3.12 \(\int \tan ^{-1}\left (x+\sqrt{1-x^2}\right ) \, dx\)

Optimal. Leaf size=141 \[ \frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} x-1}{\sqrt{1-x^2}}\right )+\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} x+1}{\sqrt{1-x^2}}\right )-\frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{2 x^2-1}{\sqrt{3}}\right )+x \tan ^{-1}\left (\sqrt{1-x^2}+x\right )-\frac{1}{4} \tanh ^{-1}\left (x \sqrt{1-x^2}\right )-\frac{1}{8} \log \left (x^4-x^2+1\right )-\frac{1}{2} \sin ^{-1}(x) \]

[Out]

-ArcSin[x]/2 + (Sqrt[3]*ArcTan[(-1 + Sqrt[3]*x)/Sqrt[1 - x^2]])/4 + (Sqrt[3]*Arc
Tan[(1 + Sqrt[3]*x)/Sqrt[1 - x^2]])/4 - (Sqrt[3]*ArcTan[(-1 + 2*x^2)/Sqrt[3]])/4
+ x*ArcTan[x + Sqrt[1 - x^2]] - ArcTanh[x*Sqrt[1 - x^2]]/4 - Log[1 - x^2 + x^4]
/8

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Rubi [C]  time = 1.47868, 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 \[ \frac{1}{4} \sqrt{3} \tan ^{-1}\left (\frac{1-2 x^2}{\sqrt{3}}\right )+\frac{1}{12} \left (-\sqrt{3}+3 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{-\frac{-\sqrt{3}+i}{\sqrt{3}+i}} \sqrt{1-x^2}}\right )+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{-\frac{-\sqrt{3}+i}{\sqrt{3}+i}} \sqrt{1-x^2}}\right )}{\sqrt{3}}-\frac{1}{12} \left (\sqrt{3}+3 i\right ) \tan ^{-1}\left (\frac{\sqrt{-\frac{-\sqrt{3}+i}{\sqrt{3}+i}} x}{\sqrt{1-x^2}}\right )+\frac{\tan ^{-1}\left (\frac{\sqrt{-\frac{-\sqrt{3}+i}{\sqrt{3}+i}} x}{\sqrt{1-x^2}}\right )}{\sqrt{3}}+x \tan ^{-1}\left (\sqrt{1-x^2}+x\right )-\frac{1}{8} \log \left (x^4-x^2+1\right )-\frac{1}{2} \sin ^{-1}(x) \]

Warning: Unable to verify antiderivative.

[In]  Int[ArcTan[x + Sqrt[1 - x^2]],x]

[Out]

-ArcSin[x]/2 + (Sqrt[3]*ArcTan[(1 - 2*x^2)/Sqrt[3]])/4 + ArcTan[x/(Sqrt[-((I - S
qrt[3])/(I + Sqrt[3]))]*Sqrt[1 - x^2])]/Sqrt[3] + ((3*I - Sqrt[3])*ArcTan[x/(Sqr
t[-((I - Sqrt[3])/(I + Sqrt[3]))]*Sqrt[1 - x^2])])/12 + 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]])/12 + x*ArcTan[x + Sqrt[1 - x^2
]] - Log[1 - x^2 + x^4]/8

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(atan(x+(-x**2+1)**(1/2)),x)

[Out]

Timed out

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Mathematica [C]  time = 8.10516, size = 1822, normalized size = 12.92 \[ \text{result too large to display} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[ArcTan[x + Sqrt[1 - x^2]],x]

[Out]

x*ArcTan[x + Sqrt[1 - x^2]] + (-8*ArcSin[x] + 2*Sqrt[2 + (2*I)*Sqrt[3]]*ArcTan[(
(1 + I*Sqrt[3] - 2*x^2)*(-1 + x^2))/(-3*I - Sqrt[3] + 2*Sqrt[3]*x^4 + x^3*(-6 -
(2*I)*Sqrt[3] - 2*Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(6 + (2*I)*Sqrt[3]
- 2*Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x^2*(3*I - Sqrt[3] + 2*Sqrt[6 - (6*
I)*Sqrt[3]]*Sqrt[1 - x^2]))] - 2*Sqrt[2 + (2*I)*Sqrt[3]]*ArcTan[((1 + I*Sqrt[3]
- 2*x^2)*(-1 + x^2))/(-3*I - Sqrt[3] + 2*Sqrt[3]*x^4 + 2*x*(-3 - I*Sqrt[3] + Sqr
t[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + 2*x^3*(3 + I*Sqrt[3] + Sqrt[2 - (2*I)*Sqrt
[3]]*Sqrt[1 - x^2]) + x^2*(3*I - Sqrt[3] + 2*Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^
2]))] - 2*Sqrt[2 - (2*I)*Sqrt[3]]*ArcTan[((-1 + x^2)*(-1 + I*Sqrt[3] + 2*x^2))/(
3*I - Sqrt[3] + 2*Sqrt[3]*x^4 + x*(6 - (2*I)*Sqrt[3] - 2*Sqrt[2 + (2*I)*Sqrt[3]]
*Sqrt[1 - x^2]) + x^3*(-6 + (2*I)*Sqrt[3] - 2*Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x
^2]) + x^2*(-3*I - Sqrt[3] + 2*Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2]))] + 2*Sqrt
[2 - (2*I)*Sqrt[3]]*ArcTan[((-1 + x^2)*(-1 + I*Sqrt[3] + 2*x^2))/(3*I - Sqrt[3]
+ 2*Sqrt[3]*x^4 + 2*x^3*(3 - I*Sqrt[3] + Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2])
+ 2*x*(-3 + I*Sqrt[3] + Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x^2*(-3*I - Sqr
t[3] + 2*Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2]))] - 2*Log[-1/2 - (I/2)*Sqrt[3] +
x^2] + (2*I)*Sqrt[3]*Log[-1/2 - (I/2)*Sqrt[3] + x^2] - 2*Log[(I/2)*(I + Sqrt[3]
) + x^2] - (2*I)*Sqrt[3]*Log[(I/2)*(I + Sqrt[3]) + x^2] - I*Sqrt[2 - (2*I)*Sqrt[
3]]*Log[16*(1 + Sqrt[3]*x + x^2)^2] + I*Sqrt[2 + (2*I)*Sqrt[3]]*Log[16*(1 + Sqrt
[3]*x + x^2)^2] + I*Sqrt[2 - (2*I)*Sqrt[3]]*Log[(4 - 4*Sqrt[3]*x + 4*x^2)^2] - I
*Sqrt[2 + (2*I)*Sqrt[3]]*Log[(4 - 4*Sqrt[3]*x + 4*x^2)^2] - I*Sqrt[2 + (2*I)*Sqr
t[3]]*Log[3*I + Sqrt[3] - (-I + Sqrt[3])*x^4 + (2*I)*Sqrt[2 - (2*I)*Sqrt[3]]*Sqr
t[1 - x^2] + (5*I)*x^2*(2 + Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(3 + (5*I
)*Sqrt[3] + (3*I)*Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2]) + I*x^3*(3*I + 3*Sqrt[3
] + Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2])] + I*Sqrt[2 + (2*I)*Sqrt[3]]*Log[3*I
+ Sqrt[3] - (-I + Sqrt[3])*x^4 + (2*I)*Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2] + (
5*I)*x^2*(2 + Sqrt[2 - (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x^3*(3 - (3*I)*Sqrt[3] -
I*Sqrt[6 - (6*I)*Sqrt[3]]*Sqrt[1 - x^2]) - I*x*(-3*I + 5*Sqrt[3] + 3*Sqrt[6 - (6
*I)*Sqrt[3]]*Sqrt[1 - x^2])] + I*Sqrt[2 - (2*I)*Sqrt[3]]*Log[-3*I + Sqrt[3] - (I
+ Sqrt[3])*x^4 - (2*I)*Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2] - (5*I)*x^2*(2 + S
qrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2]) + x*(3 - (5*I)*Sqrt[3] - (3*I)*Sqrt[6 + (6
*I)*Sqrt[3]]*Sqrt[1 - x^2]) - I*x^3*(-3*I + 3*Sqrt[3] + Sqrt[6 + (6*I)*Sqrt[3]]*
Sqrt[1 - x^2])] - I*Sqrt[2 - (2*I)*Sqrt[3]]*Log[-3*I + Sqrt[3] - (I + Sqrt[3])*x
^4 - (2*I)*Sqrt[2 + (2*I)*Sqrt[3]]*Sqrt[1 - x^2] - (5*I)*x^2*(2 + Sqrt[2 + (2*I)
*Sqrt[3]]*Sqrt[1 - x^2]) + x^3*(3 + (3*I)*Sqrt[3] + I*Sqrt[6 + (6*I)*Sqrt[3]]*Sq
rt[1 - x^2]) + I*x*(3*I + 5*Sqrt[3] + 3*Sqrt[6 + (6*I)*Sqrt[3]]*Sqrt[1 - x^2])])
/16

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Maple [C]  time = 0.086, size = 439, normalized size = 3.1 \[ x\arctan \left ( x+\sqrt{-{x}^{2}+1} \right ) -{\frac{\ln \left ({x}^{4}-{x}^{2}+1 \right ) }{8}}-{\frac{\sqrt{3}}{4}\arctan \left ({\frac{ \left ( 2\,{x}^{2}-1 \right ) \sqrt{3}}{3}} \right ) }+{\frac{i}{8}}\sqrt{3}\ln \left ({\frac{1}{{x}^{2}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) ^{2}}+{\frac{-1-i\sqrt{3}}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-1 \right ) +{\frac{1}{8}\ln \left ({\frac{1}{{x}^{2}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) ^{2}}+{\frac{-1-i\sqrt{3}}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-1 \right ) }-{\frac{i}{8}}\sqrt{3}\ln \left ({\frac{1}{{x}^{2}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) ^{2}}+{\frac{1+i\sqrt{3}}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-1 \right ) +{\frac{1}{8}\ln \left ({\frac{1}{{x}^{2}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) ^{2}}+{\frac{-1+i\sqrt{3}}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-1 \right ) }-{\frac{i}{8}}\sqrt{3}\ln \left ({\frac{1}{{x}^{2}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) ^{2}}+{\frac{-1+i\sqrt{3}}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-1 \right ) -{\frac{1}{8}\ln \left ({\frac{1}{{x}^{2}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) ^{2}}+{\frac{1+i\sqrt{3}}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-1 \right ) }+{\frac{i}{8}}\sqrt{3}\ln \left ({\frac{1}{{x}^{2}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) ^{2}}+{\frac{1-i\sqrt{3}}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-1 \right ) -{\frac{1}{8}\ln \left ({\frac{1}{{x}^{2}} \left ( \sqrt{-{x}^{2}+1}-1 \right ) ^{2}}+{\frac{1-i\sqrt{3}}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) }-1 \right ) }+\arctan \left ({\frac{1}{x} \left ( \sqrt{-{x}^{2}+1}-1 \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(arctan(x+(-x^2+1)^(1/2)),x)

[Out]

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)+1/8*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)+1/8*ln(((-x^2+1)^(1/2)-1)^2/x^2+(-1-I*3^(1/2))*((-x^2+1)^(1/2)-1)/x-1)-1
/8*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
/8*ln(((-x^2+1)^(1/2)-1)^2/x^2+(-1+I*3^(1/2))*((-x^2+1)^(1/2)-1)/x-1)-1/8*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)-1/8*ln(((
-x^2+1)^(1/2)-1)^2/x^2+(1+I*3^(1/2))*((-x^2+1)^(1/2)-1)/x-1)+1/8*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)-1/8*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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ x \arctan \left (x + \sqrt{x + 1} \sqrt{-x + 1}\right ) - \int \frac{x^{3} + x^{2} e^{\left (\frac{1}{2} \, \log \left (x + 1\right ) + \frac{1}{2} \, \log \left (-x + 1\right )\right )} - x}{x^{4} +{\left (x^{2} - 1\right )} e^{\left (\log \left (x + 1\right ) + \log \left (-x + 1\right )\right )} + 2 \,{\left (x^{3} - x\right )} e^{\left (\frac{1}{2} \, \log \left (x + 1\right ) + \frac{1}{2} \, \log \left (-x + 1\right )\right )} - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(x + sqrt(-x^2 + 1)),x, algorithm="maxima")

[Out]

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)

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Fricas [A]  time = 0.256353, size = 240, normalized size = 1.7 \[ x \arctan \left (x + \sqrt{-x^{2} + 1}\right ) - \frac{1}{4} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) - \frac{1}{8} \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (4 \, \sqrt{-x^{2} + 1} x + 1\right )}}{3 \,{\left (2 \, x^{2} - 1\right )}}\right ) - \frac{1}{8} \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (4 \, \sqrt{-x^{2} + 1} x - 1\right )}}{3 \,{\left (2 \, x^{2} - 1\right )}}\right ) - \frac{1}{2} \, \arctan \left (\frac{x}{\sqrt{-x^{2} + 1}}\right ) - \frac{1}{8} \, \log \left (x^{4} - x^{2} + 1\right ) - \frac{1}{16} \, \log \left (-x^{4} + x^{2} + 2 \, \sqrt{-x^{2} + 1} x + 1\right ) + \frac{1}{16} \, \log \left (-x^{4} + x^{2} - 2 \, \sqrt{-x^{2} + 1} x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(x + sqrt(-x^2 + 1)),x, algorithm="fricas")

[Out]

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*sqrt(3)*(4*sqrt(-x^2 + 1)*x + 1)/(2*x^2 - 1)) - 1/8*sqrt(3)*
arctan(1/3*sqrt(3)*(4*sqrt(-x^2 + 1)*x - 1)/(2*x^2 - 1)) - 1/2*arctan(x/sqrt(-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*sqrt(-x^2 + 1)*x + 1)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(atan(x+(-x**2+1)**(1/2)),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21417, size = 491, normalized size = 3.48 \[ x \arctan \left (x + \sqrt{-x^{2} + 1}\right ) - \frac{1}{4} \, \pi{\rm sign}\left (x\right ) + \frac{1}{8} \, \sqrt{3}{\left (\pi{\rm sign}\left (x\right ) + 2 \, \arctan \left (-\frac{\sqrt{3} x{\left (\frac{\sqrt{-x^{2} + 1} - 1}{x} + \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{3 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right )\right )} + \frac{1}{8} \, \sqrt{3}{\left (\pi{\rm sign}\left (x\right ) + 2 \, \arctan \left (\frac{\sqrt{3} x{\left (\frac{\sqrt{-x^{2} + 1} - 1}{x} - \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + 1\right )}}{3 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right )\right )} - \frac{1}{4} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} - 1\right )}\right ) - \frac{1}{2} \, \arctan \left (-\frac{x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right ) - \frac{1}{8} \,{\rm ln}\left (x^{4} - x^{2} + 1\right ) + \frac{1}{8} \,{\rm ln}\left ({\left (\frac{x}{\sqrt{-x^{2} + 1} - 1} - \frac{\sqrt{-x^{2} + 1} - 1}{x}\right )}^{2} + \frac{2 \, x}{\sqrt{-x^{2} + 1} - 1} - \frac{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} + 4\right ) - \frac{1}{8} \,{\rm ln}\left ({\left (\frac{x}{\sqrt{-x^{2} + 1} - 1} - \frac{\sqrt{-x^{2} + 1} - 1}{x}\right )}^{2} - \frac{2 \, x}{\sqrt{-x^{2} + 1} - 1} + \frac{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} + 4\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(arctan(x + sqrt(-x^2 + 1)),x, algorithm="giac")

[Out]

x*arctan(x + sqrt(-x^2 + 1)) - 1/4*pi*sign(x) + 1/8*sqrt(3)*(pi*sign(x) + 2*arct
an(-1/3*sqrt(3)*x*((sqrt(-x^2 + 1) - 1)/x + (sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqr
t(-x^2 + 1) - 1))) + 1/8*sqrt(3)*(pi*sign(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*sqr
t(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*ln(x^4 - x^2 + 1) + 1/8*ln((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*ln((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)