3.280 \(\int \frac{\sqrt{4+2 x+x^2}}{(-1+x)^2} \, dx\)

Optimal. Leaf size=62 \[ \frac{\sqrt{x^2+2 x+4}}{1-x}-\frac{2 \tanh ^{-1}\left (\frac{2 x+5}{\sqrt{7} \sqrt{x^2+2 x+4}}\right )}{\sqrt{7}}+\sinh ^{-1}\left (\frac{x+1}{\sqrt{3}}\right ) \]

[Out]

Sqrt[4 + 2*x + x^2]/(1 - x) + ArcSinh[(1 + x)/Sqrt[3]] - (2*ArcTanh[(5 + 2*x)/(S
qrt[7]*Sqrt[4 + 2*x + x^2])])/Sqrt[7]

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Rubi [A]  time = 0.122442, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{\sqrt{x^2+2 x+4}}{1-x}-\frac{2 \tanh ^{-1}\left (\frac{2 x+5}{\sqrt{7} \sqrt{x^2+2 x+4}}\right )}{\sqrt{7}}+\sinh ^{-1}\left (\frac{x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[4 + 2*x + x^2]/(1 - x) + ArcSinh[(1 + x)/Sqrt[3]] - (2*ArcTanh[(5 + 2*x)/(S
qrt[7]*Sqrt[4 + 2*x + x^2])])/Sqrt[7]

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Rubi in Sympy [A]  time = 6.87574, size = 68, normalized size = 1.1 \[ \operatorname{atanh}{\left (\frac{2 x + 2}{2 \sqrt{x^{2} + 2 x + 4}} \right )} - \frac{2 \sqrt{7} \operatorname{atanh}{\left (\frac{\sqrt{7} \left (4 x + 10\right )}{14 \sqrt{x^{2} + 2 x + 4}} \right )}}{7} + \frac{\sqrt{x^{2} + 2 x + 4}}{- x + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh((2*x + 2)/(2*sqrt(x**2 + 2*x + 4))) - 2*sqrt(7)*atanh(sqrt(7)*(4*x + 10)/(
14*sqrt(x**2 + 2*x + 4)))/7 + sqrt(x**2 + 2*x + 4)/(-x + 1)

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Mathematica [A]  time = 0.0890938, size = 72, normalized size = 1.16 \[ -\frac{\sqrt{x^2+2 x+4}}{x-1}-\frac{2 \log \left (\sqrt{7} \sqrt{x^2+2 x+4}+2 x+5\right )}{\sqrt{7}}+\frac{2 \log (x-1)}{\sqrt{7}}+\sinh ^{-1}\left (\frac{x+1}{\sqrt{3}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[4 + 2*x + x^2]/(-1 + x)) + ArcSinh[(1 + x)/Sqrt[3]] + (2*Log[-1 + x])/Sqr
t[7] - (2*Log[5 + 2*x + Sqrt[7]*Sqrt[4 + 2*x + x^2]])/Sqrt[7]

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Maple [A]  time = 0.01, size = 91, normalized size = 1.5 \[ -{\frac{1}{-7+7\,x} \left ( \left ( -1+x \right ) ^{2}+3+4\,x \right ) ^{{\frac{3}{2}}}}+{\frac{2}{7}\sqrt{ \left ( -1+x \right ) ^{2}+3+4\,x}}+{\it Arcsinh} \left ({\frac{ \left ( 1+x \right ) \sqrt{3}}{3}} \right ) -{\frac{2\,\sqrt{7}}{7}{\it Artanh} \left ({\frac{ \left ( 10+4\,x \right ) \sqrt{7}}{14}{\frac{1}{\sqrt{ \left ( -1+x \right ) ^{2}+3+4\,x}}}} \right ) }+{\frac{2\,x+2}{14}\sqrt{ \left ( -1+x \right ) ^{2}+3+4\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.58115, size = 82, normalized size = 1.32 \[ -\frac{2}{7} \, \sqrt{7} \operatorname{arsinh}\left (\frac{2 \, \sqrt{3} x}{3 \,{\left | x - 1 \right |}} + \frac{5 \, \sqrt{3}}{3 \,{\left | x - 1 \right |}}\right ) - \frac{\sqrt{x^{2} + 2 \, x + 4}}{x - 1} + \operatorname{arsinh}\left (\frac{1}{3} \, \sqrt{3} x + \frac{1}{3} \, \sqrt{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/7*sqrt(7)*arcsinh(2/3*sqrt(3)*x/abs(x - 1) + 5/3*sqrt(3)/abs(x - 1)) - sqrt(x
^2 + 2*x + 4)/(x - 1) + arcsinh(1/3*sqrt(3)*x + 1/3*sqrt(3))

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Fricas [A]  time = 0.226483, size = 247, normalized size = 3.98 \[ -\frac{{\left (\sqrt{7} \sqrt{x^{2} + 2 \, x + 4}{\left (x - 1\right )} - \sqrt{7}{\left (x^{2} - 1\right )}\right )} \log \left (-x + \sqrt{x^{2} + 2 \, x + 4} - 1\right ) + 2 \,{\left (x^{2} - \sqrt{x^{2} + 2 \, x + 4}{\left (x - 1\right )} - 1\right )} \log \left (\frac{\sqrt{7}{\left (x^{2} + 6\right )} - \sqrt{x^{2} + 2 \, x + 4}{\left (\sqrt{7}{\left (x - 1\right )} + 7\right )} + 7 \, x - 7}{x^{2} - \sqrt{x^{2} + 2 \, x + 4}{\left (x - 1\right )} - 1}\right ) + \sqrt{7}{\left (2 \, x + 5\right )} - 2 \, \sqrt{7} \sqrt{x^{2} + 2 \, x + 4}}{\sqrt{7} \sqrt{x^{2} + 2 \, x + 4}{\left (x - 1\right )} - \sqrt{7}{\left (x^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-((sqrt(7)*sqrt(x^2 + 2*x + 4)*(x - 1) - sqrt(7)*(x^2 - 1))*log(-x + sqrt(x^2 +
2*x + 4) - 1) + 2*(x^2 - sqrt(x^2 + 2*x + 4)*(x - 1) - 1)*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)) + sqrt(7)*(2*x + 5) - 2*sqrt(7)*sqrt(x^2 + 2*x + 4))/(sqrt(7)*s
qrt(x^2 + 2*x + 4)*(x - 1) - sqrt(7)*(x^2 - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.23889, size = 204, normalized size = 3.29 \[ -\frac{2}{7} \, \sqrt{7}{\rm ln}\left (2 \, \sqrt{7} + 7 \, \sqrt{\frac{4}{x - 1} + \frac{7}{{\left (x - 1\right )}^{2}} + 1} + \frac{7 \, \sqrt{7}}{x - 1}\right ){\rm sign}\left (\frac{1}{x - 1}\right ) +{\rm ln}\left (\sqrt{\frac{4}{x - 1} + \frac{7}{{\left (x - 1\right )}^{2}} + 1} + \frac{\sqrt{7}}{x - 1} + 1\right ){\rm sign}\left (\frac{1}{x - 1}\right ) -{\rm ln}\left ({\left | \sqrt{\frac{4}{x - 1} + \frac{7}{{\left (x - 1\right )}^{2}} + 1} + \frac{\sqrt{7}}{x - 1} - 1 \right |}\right ){\rm sign}\left (\frac{1}{x - 1}\right ) - \sqrt{\frac{4}{x - 1} + \frac{7}{{\left (x - 1\right )}^{2}} + 1}{\rm sign}\left (\frac{1}{x - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/7*sqrt(7)*ln(2*sqrt(7) + 7*sqrt(4/(x - 1) + 7/(x - 1)^2 + 1) + 7*sqrt(7)/(x -
1))*sign(1/(x - 1)) + ln(sqrt(4/(x - 1) + 7/(x - 1)^2 + 1) + sqrt(7)/(x - 1) +
1)*sign(1/(x - 1)) - ln(abs(sqrt(4/(x - 1) + 7/(x - 1)^2 + 1) + sqrt(7)/(x - 1)
- 1))*sign(1/(x - 1)) - sqrt(4/(x - 1) + 7/(x - 1)^2 + 1)*sign(1/(x - 1))