left (\frac {\sqrt {-x^{2} + 1} - 1}{x} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} + \frac {1}{8} \, \sqrt {3} {\left (\pi \mathrm {sgn}\left (x\right ) + 2 \, \arctan \left (\frac {\sqrt {3} x {\left (\frac {\sqrt {-x^{2} + 1} - 1}{x} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{3 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right )\right )} - \frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x^{2} - 1\right )}\right ) - \frac {1}{2} \, \arctan \left (-\frac {x {\left (\frac {{\left (\sqrt {-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}\right ) - \frac {1}{8} \, \log \left (x^{4} - x^{2} + 1\right ) + \frac {1}{8} \, \log \left ({\left (\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}\right )}^{2} + \frac {2 \, x}{\sqrt {-x^{2} + 1} - 1} - \frac {2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} + 4\right ) - \frac {1}{8} \, \log \left ({\left (\frac {x}{\sqrt {-x^{2} + 1} - 1} - \frac {\sqrt {-x^{2} + 1} - 1}{x}\right )}^{2} - \frac {2 \, x}{\sqrt {-x^{2} + 1} - 1} + \frac {2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}}{x} + 4\right ) \] Input:
integrate(arctan(x+(-x^2+1)^(1/2)),x, algorithm="giac")
Output:
x*arctan(x + sqrt(-x^2 + 1)) - 1/4*pi*sgn(x) + 1/8*sqrt(3)*(pi*sgn(x) + 2* arctan(-1/3*sqrt(3)*x*((sqrt(-x^2 + 1) - 1)/x + (sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))) + 1/8*sqrt(3)*(pi*sgn(x) + 2*arctan(1/3*sqrt( 3)*x*((sqrt(-x^2 + 1) - 1)/x - (sqrt(-x^2 + 1) - 1)^2/x^2 + 1)/(sqrt(-x^2 + 1) - 1))) - 1/4*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 - 1)) - 1/2*arctan(-1/ 2*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1)) - 1/8*log(x^4 - x^2 + 1) + 1/8*log((x/(sqrt(-x^2 + 1) - 1) - (sqrt(-x^2 + 1) - 1)/x)^2 + 2*x/(sqrt(-x^2 + 1) - 1) - 2*(sqrt(-x^2 + 1) - 1)/x + 4) - 1/8*log((x/(sqr t(-x^2 + 1) - 1) - (sqrt(-x^2 + 1) - 1)/x)^2 - 2*x/(sqrt(-x^2 + 1) - 1) + 2*(sqrt(-x^2 + 1) - 1)/x + 4)
Time = 0.79 (sec) , antiderivative size = 661, normalized size of antiderivative = 4.69 \[ \int \arctan \left (x+\sqrt {1-x^2}\right ) \, dx=\text {Too large to display} \] Input:
int(atan(x + (1 - x^2)^(1/2)),x)
Output:
x*atan(x + (1 - x^2)^(1/2)) - asin(x)/2 + (log(x - 3^(1/2)/2 - 1i/2)*(3^(1 /2)/2 + (3^(1/2)/2 + 1i/2)^3 + 1i/2))/(2*3^(1/2) - 8*(3^(1/2)/2 + 1i/2)^3 + 2i) - (log(x - 3^(1/2)/2 + 1i/2)*(3^(1/2)/2 + (3^(1/2)/2 - 1i/2)^3 - 1i/ 2))/(8*(3^(1/2)/2 - 1i/2)^3 - 2*3^(1/2) + 2i) - (log(x + 3^(1/2)/2 - 1i/2) *(3^(1/2)/2 + (3^(1/2)/2 - 1i/2)^3 - 1i/2))/(8*(3^(1/2)/2 - 1i/2)^3 - 2*3^ (1/2) + 2i) + (log(x + 3^(1/2)/2 + 1i/2)*(3^(1/2)/2 + (3^(1/2)/2 + 1i/2)^3 + 1i/2))/(2*3^(1/2) - 8*(3^(1/2)/2 + 1i/2)^3 + 2i) + (log((((x*(3^(1/2)/2 + 1i/2) - 1)*1i)/(1 - (3^(1/2)/2 + 1i/2)^2)^(1/2) - (1 - x^2)^(1/2)*1i)/( 3^(1/2)/2 - x + 1i/2))*((3^(1/2)/2 + 1i/2)^2 + 1))/((1 - (3^(1/2)/2 + 1i/2 )^2)^(1/2)*(2*3^(1/2) - 8*(3^(1/2)/2 + 1i/2)^3 + 2i)) - (log((((x*(3^(1/2) /2 - 1i/2) - 1)*1i)/(1 - (3^(1/2)/2 - 1i/2)^2)^(1/2) - (1 - x^2)^(1/2)*1i) /(x - 3^(1/2)/2 + 1i/2))*((3^(1/2)/2 - 1i/2)^2 + 1))/((1 - (3^(1/2)/2 - 1i /2)^2)^(1/2)*(8*(3^(1/2)/2 - 1i/2)^3 - 2*3^(1/2) + 2i)) + (log((((x*(3^(1/ 2)/2 - 1i/2) + 1)*1i)/(1 - (3^(1/2)/2 - 1i/2)^2)^(1/2) + (1 - x^2)^(1/2)*1 i)/(x + 3^(1/2)/2 - 1i/2))*((3^(1/2)/2 - 1i/2)^2 + 1))/((1 - (3^(1/2)/2 - 1i/2)^2)^(1/2)*(8*(3^(1/2)/2 - 1i/2)^3 - 2*3^(1/2) + 2i)) - (log((((x*(3^( 1/2)/2 + 1i/2) + 1)*1i)/(1 - (3^(1/2)/2 + 1i/2)^2)^(1/2) + (1 - x^2)^(1/2) *1i)/(x + 3^(1/2)/2 + 1i/2))*((3^(1/2)/2 + 1i/2)^2 + 1))/((1 - (3^(1/2)/2 + 1i/2)^2)^(1/2)*(2*3^(1/2) - 8*(3^(1/2)/2 + 1i/2)^3 + 2i))
\[ \int \arctan \left (x+\sqrt {1-x^2}\right ) \, dx=-\frac {\mathit {asin} \left (x \right )}{2}+\mathit {atan} \left (\sqrt {-x^{2}+1}+x \right ) x +\frac {3 \left (\int \frac {x^{2}}{\sqrt {-x^{2}+1}-x^{3}+x}d x \right )}{2}+2 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{4}-2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{3}+2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+2 \tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )+1\right )-4 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (x \right )}{2}\right )^{2}+1\right )-\frac {3 \,\mathrm {log}\left (\sqrt {-x^{2}+1}\, x +1\right )}{2} \] Input:
int(atan(x+(-x^2+1)^(1/2)),x)
Output:
( - asin(x) + 2*atan(sqrt( - x**2 + 1) + x)*x + 3*int(x**2/(sqrt( - x**2 + 1) - x**3 + x),x) + 4*log(tan(asin(x)/2)**4 - 2*tan(asin(x)/2)**3 + 2*tan (asin(x)/2)**2 + 2*tan(asin(x)/2) + 1) - 8*log(tan(asin(x)/2)**2 + 1) - 3* log(sqrt( - x**2 + 1)*x + 1))/2