3.62 \(\int \frac{(1+x)^2}{x^5 \sqrt{1-x^2}} \, dx\)

Optimal. Leaf size=89 \[ -\frac{4 \sqrt{1-x^2}}{3 x}-\frac{7 \sqrt{1-x^2}}{8 x^2}-\frac{7}{8} \tanh ^{-1}\left (\sqrt{1-x^2}\right )-\frac{\sqrt{1-x^2}}{4 x^4}-\frac{2 \sqrt{1-x^2}}{3 x^3} \]

[Out]

-Sqrt[1 - x^2]/(4*x^4) - (2*Sqrt[1 - x^2])/(3*x^3) - (7*Sqrt[1 - x^2])/(8*x^2) -
(4*Sqrt[1 - x^2])/(3*x) - (7*ArcTanh[Sqrt[1 - x^2]])/8

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Rubi [A]  time = 0.20857, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ -\frac{4 \sqrt{1-x^2}}{3 x}-\frac{7 \sqrt{1-x^2}}{8 x^2}-\frac{7}{8} \tanh ^{-1}\left (\sqrt{1-x^2}\right )-\frac{\sqrt{1-x^2}}{4 x^4}-\frac{2 \sqrt{1-x^2}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)^2/(x^5*Sqrt[1 - x^2]),x]

[Out]

-Sqrt[1 - x^2]/(4*x^4) - (2*Sqrt[1 - x^2])/(3*x^3) - (7*Sqrt[1 - x^2])/(8*x^2) -
(4*Sqrt[1 - x^2])/(3*x) - (7*ArcTanh[Sqrt[1 - x^2]])/8

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Rubi in Sympy [A]  time = 13.5497, size = 71, normalized size = 0.8 \[ - \frac{7 \operatorname{atanh}{\left (\sqrt{- x^{2} + 1} \right )}}{8} - \frac{4 \sqrt{- x^{2} + 1}}{3 x} - \frac{7 \sqrt{- x^{2} + 1}}{8 x^{2}} - \frac{2 \sqrt{- x^{2} + 1}}{3 x^{3}} - \frac{\sqrt{- x^{2} + 1}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)**2/x**5/(-x**2+1)**(1/2),x)

[Out]

-7*atanh(sqrt(-x**2 + 1))/8 - 4*sqrt(-x**2 + 1)/(3*x) - 7*sqrt(-x**2 + 1)/(8*x**
2) - 2*sqrt(-x**2 + 1)/(3*x**3) - sqrt(-x**2 + 1)/(4*x**4)

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Mathematica [A]  time = 0.0561333, size = 58, normalized size = 0.65 \[ -\frac{7}{8} \log \left (\sqrt{1-x^2}+1\right )-\frac{\sqrt{1-x^2} \left (32 x^3+21 x^2+16 x+6\right )}{24 x^4}+\frac{7 \log (x)}{8} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x)^2/(x^5*Sqrt[1 - x^2]),x]

[Out]

-(Sqrt[1 - x^2]*(6 + 16*x + 21*x^2 + 32*x^3))/(24*x^4) + (7*Log[x])/8 - (7*Log[1
+ Sqrt[1 - x^2]])/8

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Maple [A]  time = 0.011, size = 70, normalized size = 0.8 \[ -{\frac{1}{4\,{x}^{4}}\sqrt{-{x}^{2}+1}}-{\frac{7}{8\,{x}^{2}}\sqrt{-{x}^{2}+1}}-{\frac{7}{8}{\it Artanh} \left ({\frac{1}{\sqrt{-{x}^{2}+1}}} \right ) }-{\frac{2}{3\,{x}^{3}}\sqrt{-{x}^{2}+1}}-{\frac{4}{3\,x}\sqrt{-{x}^{2}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)^2/x^5/(-x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.792891, size = 111, normalized size = 1.25 \[ -\frac{4 \, \sqrt{-x^{2} + 1}}{3 \, x} - \frac{7 \, \sqrt{-x^{2} + 1}}{8 \, x^{2}} - \frac{2 \, \sqrt{-x^{2} + 1}}{3 \, x^{3}} - \frac{\sqrt{-x^{2} + 1}}{4 \, x^{4}} - \frac{7}{8} \, \log \left (\frac{2 \, \sqrt{-x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^2/(sqrt(-x^2 + 1)*x^5),x, algorithm="maxima")

[Out]

-4/3*sqrt(-x^2 + 1)/x - 7/8*sqrt(-x^2 + 1)/x^2 - 2/3*sqrt(-x^2 + 1)/x^3 - 1/4*sq
rt(-x^2 + 1)/x^4 - 7/8*log(2*sqrt(-x^2 + 1)/abs(x) + 2/abs(x))

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Fricas [A]  time = 0.268864, size = 231, normalized size = 2.6 \[ \frac{128 \, x^{7} + 84 \, x^{6} - 320 \, x^{5} - 228 \, x^{4} + 64 \, x^{3} + 96 \, x^{2} + 21 \,{\left (x^{8} - 8 \, x^{6} + 8 \, x^{4} + 4 \,{\left (x^{6} - 2 \, x^{4}\right )} \sqrt{-x^{2} + 1}\right )} \log \left (\frac{\sqrt{-x^{2} + 1} - 1}{x}\right ) -{\left (32 \, x^{7} + 21 \, x^{6} - 240 \, x^{5} - 162 \, x^{4} + 128 \, x^{3} + 120 \, x^{2} + 128 \, x + 48\right )} \sqrt{-x^{2} + 1} + 128 \, x + 48}{24 \,{\left (x^{8} - 8 \, x^{6} + 8 \, x^{4} + 4 \,{\left (x^{6} - 2 \, x^{4}\right )} \sqrt{-x^{2} + 1}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^2/(sqrt(-x^2 + 1)*x^5),x, algorithm="fricas")

[Out]

1/24*(128*x^7 + 84*x^6 - 320*x^5 - 228*x^4 + 64*x^3 + 96*x^2 + 21*(x^8 - 8*x^6 +
8*x^4 + 4*(x^6 - 2*x^4)*sqrt(-x^2 + 1))*log((sqrt(-x^2 + 1) - 1)/x) - (32*x^7 +
21*x^6 - 240*x^5 - 162*x^4 + 128*x^3 + 120*x^2 + 128*x + 48)*sqrt(-x^2 + 1) + 1
28*x + 48)/(x^8 - 8*x^6 + 8*x^4 + 4*(x^6 - 2*x^4)*sqrt(-x^2 + 1))

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Sympy [A]  time = 33.1151, size = 223, normalized size = 2.51 \[ 2 \left (\begin{cases} - \frac{\sqrt{- x^{2} + 1}}{x} - \frac{\left (- x^{2} + 1\right )^{\frac{3}{2}}}{3 x^{3}} & \text{for}\: x > -1 \wedge x < 1 \end{cases}\right ) + \begin{cases} - \frac{\operatorname{acosh}{\left (\frac{1}{x} \right )}}{2} - \frac{\sqrt{-1 + \frac{1}{x^{2}}}}{2 x} & \text{for}\: \left |{\frac{1}{x^{2}}}\right | > 1 \\\frac{i \operatorname{asin}{\left (\frac{1}{x} \right )}}{2} - \frac{i}{2 x \sqrt{1 - \frac{1}{x^{2}}}} + \frac{i}{2 x^{3} \sqrt{1 - \frac{1}{x^{2}}}} & \text{otherwise} \end{cases} + \begin{cases} - \frac{3 \operatorname{acosh}{\left (\frac{1}{x} \right )}}{8} + \frac{3}{8 x \sqrt{-1 + \frac{1}{x^{2}}}} - \frac{1}{8 x^{3} \sqrt{-1 + \frac{1}{x^{2}}}} - \frac{1}{4 x^{5} \sqrt{-1 + \frac{1}{x^{2}}}} & \text{for}\: \left |{\frac{1}{x^{2}}}\right | > 1 \\\frac{3 i \operatorname{asin}{\left (\frac{1}{x} \right )}}{8} - \frac{3 i}{8 x \sqrt{1 - \frac{1}{x^{2}}}} + \frac{i}{8 x^{3} \sqrt{1 - \frac{1}{x^{2}}}} + \frac{i}{4 x^{5} \sqrt{1 - \frac{1}{x^{2}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)**2/x**5/(-x**2+1)**(1/2),x)

[Out]

2*Piecewise((-sqrt(-x**2 + 1)/x - (-x**2 + 1)**(3/2)/(3*x**3), (x > -1) & (x < 1
))) + Piecewise((-acosh(1/x)/2 - sqrt(-1 + x**(-2))/(2*x), Abs(x**(-2)) > 1), (I
*asin(1/x)/2 - I/(2*x*sqrt(1 - 1/x**2)) + I/(2*x**3*sqrt(1 - 1/x**2)), True)) +
Piecewise((-3*acosh(1/x)/8 + 3/(8*x*sqrt(-1 + x**(-2))) - 1/(8*x**3*sqrt(-1 + x*
*(-2))) - 1/(4*x**5*sqrt(-1 + x**(-2))), Abs(x**(-2)) > 1), (3*I*asin(1/x)/8 - 3
*I/(8*x*sqrt(1 - 1/x**2)) + I/(8*x**3*sqrt(1 - 1/x**2)) + I/(4*x**5*sqrt(1 - 1/x
**2)), True))

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GIAC/XCAS [A]  time = 0.29309, size = 220, normalized size = 2.47 \[ \frac{x^{4}{\left (\frac{16 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{x} - \frac{48 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} + \frac{144 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}}{x^{3}} - 3\right )}}{192 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}^{4}} - \frac{3 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}{4 \, x} + \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{4 \, x^{2}} - \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{3}}{12 \, x^{3}} + \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{4}}{64 \, x^{4}} + \frac{7}{8} \,{\rm ln}\left (-\frac{\sqrt{-x^{2} + 1} - 1}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)^2/(sqrt(-x^2 + 1)*x^5),x, algorithm="giac")

[Out]

1/192*x^4*(16*(sqrt(-x^2 + 1) - 1)/x - 48*(sqrt(-x^2 + 1) - 1)^2/x^2 + 144*(sqrt
(-x^2 + 1) - 1)^3/x^3 - 3)/(sqrt(-x^2 + 1) - 1)^4 - 3/4*(sqrt(-x^2 + 1) - 1)/x +
1/4*(sqrt(-x^2 + 1) - 1)^2/x^2 - 1/12*(sqrt(-x^2 + 1) - 1)^3/x^3 + 1/64*(sqrt(-
x^2 + 1) - 1)^4/x^4 + 7/8*ln(-(sqrt(-x^2 + 1) - 1)/abs(x))