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

Optimal. Leaf size=65 \[ -\frac{1}{2} (1-i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1-i} \sqrt{x}}{\sqrt{x+1}}\right )-\frac{1}{2} (1+i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1+i} \sqrt{x}}{\sqrt{x+1}}\right ) \]

[Out]

-((1 - I)^(3/2)*ArcTanh[(Sqrt[1 - I]*Sqrt[x])/Sqrt[1 + x]])/2 - ((1 + I)^(3/2)*A
rcTanh[(Sqrt[1 + I]*Sqrt[x])/Sqrt[1 + x]])/2

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Rubi [A]  time = 0.126082, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{1}{2} (1-i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1-i} \sqrt{x}}{\sqrt{x+1}}\right )-\frac{1}{2} (1+i)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{1+i} \sqrt{x}}{\sqrt{x+1}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-((1 - I)^(3/2)*ArcTanh[(Sqrt[1 - I]*Sqrt[x])/Sqrt[1 + x]])/2 - ((1 + I)^(3/2)*A
rcTanh[(Sqrt[1 + I]*Sqrt[x])/Sqrt[1 + x]])/2

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Rubi in Sympy [A]  time = 19.2993, size = 121, normalized size = 1.86 \[ - \frac{i \left (\sqrt{1 + \sqrt{2}} - \sqrt{- \sqrt{2} + 1}\right ) \operatorname{atanh}{\left (\frac{2 \sqrt{x}}{\sqrt{x + 1} \left (\sqrt{1 + \sqrt{2}} - i \sqrt{-1 + \sqrt{2}}\right )} \right )}}{2} + \frac{i \left (\sqrt{1 + \sqrt{2}} + \sqrt{- \sqrt{2} + 1}\right ) \operatorname{atanh}{\left (\frac{2 \sqrt{x}}{\sqrt{x + 1} \left (\sqrt{1 + \sqrt{2}} + i \sqrt{-1 + \sqrt{2}}\right )} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-I*(sqrt(1 + sqrt(2)) - sqrt(-sqrt(2) + 1))*atanh(2*sqrt(x)/(sqrt(x + 1)*(sqrt(1
+ sqrt(2)) - I*sqrt(-1 + sqrt(2)))))/2 + I*(sqrt(1 + sqrt(2)) + sqrt(-sqrt(2) +
1))*atanh(2*sqrt(x)/(sqrt(x + 1)*(sqrt(1 + sqrt(2)) + I*sqrt(-1 + sqrt(2)))))/2

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Mathematica [A]  time = 0.126005, size = 65, normalized size = 1. \[ \frac{1}{2} \left (\sqrt{2-2 i} \tan ^{-1}\left ((1-i)^{3/2} \sqrt{\frac{x}{2 x+2}}\right )+\sqrt{2+2 i} \tan ^{-1}\left ((1+i)^{3/2} \sqrt{\frac{x}{2 x+2}}\right )\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.19, size = 305, normalized size = 4.7 \[{\frac{ \left ( \sqrt{2}-1+x \right ) \sqrt{2}}{ \left ( -16+12\,\sqrt{2} \right ) \sqrt{1+\sqrt{2}}}\sqrt{{\frac{x \left ( 1+x \right ) }{ \left ( \sqrt{2}-1+x \right ) ^{2}}}} \left ( \sqrt{-2+2\,\sqrt{2}}\arctan \left ({\frac{ \left ( \sqrt{2}+1-x \right ) \left ( -4+3\,\sqrt{2} \right ) \left ( 3+2\,\sqrt{2} \right ) \sqrt{-2+2\,\sqrt{2}} \left ( \sqrt{2}-1+x \right ) }{4\,x \left ( 1+x \right ) }\sqrt{{\frac{x \left ( 1+x \right ) \left ( -4+3\,\sqrt{2} \right ) \left ( 3\,\sqrt{2}+4 \right ) }{ \left ( \sqrt{2}-1+x \right ) ^{2}}}}} \right ) \sqrt{1+\sqrt{2}}\sqrt{2}-2\,\sqrt{-2+2\,\sqrt{2}}\arctan \left ( 1/4\,{\frac{ \left ( \sqrt{2}+1-x \right ) \left ( -4+3\,\sqrt{2} \right ) \left ( 3+2\,\sqrt{2} \right ) \sqrt{-2+2\,\sqrt{2}} \left ( \sqrt{2}-1+x \right ) }{x \left ( 1+x \right ) }\sqrt{{\frac{x \left ( 1+x \right ) \left ( -4+3\,\sqrt{2} \right ) \left ( 3\,\sqrt{2}+4 \right ) }{ \left ( \sqrt{2}-1+x \right ) ^{2}}}}} \right ) \sqrt{1+\sqrt{2}}+4\,{\it Artanh} \left ({\frac{\sqrt{2}}{\sqrt{1+\sqrt{2}}}\sqrt{{\frac{x \left ( 1+x \right ) }{ \left ( \sqrt{2}-1+x \right ) ^{2}}}}} \right ) \sqrt{2}-6\,{\it Artanh} \left ({\frac{\sqrt{2}}{\sqrt{1+\sqrt{2}}}\sqrt{{\frac{x \left ( 1+x \right ) }{ \left ( \sqrt{2}-1+x \right ) ^{2}}}}} \right ) \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{1+x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4/x^(1/2)/(1+x)^(1/2)*(x*(1+x)/(2^(1/2)-1+x)^2)^(1/2)*(2^(1/2)-1+x)*((-2+2*2^(
1/2))^(1/2)*arctan(1/4*(2^(1/2)+1-x)*(-4+3*2^(1/2))*(3+2*2^(1/2))*(-2+2*2^(1/2))
^(1/2)*(x*(1+x)*(-4+3*2^(1/2))*(3*2^(1/2)+4)/(2^(1/2)-1+x)^2)^(1/2)*(2^(1/2)-1+x
)/x/(1+x))*(1+2^(1/2))^(1/2)*2^(1/2)-2*(-2+2*2^(1/2))^(1/2)*arctan(1/4*(2^(1/2)+
1-x)*(-4+3*2^(1/2))*(3+2*2^(1/2))*(-2+2*2^(1/2))^(1/2)*(x*(1+x)*(-4+3*2^(1/2))*(
3*2^(1/2)+4)/(2^(1/2)-1+x)^2)^(1/2)*(2^(1/2)-1+x)/x/(1+x))*(1+2^(1/2))^(1/2)+4*a
rctanh(2^(1/2)*(x*(1+x)/(2^(1/2)-1+x)^2)^(1/2)/(1+2^(1/2))^(1/2))*2^(1/2)-6*arct
anh(2^(1/2)*(x*(1+x)/(2^(1/2)-1+x)^2)^(1/2)/(1+2^(1/2))^(1/2)))*2^(1/2)/(-4+3*2^
(1/2))/(1+2^(1/2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.32074, size = 1146, normalized size = 17.63 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/((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(2)*(12*x + 5) - 17*x -
7)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - (2^(3/4)*(12*sqrt(2) - 17)*sqrt((sqrt(2
) - 2)/(2*sqrt(2) - 3)) + 14*sqrt(2)*x - 20*x)*sqrt(x + 1)*sqrt(x) - 20*x^2 + 7*
sqrt(2)*(2*x^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(2)*(12*x + 5) - 17*x - 7)*sqrt((
sqrt(2) - 2)/(2*sqrt(2) - 3)) - (2^(3/4)*(12*sqrt(2) - 17)*sqrt((sqrt(2) - 2)/(2
*sqrt(2) - 3)) - 14*sqrt(2)*x + 20*x)*sqrt(x + 1)*sqrt(x) + 20*x^2 - 7*sqrt(2)*(
2*x^2 + x + 1) - sqrt(2)*(7*sqrt(2) - 10) + 10*x + 10)/(7*sqrt(2) - 10)) - 4*2^(
1/4)*arctan(-(sqrt(2)*(sqrt(2) - 2)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 2^(3/4
))/(sqrt(2)*sqrt(x + 1)*sqrt(x)*(sqrt(2) - 2)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)
) - sqrt(2)*(sqrt(2)*x - 2*x)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - 2*(sqrt(2) -
1)*sqrt((2^(3/4)*(sqrt(2)*(12*x + 5) - 17*x - 7)*sqrt((sqrt(2) - 2)/(2*sqrt(2)
- 3)) - (2^(3/4)*(12*sqrt(2) - 17)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) + 14*sqrt
(2)*x - 20*x)*sqrt(x + 1)*sqrt(x) - 20*x^2 + 7*sqrt(2)*(2*x^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^(1/4)*(sqrt(2) - 2))) + 4*2^(1/4)*arctan(-(sqrt(2)*(sqrt(2) - 2)*sqrt((
sqrt(2) - 2)/(2*sqrt(2) - 3)) - 2^(3/4))/(sqrt(2)*sqrt(x + 1)*sqrt(x)*(sqrt(2) -
2)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - sqrt(2)*(sqrt(2)*x - 2*x)*sqrt((sqrt(2
) - 2)/(2*sqrt(2) - 3)) - 2*(sqrt(2) - 1)*sqrt(-(2^(3/4)*(sqrt(2)*(12*x + 5) - 1
7*x - 7)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)) - (2^(3/4)*(12*sqrt(2) - 17)*sqrt((
sqrt(2) - 2)/(2*sqrt(2) - 3)) - 14*sqrt(2)*x + 20*x)*sqrt(x + 1)*sqrt(x) + 20*x^
2 - 7*sqrt(2)*(2*x^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^(1/4)*(sqrt(2) - 2))))/((sqrt(2)
- 2)*sqrt((sqrt(2) - 2)/(2*sqrt(2) - 3)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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