3.745 \(\int \frac{1}{\left (1+\frac{2 x}{1+x^2}\right )^{3/2}} \, dx\)

Optimal. Leaf size=144 \[ \frac{3 (x+2)}{2 \sqrt{\frac{2 x}{x^2+1}+1}}-\frac{x^2+1}{2 (x+1) \sqrt{\frac{2 x}{x^2+1}+1}}-\frac{3 (x+1) \sinh ^{-1}(x)}{\sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1}}-\frac{9 (x+1) \tanh ^{-1}\left (\frac{1-x}{\sqrt{2} \sqrt{x^2+1}}\right )}{2 \sqrt{2} \sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1}} \]

[Out]

(3*(2 + x))/(2*Sqrt[1 + (2*x)/(1 + x^2)]) - (1 + x^2)/(2*(1 + x)*Sqrt[1 + (2*x)/
(1 + x^2)]) - (3*(1 + x)*ArcSinh[x])/(Sqrt[1 + x^2]*Sqrt[1 + (2*x)/(1 + x^2)]) -
(9*(1 + x)*ArcTanh[(1 - x)/(Sqrt[2]*Sqrt[1 + x^2])])/(2*Sqrt[2]*Sqrt[1 + x^2]*S
qrt[1 + (2*x)/(1 + x^2)])

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Rubi [A]  time = 0.212711, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ \frac{3 (x+2)}{2 \sqrt{\frac{2 x}{x^2+1}+1}}-\frac{x^2+1}{2 (x+1) \sqrt{\frac{2 x}{x^2+1}+1}}-\frac{3 (x+1) \sinh ^{-1}(x)}{\sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1}}-\frac{9 (x+1) \tanh ^{-1}\left (\frac{1-x}{\sqrt{2} \sqrt{x^2+1}}\right )}{2 \sqrt{2} \sqrt{x^2+1} \sqrt{\frac{2 x}{x^2+1}+1}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + (2*x)/(1 + x^2))^(-3/2),x]

[Out]

(3*(2 + x))/(2*Sqrt[1 + (2*x)/(1 + x^2)]) - (1 + x^2)/(2*(1 + x)*Sqrt[1 + (2*x)/
(1 + x^2)]) - (3*(1 + x)*ArcSinh[x])/(Sqrt[1 + x^2]*Sqrt[1 + (2*x)/(1 + x^2)]) -
(9*(1 + x)*ArcTanh[(1 - x)/(Sqrt[2]*Sqrt[1 + x^2])])/(2*Sqrt[2]*Sqrt[1 + x^2]*S
qrt[1 + (2*x)/(1 + x^2)])

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Rubi in Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RecursionError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(1+2*x/(x**2+1))**(3/2),x)

[Out]

Exception raised: RecursionError

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Mathematica [A]  time = 0.180511, size = 184, normalized size = 1.28 \[ \frac{(x+1) \left (4 \sqrt{x^2+1} x^2+18 \sqrt{x^2+1} x+10 \sqrt{x^2+1}-9 \sqrt{2} x^2 \log \left (\sqrt{2} \sqrt{x^2+1}-x+1\right )-18 \sqrt{2} x \log \left (\sqrt{2} \sqrt{x^2+1}-x+1\right )-9 \sqrt{2} \log \left (\sqrt{2} \sqrt{x^2+1}-x+1\right )+9 \sqrt{2} (x+1)^2 \log (x+1)-12 (x+1)^2 \sinh ^{-1}(x)\right )}{4 \left (\frac{(x+1)^2}{x^2+1}\right )^{3/2} \left (x^2+1\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + (2*x)/(1 + x^2))^(-3/2),x]

[Out]

((1 + x)*(10*Sqrt[1 + x^2] + 18*x*Sqrt[1 + x^2] + 4*x^2*Sqrt[1 + x^2] - 12*(1 +
x)^2*ArcSinh[x] + 9*Sqrt[2]*(1 + x)^2*Log[1 + x] - 9*Sqrt[2]*Log[1 - x + Sqrt[2]
*Sqrt[1 + x^2]] - 18*Sqrt[2]*x*Log[1 - x + Sqrt[2]*Sqrt[1 + x^2]] - 9*Sqrt[2]*x^
2*Log[1 - x + Sqrt[2]*Sqrt[1 + x^2]]))/(4*((1 + x)^2/(1 + x^2))^(3/2)*(1 + x^2)^
(3/2))

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Maple [A]  time = 0.015, size = 217, normalized size = 1.5 \[ -{\frac{1+x}{8} \left ( - \left ({x}^{2}+1 \right ) ^{{\frac{5}{2}}}x+ \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}{x}^{3}+ \left ({x}^{2}+1 \right ) ^{{\frac{5}{2}}}- \left ({x}^{2}+1 \right ) ^{{\frac{3}{2}}}{x}^{2}-18\,{\it Artanh} \left ( 1/2\,{\frac{ \left ( -1+x \right ) \sqrt{2}}{\sqrt{{x}^{2}+1}}} \right ) \sqrt{2}{x}^{2}-5\,x \left ({x}^{2}+1 \right ) ^{3/2}+6\,\sqrt{{x}^{2}+1}{x}^{3}+24\,{\it Arcsinh} \left ( x \right ){x}^{2}-36\,{\it Artanh} \left ( 1/2\,{\frac{ \left ( -1+x \right ) \sqrt{2}}{\sqrt{{x}^{2}+1}}} \right ) \sqrt{2}x-3\, \left ({x}^{2}+1 \right ) ^{3/2}-6\,\sqrt{{x}^{2}+1}{x}^{2}+48\,{\it Arcsinh} \left ( x \right ) x-18\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{ \left ( -1+x \right ) \sqrt{2}}{\sqrt{{x}^{2}+1}}} \right ) -30\,x\sqrt{{x}^{2}+1}+24\,{\it Arcsinh} \left ( x \right ) -18\,\sqrt{{x}^{2}+1} \right ) \left ({\frac{{x}^{2}+2\,x+1}{{x}^{2}+1}} \right ) ^{-{\frac{3}{2}}} \left ({x}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+2*x/(x^2+1))^(3/2),x)

[Out]

-1/8/((x^2+2*x+1)/(x^2+1))^(3/2)*(1+x)*(-(x^2+1)^(5/2)*x+(x^2+1)^(3/2)*x^3+(x^2+
1)^(5/2)-(x^2+1)^(3/2)*x^2-18*arctanh(1/2*(-1+x)*2^(1/2)/(x^2+1)^(1/2))*2^(1/2)*
x^2-5*x*(x^2+1)^(3/2)+6*(x^2+1)^(1/2)*x^3+24*arcsinh(x)*x^2-36*arctanh(1/2*(-1+x
)*2^(1/2)/(x^2+1)^(1/2))*2^(1/2)*x-3*(x^2+1)^(3/2)-6*(x^2+1)^(1/2)*x^2+48*arcsin
h(x)*x-18*2^(1/2)*arctanh(1/2*(-1+x)*2^(1/2)/(x^2+1)^(1/2))-30*x*(x^2+1)^(1/2)+2
4*arcsinh(x)-18*(x^2+1)^(1/2))/(x^2+1)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x/(x^2 + 1) + 1)^(-3/2),x, algorithm="maxima")

[Out]

integrate((2*x/(x^2 + 1) + 1)^(-3/2), x)

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Fricas [A]  time = 0.28165, size = 377, normalized size = 2.62 \[ \frac{10 \, x^{3} + 9 \, \sqrt{2}{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (\frac{4 \, x^{2} + 2 \, \sqrt{2}{\left (x^{3} + 2 \, x^{2} + 2 \, x + 1\right )} -{\left (4 \, x^{2} + \sqrt{2}{\left (2 \, x^{3} + 2 \, x^{2} + 3 \, x + 1\right )} + 2 \, x + 2\right )} \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}} + 6 \, x + 2}{2 \, x^{3} + 4 \, x^{2} -{\left (2 \, x^{3} + 2 \, x^{2} + x + 1\right )} \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}} + 2 \, x}\right ) + 30 \, x^{2} + 12 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )} \log \left (-\frac{x \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}} - x - 1}{\sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}}}\right ) + 2 \,{\left (2 \, x^{4} + 9 \, x^{3} + 7 \, x^{2} + 9 \, x + 5\right )} \sqrt{\frac{x^{2} + 2 \, x + 1}{x^{2} + 1}} + 30 \, x + 10}{4 \,{\left (x^{3} + 3 \, x^{2} + 3 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x/(x^2 + 1) + 1)^(-3/2),x, algorithm="fricas")

[Out]

1/4*(10*x^3 + 9*sqrt(2)*(x^3 + 3*x^2 + 3*x + 1)*log((4*x^2 + 2*sqrt(2)*(x^3 + 2*
x^2 + 2*x + 1) - (4*x^2 + sqrt(2)*(2*x^3 + 2*x^2 + 3*x + 1) + 2*x + 2)*sqrt((x^2
+ 2*x + 1)/(x^2 + 1)) + 6*x + 2)/(2*x^3 + 4*x^2 - (2*x^3 + 2*x^2 + x + 1)*sqrt(
(x^2 + 2*x + 1)/(x^2 + 1)) + 2*x)) + 30*x^2 + 12*(x^3 + 3*x^2 + 3*x + 1)*log(-(x
*sqrt((x^2 + 2*x + 1)/(x^2 + 1)) - x - 1)/sqrt((x^2 + 2*x + 1)/(x^2 + 1))) + 2*(
2*x^4 + 9*x^3 + 7*x^2 + 9*x + 5)*sqrt((x^2 + 2*x + 1)/(x^2 + 1)) + 30*x + 10)/(x
^3 + 3*x^2 + 3*x + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (\frac{2 x}{x^{2} + 1} + 1\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(1+2*x/(x**2+1))**(3/2),x)

[Out]

Integral((2*x/(x**2 + 1) + 1)**(-3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (\frac{2 \, x}{x^{2} + 1} + 1\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x/(x^2 + 1) + 1)^(-3/2),x, algorithm="giac")

[Out]

integrate((2*x/(x^2 + 1) + 1)^(-3/2), x)