3.279 \(\int \frac{1}{\sqrt{4+2 x+x^2} \left (-x+x^3\right )} \, dx\)

Optimal. Leaf size=86 \[ \frac{1}{2} \tanh ^{-1}\left (\frac{x+4}{2 \sqrt{x^2+2 x+4}}\right )-\frac{\tanh ^{-1}\left (\frac{2 x+5}{\sqrt{7} \sqrt{x^2+2 x+4}}\right )}{2 \sqrt{7}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^2+2 x+4}}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

[Out]

ArcTanh[(4 + x)/(2*Sqrt[4 + 2*x + x^2])]/2 - ArcTanh[(5 + 2*x)/(Sqrt[7]*Sqrt[4 +
2*x + x^2])]/(2*Sqrt[7]) - ArcTanh[Sqrt[4 + 2*x + x^2]/Sqrt[3]]/(2*Sqrt[3])

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Rubi [A]  time = 0.493948, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{1}{2} \tanh ^{-1}\left (\frac{x+4}{2 \sqrt{x^2+2 x+4}}\right )-\frac{\tanh ^{-1}\left (\frac{2 x+5}{\sqrt{7} \sqrt{x^2+2 x+4}}\right )}{2 \sqrt{7}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^2+2 x+4}}{\sqrt{3}}\right )}{2 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[4 + 2*x + x^2]*(-x + x^3)),x]

[Out]

ArcTanh[(4 + x)/(2*Sqrt[4 + 2*x + x^2])]/2 - ArcTanh[(5 + 2*x)/(Sqrt[7]*Sqrt[4 +
2*x + x^2])]/(2*Sqrt[7]) - ArcTanh[Sqrt[4 + 2*x + x^2]/Sqrt[3]]/(2*Sqrt[3])

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Rubi in Sympy [A]  time = 21.054, size = 80, normalized size = 0.93 \[ - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{\sqrt{3} \sqrt{x^{2} + 2 x + 4}}{3} \right )}}{6} + \frac{\operatorname{atanh}{\left (\frac{2 x + 8}{4 \sqrt{x^{2} + 2 x + 4}} \right )}}{2} - \frac{\sqrt{7} \operatorname{atanh}{\left (\frac{\sqrt{7} \left (4 x + 10\right )}{14 \sqrt{x^{2} + 2 x + 4}} \right )}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**3-x)/(x**2+2*x+4)**(1/2),x)

[Out]

-sqrt(3)*atanh(sqrt(3)*sqrt(x**2 + 2*x + 4)/3)/6 + atanh((2*x + 8)/(4*sqrt(x**2
+ 2*x + 4)))/2 - sqrt(7)*atanh(sqrt(7)*(4*x + 10)/(14*sqrt(x**2 + 2*x + 4)))/14

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Mathematica [A]  time = 0.0635976, size = 112, normalized size = 1.3 \[ \frac{1}{42} \left (21 \log \left (2 \sqrt{x^2+2 x+4}+x+4\right )-7 \sqrt{3} \log \left (\sqrt{3} \sqrt{x^2+2 x+4}+3\right )-3 \sqrt{7} \log \left (\sqrt{7} \sqrt{x^2+2 x+4}+2 x+5\right )+3 \sqrt{7} \log (1-x)-21 \log (x)+7 \sqrt{3} \log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(Sqrt[4 + 2*x + x^2]*(-x + x^3)),x]

[Out]

(3*Sqrt[7]*Log[1 - x] - 21*Log[x] + 7*Sqrt[3]*Log[1 + x] + 21*Log[4 + x + 2*Sqrt
[4 + 2*x + x^2]] - 7*Sqrt[3]*Log[3 + Sqrt[3]*Sqrt[4 + 2*x + x^2]] - 3*Sqrt[7]*Lo
g[5 + 2*x + Sqrt[7]*Sqrt[4 + 2*x + x^2]])/42

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Maple [A]  time = 0.017, size = 69, normalized size = 0.8 \[{\frac{1}{2}{\it Artanh} \left ({\frac{8+2\,x}{4}{\frac{1}{\sqrt{{x}^{2}+2\,x+4}}}} \right ) }-{\frac{\sqrt{7}}{14}{\it Artanh} \left ({\frac{ \left ( 10+4\,x \right ) \sqrt{7}}{14}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}+3+4\,x}}}} \right ) }-{\frac{\sqrt{3}}{6}{\it Artanh} \left ({\sqrt{3}{\frac{1}{\sqrt{ \left ( 1+x \right ) ^{2}+3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^3-x)/(x^2+2*x+4)^(1/2),x)

[Out]

1/2*arctanh(1/4*(8+2*x)/(x^2+2*x+4)^(1/2))-1/14*7^(1/2)*arctanh(1/14*(10+4*x)*7^
(1/2)/((-1+x)^2+3+4*x)^(1/2))-1/6*3^(1/2)*arctanh(3^(1/2)/((1+x)^2+3)^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 - x)*sqrt(x^2 + 2*x + 4)),x, algorithm="maxima")

[Out]

integrate(1/((x^3 - x)*sqrt(x^2 + 2*x + 4)), x)

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Fricas [A]  time = 0.232557, size = 254, normalized size = 2.95 \[ \frac{1}{42} \, \sqrt{7} \sqrt{3}{\left (\sqrt{7} \sqrt{3} \log \left (-x + \sqrt{x^{2} + 2 \, x + 4} + 2\right ) - \sqrt{7} \sqrt{3} \log \left (-x + \sqrt{x^{2} + 2 \, x + 4} - 2\right ) + \sqrt{7} \log \left (\frac{\sqrt{3}{\left (x^{2} + 2 \, x + 4\right )} - \sqrt{x^{2} + 2 \, x + 4}{\left (\sqrt{3}{\left (x + 1\right )} + 3\right )} + 3 \, x + 3}{x^{2} - \sqrt{x^{2} + 2 \, x + 4}{\left (x + 1\right )} + 2 \, x + 1}\right ) + \sqrt{3} \log \left (\frac{\sqrt{7}{\left (x^{2} + 6\right )} - \sqrt{x^{2} + 2 \, x + 4}{\left (\sqrt{7}{\left (x - 1\right )} + 7\right )} + 7 \, x - 7}{x^{2} - \sqrt{x^{2} + 2 \, x + 4}{\left (x - 1\right )} - 1}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 - x)*sqrt(x^2 + 2*x + 4)),x, algorithm="fricas")

[Out]

1/42*sqrt(7)*sqrt(3)*(sqrt(7)*sqrt(3)*log(-x + sqrt(x^2 + 2*x + 4) + 2) - sqrt(7
)*sqrt(3)*log(-x + sqrt(x^2 + 2*x + 4) - 2) + sqrt(7)*log((sqrt(3)*(x^2 + 2*x +
4) - sqrt(x^2 + 2*x + 4)*(sqrt(3)*(x + 1) + 3) + 3*x + 3)/(x^2 - sqrt(x^2 + 2*x
+ 4)*(x + 1) + 2*x + 1)) + sqrt(3)*log((sqrt(7)*(x^2 + 6) - sqrt(x^2 + 2*x + 4)*
(sqrt(7)*(x - 1) + 7) + 7*x - 7)/(x^2 - sqrt(x^2 + 2*x + 4)*(x - 1) - 1)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (x - 1\right ) \left (x + 1\right ) \sqrt{x^{2} + 2 x + 4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**3-x)/(x**2+2*x+4)**(1/2),x)

[Out]

Integral(1/(x*(x - 1)*(x + 1)*sqrt(x**2 + 2*x + 4)), x)

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GIAC/XCAS [A]  time = 0.247236, size = 198, normalized size = 2.3 \[ \frac{1}{14} \, \sqrt{7}{\rm ln}\left (\frac{{\left | -2 \, x - 2 \, \sqrt{7} + 2 \, \sqrt{x^{2} + 2 \, x + 4} + 2 \right |}}{{\left | -2 \, x + 2 \, \sqrt{7} + 2 \, \sqrt{x^{2} + 2 \, x + 4} + 2 \right |}}\right ) + \frac{1}{6} \, \sqrt{3}{\rm ln}\left (-\frac{{\left | -2 \, x - 2 \, \sqrt{3} + 2 \, \sqrt{x^{2} + 2 \, x + 4} - 2 \right |}}{2 \,{\left (x - \sqrt{3} - \sqrt{x^{2} + 2 \, x + 4} + 1\right )}}\right ) + \frac{1}{2} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 2 \, x + 4} + 2 \right |}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | -x + \sqrt{x^{2} + 2 \, x + 4} - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^3 - x)*sqrt(x^2 + 2*x + 4)),x, algorithm="giac")

[Out]

1/14*sqrt(7)*ln(abs(-2*x - 2*sqrt(7) + 2*sqrt(x^2 + 2*x + 4) + 2)/abs(-2*x + 2*s
qrt(7) + 2*sqrt(x^2 + 2*x + 4) + 2)) + 1/6*sqrt(3)*ln(-1/2*abs(-2*x - 2*sqrt(3)
+ 2*sqrt(x^2 + 2*x + 4) - 2)/(x - sqrt(3) - sqrt(x^2 + 2*x + 4) + 1)) + 1/2*ln(a
bs(-x + sqrt(x^2 + 2*x + 4) + 2)) - 1/2*ln(abs(-x + sqrt(x^2 + 2*x + 4) - 2))