3.643 \(\int \frac{\sqrt{1+x}}{1+x^2} \, dx\)

Optimal. Leaf size=205 \[ \frac{\log \left (x-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )}}-\frac{\log \left (x+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )}}-\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{x+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )+\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{2 \sqrt{x+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right ) \]

[Out]

-(Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + x])/Sqrt[2*(-
1 + Sqrt[2])]]) + Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1
+ x])/Sqrt[2*(-1 + Sqrt[2])]] + Log[1 + Sqrt[2] + x - Sqrt[2*(1 + Sqrt[2])]*Sqr
t[1 + x]]/(2*Sqrt[2*(1 + Sqrt[2])]) - Log[1 + Sqrt[2] + x + Sqrt[2*(1 + Sqrt[2])
]*Sqrt[1 + x]]/(2*Sqrt[2*(1 + Sqrt[2])])

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Rubi [A]  time = 0.478884, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ \frac{\log \left (x-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )}}-\frac{\log \left (x+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )}{2 \sqrt{2 \left (1+\sqrt{2}\right )}}-\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{x+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )+\sqrt{\frac{1}{2} \left (1+\sqrt{2}\right )} \tan ^{-1}\left (\frac{2 \sqrt{x+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[1 + x]/(1 + x^2),x]

[Out]

-(Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + x])/Sqrt[2*(-
1 + Sqrt[2])]]) + Sqrt[(1 + Sqrt[2])/2]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1
+ x])/Sqrt[2*(-1 + Sqrt[2])]] + Log[1 + Sqrt[2] + x - Sqrt[2*(1 + Sqrt[2])]*Sqr
t[1 + x]]/(2*Sqrt[2*(1 + Sqrt[2])]) - Log[1 + Sqrt[2] + x + Sqrt[2*(1 + Sqrt[2])
]*Sqrt[1 + x]]/(2*Sqrt[2*(1 + Sqrt[2])])

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Rubi in Sympy [A]  time = 41.9041, size = 206, normalized size = 1. \[ \frac{\sqrt{2} \log{\left (x - \sqrt{2} \sqrt{1 + \sqrt{2}} \sqrt{x + 1} + 1 + \sqrt{2} \right )}}{4 \sqrt{1 + \sqrt{2}}} - \frac{\sqrt{2} \log{\left (x + \sqrt{2} \sqrt{1 + \sqrt{2}} \sqrt{x + 1} + 1 + \sqrt{2} \right )}}{4 \sqrt{1 + \sqrt{2}}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt{x + 1} - \frac{\sqrt{2 + 2 \sqrt{2}}}{2}\right )}{\sqrt{-1 + \sqrt{2}}} \right )}}{2 \sqrt{-1 + \sqrt{2}}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt{x + 1} + \frac{\sqrt{2 + 2 \sqrt{2}}}{2}\right )}{\sqrt{-1 + \sqrt{2}}} \right )}}{2 \sqrt{-1 + \sqrt{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*log(x - sqrt(2)*sqrt(1 + sqrt(2))*sqrt(x + 1) + 1 + sqrt(2))/(4*sqrt(1 +
sqrt(2))) - sqrt(2)*log(x + sqrt(2)*sqrt(1 + sqrt(2))*sqrt(x + 1) + 1 + sqrt(2)
)/(4*sqrt(1 + sqrt(2))) + sqrt(2)*atan(sqrt(2)*(sqrt(x + 1) - sqrt(2 + 2*sqrt(2)
)/2)/sqrt(-1 + sqrt(2)))/(2*sqrt(-1 + sqrt(2))) + sqrt(2)*atan(sqrt(2)*(sqrt(x +
1) + sqrt(2 + 2*sqrt(2))/2)/sqrt(-1 + sqrt(2)))/(2*sqrt(-1 + sqrt(2)))

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Mathematica [C]  time = 0.0546473, size = 51, normalized size = 0.25 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{-1-i}}\right )}{(-1-i)^{3/2}}-\frac{2 \tan ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{-1+i}}\right )}{(-1+i)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[1 + x]/(1 + x^2),x]

[Out]

(-2*ArcTan[Sqrt[1 + x]/Sqrt[-1 - I]])/(-1 - I)^(3/2) - (2*ArcTan[Sqrt[1 + x]/Sqr
t[-1 + I]])/(-1 + I)^(3/2)

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Maple [B]  time = 0.116, size = 336, normalized size = 1.6 \[{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}}{4}\ln \left ( 1+x+\sqrt{2}-\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{\sqrt{2} \left ( 2+2\,\sqrt{2} \right ) }{2\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{\sqrt{2+2\,\sqrt{2}}}{4}\ln \left ( 1+x+\sqrt{2}-\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }-{\frac{2+2\,\sqrt{2}}{2\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}}{4}\ln \left ( 1+x+\sqrt{2}+\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{\sqrt{2} \left ( 2+2\,\sqrt{2} \right ) }{2\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }+{\frac{\sqrt{2+2\,\sqrt{2}}}{4}\ln \left ( 1+x+\sqrt{2}+\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }-{\frac{2+2\,\sqrt{2}}{2\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^(1/2)/(x^2+1),x)

[Out]

1/4*(2+2*2^(1/2))^(1/2)*2^(1/2)*ln(1+x+2^(1/2)-(1+x)^(1/2)*(2+2*2^(1/2))^(1/2))+
1/2*2^(1/2)*(2+2*2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan((2*(1+x)^(1/2)-(2+2*2^(1/2
))^(1/2))/(-2+2*2^(1/2))^(1/2))-1/4*(2+2*2^(1/2))^(1/2)*ln(1+x+2^(1/2)-(1+x)^(1/
2)*(2+2*2^(1/2))^(1/2))-1/2*(2+2*2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan((2*(1+x)^(
1/2)-(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))-1/4*(2+2*2^(1/2))^(1/2)*2^(1/2)*
ln(1+x+2^(1/2)+(1+x)^(1/2)*(2+2*2^(1/2))^(1/2))+1/2*2^(1/2)*(2+2*2^(1/2))/(-2+2*
2^(1/2))^(1/2)*arctan((2*(1+x)^(1/2)+(2+2*2^(1/2))^(1/2))/(-2+2*2^(1/2))^(1/2))+
1/4*(2+2*2^(1/2))^(1/2)*ln(1+x+2^(1/2)+(1+x)^(1/2)*(2+2*2^(1/2))^(1/2))-1/2*(2+2
*2^(1/2))/(-2+2*2^(1/2))^(1/2)*arctan((2*(1+x)^(1/2)+(2+2*2^(1/2))^(1/2))/(-2+2*
2^(1/2))^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x + 1}}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x + 1)/(x^2 + 1), x)

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Fricas [A]  time = 0.233305, size = 667, normalized size = 3.25 \[ \frac{\sqrt{2}{\left (2^{\frac{1}{4}}{\left (\sqrt{2} - 1\right )} \log \left (\frac{2^{\frac{3}{4}} \sqrt{x + 1}{\left (17 \, \sqrt{2} - 24\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + 24 \, \sqrt{2}{\left (x + 1\right )} + 2 \, \sqrt{2}{\left (12 \, \sqrt{2} - 17\right )} - 34 \, x - 34}{2 \,{\left (12 \, \sqrt{2} - 17\right )}}\right ) - 2^{\frac{1}{4}}{\left (\sqrt{2} - 1\right )} \log \left (-\frac{2^{\frac{3}{4}} \sqrt{x + 1}{\left (17 \, \sqrt{2} - 24\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 24 \, \sqrt{2}{\left (x + 1\right )} - 2 \, \sqrt{2}{\left (12 \, \sqrt{2} - 17\right )} + 34 \, x + 34}{2 \,{\left (12 \, \sqrt{2} - 17\right )}}\right ) - 4 \cdot 2^{\frac{1}{4}} \arctan \left (\frac{2^{\frac{1}{4}}{\left (\sqrt{2} - 1\right )}}{\sqrt{2} \sqrt{\frac{1}{2}}{\left (\sqrt{2} - 1\right )} \sqrt{\frac{2^{\frac{3}{4}} \sqrt{x + 1}{\left (17 \, \sqrt{2} - 24\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + 24 \, \sqrt{2}{\left (x + 1\right )} + 2 \, \sqrt{2}{\left (12 \, \sqrt{2} - 17\right )} - 34 \, x - 34}{12 \, \sqrt{2} - 17}} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + \sqrt{2} \sqrt{x + 1}{\left (\sqrt{2} - 1\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 2^{\frac{1}{4}}}\right ) - 4 \cdot 2^{\frac{1}{4}} \arctan \left (\frac{2^{\frac{1}{4}}{\left (\sqrt{2} - 1\right )}}{\sqrt{2} \sqrt{\frac{1}{2}}{\left (\sqrt{2} - 1\right )} \sqrt{-\frac{2^{\frac{3}{4}} \sqrt{x + 1}{\left (17 \, \sqrt{2} - 24\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 24 \, \sqrt{2}{\left (x + 1\right )} - 2 \, \sqrt{2}{\left (12 \, \sqrt{2} - 17\right )} + 34 \, x + 34}{12 \, \sqrt{2} - 17}} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + \sqrt{2} \sqrt{x + 1}{\left (\sqrt{2} - 1\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + 2^{\frac{1}{4}}}\right )\right )}}{4 \,{\left (\sqrt{2} - 1\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*(2^(1/4)*(sqrt(2) - 1)*log(1/2*(2^(3/4)*sqrt(x + 1)*(17*sqrt(2) - 24
)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 24*sqrt(2)*(x + 1) + 2*sqrt(2)*(12*sqrt(
2) - 17) - 34*x - 34)/(12*sqrt(2) - 17)) - 2^(1/4)*(sqrt(2) - 1)*log(-1/2*(2^(3/
4)*sqrt(x + 1)*(17*sqrt(2) - 24)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 24*sqrt(2
)*(x + 1) - 2*sqrt(2)*(12*sqrt(2) - 17) + 34*x + 34)/(12*sqrt(2) - 17)) - 4*2^(1
/4)*arctan(2^(1/4)*(sqrt(2) - 1)/(sqrt(2)*sqrt(1/2)*(sqrt(2) - 1)*sqrt((2^(3/4)*
sqrt(x + 1)*(17*sqrt(2) - 24)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 24*sqrt(2)*(
x + 1) + 2*sqrt(2)*(12*sqrt(2) - 17) - 34*x - 34)/(12*sqrt(2) - 17))*sqrt((sqrt(
2) - 2)/(2*sqrt(2) - 3)) + sqrt(2)*sqrt(x + 1)*(sqrt(2) - 1)*sqrt((sqrt(2) - 2)/
(2*sqrt(2) - 3)) - 2^(1/4))) - 4*2^(1/4)*arctan(2^(1/4)*(sqrt(2) - 1)/(sqrt(2)*s
qrt(1/2)*(sqrt(2) - 1)*sqrt(-(2^(3/4)*sqrt(x + 1)*(17*sqrt(2) - 24)*sqrt((sqrt(2
) - 2)/(2*sqrt(2) - 3)) - 24*sqrt(2)*(x + 1) - 2*sqrt(2)*(12*sqrt(2) - 17) + 34*
x + 34)/(12*sqrt(2) - 17))*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + sqrt(2)*sqrt(x
+ 1)*(sqrt(2) - 1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 2^(1/4))))/((sqrt(2) -
1)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)))

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Sympy [A]  time = 8.73062, size = 31, normalized size = 0.15 \[ 2 \operatorname{RootSum}{\left (128 t^{4} + 16 t^{2} + 1, \left ( t \mapsto t \log{\left (64 t^{3} + 4 t + \sqrt{x + 1} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*RootSum(128*_t**4 + 16*_t**2 + 1, Lambda(_t, _t*log(64*_t**3 + 4*_t + sqrt(x +
1))))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x + 1}}{x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(x + 1)/(x^2 + 1), x)