3.644 \(\int \frac{1}{\sqrt{1+x} \left (1+x^2\right )} \, dx\)

Optimal. Leaf size=198 \[ -\frac{\log \left (x-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )}{4 \sqrt{1+\sqrt{2}}}+\frac{\log \left (x+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )}{4 \sqrt{1+\sqrt{2}}}-\frac{1}{2} \sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{x+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )+\frac{1}{2} \sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{2 \sqrt{x+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right ) \]

[Out]

-(Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + x])/Sqrt[2*(-1 +
Sqrt[2])]])/2 + (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + x]
)/Sqrt[2*(-1 + Sqrt[2])]])/2 - Log[1 + Sqrt[2] + x - Sqrt[2*(1 + Sqrt[2])]*Sqrt[
1 + x]]/(4*Sqrt[1 + Sqrt[2]]) + Log[1 + Sqrt[2] + x + Sqrt[2*(1 + Sqrt[2])]*Sqrt
[1 + x]]/(4*Sqrt[1 + Sqrt[2]])

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Rubi [A]  time = 0.39918, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{\log \left (x-\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )}{4 \sqrt{1+\sqrt{2}}}+\frac{\log \left (x+\sqrt{2 \left (1+\sqrt{2}\right )} \sqrt{x+1}+\sqrt{2}+1\right )}{4 \sqrt{1+\sqrt{2}}}-\frac{1}{2} \sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{\sqrt{2 \left (1+\sqrt{2}\right )}-2 \sqrt{x+1}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right )+\frac{1}{2} \sqrt{1+\sqrt{2}} \tan ^{-1}\left (\frac{2 \sqrt{x+1}+\sqrt{2 \left (1+\sqrt{2}\right )}}{\sqrt{2 \left (\sqrt{2}-1\right )}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] - 2*Sqrt[1 + x])/Sqrt[2*(-1 +
Sqrt[2])]])/2 + (Sqrt[1 + Sqrt[2]]*ArcTan[(Sqrt[2*(1 + Sqrt[2])] + 2*Sqrt[1 + x]
)/Sqrt[2*(-1 + Sqrt[2])]])/2 - Log[1 + Sqrt[2] + x - Sqrt[2*(1 + Sqrt[2])]*Sqrt[
1 + x]]/(4*Sqrt[1 + Sqrt[2]]) + Log[1 + Sqrt[2] + x + Sqrt[2*(1 + Sqrt[2])]*Sqrt
[1 + x]]/(4*Sqrt[1 + Sqrt[2]])

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Rubi in Sympy [A]  time = 34.3727, size = 185, normalized size = 0.93 \[ - \frac{\log{\left (x - \sqrt{2} \sqrt{1 + \sqrt{2}} \sqrt{x + 1} + 1 + \sqrt{2} \right )}}{4 \sqrt{1 + \sqrt{2}}} + \frac{\log{\left (x + \sqrt{2} \sqrt{1 + \sqrt{2}} \sqrt{x + 1} + 1 + \sqrt{2} \right )}}{4 \sqrt{1 + \sqrt{2}}} + \frac{\operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt{x + 1} - \frac{\sqrt{2 + 2 \sqrt{2}}}{2}\right )}{\sqrt{-1 + \sqrt{2}}} \right )}}{2 \sqrt{-1 + \sqrt{2}}} + \frac{\operatorname{atan}{\left (\frac{\sqrt{2} \left (\sqrt{x + 1} + \frac{\sqrt{2 + 2 \sqrt{2}}}{2}\right )}{\sqrt{-1 + \sqrt{2}}} \right )}}{2 \sqrt{-1 + \sqrt{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x - sqrt(2)*sqrt(1 + sqrt(2))*sqrt(x + 1) + 1 + sqrt(2))/(4*sqrt(1 + sqrt(2
))) + log(x + sqrt(2)*sqrt(1 + sqrt(2))*sqrt(x + 1) + 1 + sqrt(2))/(4*sqrt(1 + s
qrt(2))) + atan(sqrt(2)*(sqrt(x + 1) - sqrt(2 + 2*sqrt(2))/2)/sqrt(-1 + sqrt(2))
)/(2*sqrt(-1 + sqrt(2))) + atan(sqrt(2)*(sqrt(x + 1) + sqrt(2 + 2*sqrt(2))/2)/sq
rt(-1 + sqrt(2)))/(2*sqrt(-1 + sqrt(2)))

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Mathematica [C]  time = 0.0583569, size = 55, normalized size = 0.28 \[ \frac{i \tan ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{-1+i}}\right )}{\sqrt{-1+i}}-\frac{i \tan ^{-1}\left (\frac{\sqrt{x+1}}{\sqrt{-1-i}}\right )}{\sqrt{-1-i}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.039, size = 420, normalized size = 2.1 \[ -{\frac{\sqrt{2+2\,\sqrt{2}}}{4}\ln \left ( 1+x+\sqrt{2}-\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}}{8}\ln \left ( 1+x+\sqrt{2}-\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{ \left ( 2+2\,\sqrt{2} \right ) \sqrt{2}}{4\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{2+2\,\sqrt{2}}{2\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }+{\frac{\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}-\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }+{\frac{\sqrt{2+2\,\sqrt{2}}}{4}\ln \left ( 1+x+\sqrt{2}+\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }-{\frac{\sqrt{2+2\,\sqrt{2}}\sqrt{2}}{8}\ln \left ( 1+x+\sqrt{2}+\sqrt{1+x}\sqrt{2+2\,\sqrt{2}} \right ) }+{\frac{ \left ( 2+2\,\sqrt{2} \right ) \sqrt{2}}{4\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }-{\frac{2+2\,\sqrt{2}}{2\,\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) }+{\frac{\sqrt{2}}{\sqrt{-2+2\,\sqrt{2}}}\arctan \left ({\frac{1}{\sqrt{-2+2\,\sqrt{2}}} \left ( 2\,\sqrt{1+x}+\sqrt{2+2\,\sqrt{2}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2+1)/(1+x)^(1/2),x)

[Out]

-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))+1/8*(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))+1/4*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))-1/2*(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/(-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))*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))-1/8*(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))+1/4*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))-1/2*(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/(-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))*2^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((x^2 + 1)*sqrt(x + 1)), x)

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Fricas [A]  time = 0.238213, size = 653, normalized size = 3.3 \[ \frac{\sqrt{2}{\left (2^{\frac{1}{4}}{\left (\sqrt{2} - 1\right )} \log \left (\frac{2 \,{\left (2^{\frac{3}{4}} \sqrt{x + 1}{\left (5 \, \sqrt{2} - 7\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + 7 \, \sqrt{2}{\left (x + 1\right )} + \sqrt{2}{\left (7 \, \sqrt{2} - 10\right )} - 10 \, x - 10\right )}}{7 \, \sqrt{2} - 10}\right ) - 2^{\frac{1}{4}}{\left (\sqrt{2} - 1\right )} \log \left (-\frac{2 \,{\left (2^{\frac{3}{4}} \sqrt{x + 1}{\left (5 \, \sqrt{2} - 7\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 7 \, \sqrt{2}{\left (x + 1\right )} - \sqrt{2}{\left (7 \, \sqrt{2} - 10\right )} + 10 \, x + 10\right )}}{7 \, \sqrt{2} - 10}\right ) - 4 \cdot 2^{\frac{1}{4}} \arctan \left (-\frac{2^{\frac{1}{4}}{\left (\sqrt{2} - 2\right )}}{\sqrt{2} \sqrt{x + 1}{\left (\sqrt{2} - 2\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 2 \,{\left (\sqrt{2} - 1\right )} \sqrt{\frac{2^{\frac{3}{4}} \sqrt{x + 1}{\left (5 \, \sqrt{2} - 7\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + 7 \, \sqrt{2}{\left (x + 1\right )} + \sqrt{2}{\left (7 \, \sqrt{2} - 10\right )} - 10 \, x - 10}{7 \, \sqrt{2} - 10}} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} + 2^{\frac{3}{4}}}\right ) - 4 \cdot 2^{\frac{1}{4}} \arctan \left (-\frac{2^{\frac{1}{4}}{\left (\sqrt{2} - 2\right )}}{\sqrt{2} \sqrt{x + 1}{\left (\sqrt{2} - 2\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 2 \,{\left (\sqrt{2} - 1\right )} \sqrt{-\frac{2^{\frac{3}{4}} \sqrt{x + 1}{\left (5 \, \sqrt{2} - 7\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 7 \, \sqrt{2}{\left (x + 1\right )} - \sqrt{2}{\left (7 \, \sqrt{2} - 10\right )} + 10 \, x + 10}{7 \, \sqrt{2} - 10}} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}} - 2^{\frac{3}{4}}}\right )\right )}}{4 \,{\left (\sqrt{2} - 2\right )} \sqrt{\frac{\sqrt{2} - 2}{2 \, \sqrt{2} - 3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(2)*(2^(1/4)*(sqrt(2) - 1)*log(2*(2^(3/4)*sqrt(x + 1)*(5*sqrt(2) - 7)*sq
rt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 7*sqrt(2)*(x + 1) + sqrt(2)*(7*sqrt(2) - 10)
- 10*x - 10)/(7*sqrt(2) - 10)) - 2^(1/4)*(sqrt(2) - 1)*log(-2*(2^(3/4)*sqrt(x +
1)*(5*sqrt(2) - 7)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 7*sqrt(2)*(x + 1) - sq
rt(2)*(7*sqrt(2) - 10) + 10*x + 10)/(7*sqrt(2) - 10)) - 4*2^(1/4)*arctan(-2^(1/4
)*(sqrt(2) - 2)/(sqrt(2)*sqrt(x + 1)*(sqrt(2) - 2)*sqrt((sqrt(2) - 2)/(2*sqrt(2)
- 3)) - 2*(sqrt(2) - 1)*sqrt((2^(3/4)*sqrt(x + 1)*(5*sqrt(2) - 7)*sqrt((sqrt(2)
- 2)/(2*sqrt(2) - 3)) + 7*sqrt(2)*(x + 1) + sqrt(2)*(7*sqrt(2) - 10) - 10*x - 1
0)/(7*sqrt(2) - 10))*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 2^(3/4))) - 4*2^(1/4)
*arctan(-2^(1/4)*(sqrt(2) - 2)/(sqrt(2)*sqrt(x + 1)*(sqrt(2) - 2)*sqrt((sqrt(2)
- 2)/(2*sqrt(2) - 3)) - 2*(sqrt(2) - 1)*sqrt(-(2^(3/4)*sqrt(x + 1)*(5*sqrt(2) -
7)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 7*sqrt(2)*(x + 1) - sqrt(2)*(7*sqrt(2)
- 10) + 10*x + 10)/(7*sqrt(2) - 10))*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 2^(3/
4))))/((sqrt(2) - 2)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x + 1} \left (x^{2} + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(x + 1)*(x**2 + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((x^2 + 1)*sqrt(x + 1)), x)