3.463 \(\int \frac{1}{x^{10} \sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=71 \[ -\frac{5 \sqrt{1-x^3}}{24 x^3}-\frac{5}{24} \tanh ^{-1}\left (\sqrt{1-x^3}\right )-\frac{\sqrt{1-x^3}}{9 x^9}-\frac{5 \sqrt{1-x^3}}{36 x^6} \]

[Out]

-Sqrt[1 - x^3]/(9*x^9) - (5*Sqrt[1 - x^3])/(36*x^6) - (5*Sqrt[1 - x^3])/(24*x^3)
- (5*ArcTanh[Sqrt[1 - x^3]])/24

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Rubi [A]  time = 0.0816325, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{5 \sqrt{1-x^3}}{24 x^3}-\frac{5}{24} \tanh ^{-1}\left (\sqrt{1-x^3}\right )-\frac{\sqrt{1-x^3}}{9 x^9}-\frac{5 \sqrt{1-x^3}}{36 x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^10*Sqrt[1 - x^3]),x]

[Out]

-Sqrt[1 - x^3]/(9*x^9) - (5*Sqrt[1 - x^3])/(36*x^6) - (5*Sqrt[1 - x^3])/(24*x^3)
- (5*ArcTanh[Sqrt[1 - x^3]])/24

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Rubi in Sympy [A]  time = 6.81766, size = 58, normalized size = 0.82 \[ - \frac{5 \operatorname{atanh}{\left (\sqrt{- x^{3} + 1} \right )}}{24} - \frac{5 \sqrt{- x^{3} + 1}}{24 x^{3}} - \frac{5 \sqrt{- x^{3} + 1}}{36 x^{6}} - \frac{\sqrt{- x^{3} + 1}}{9 x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**10/(-x**3+1)**(1/2),x)

[Out]

-5*atanh(sqrt(-x**3 + 1))/24 - 5*sqrt(-x**3 + 1)/(24*x**3) - 5*sqrt(-x**3 + 1)/(
36*x**6) - sqrt(-x**3 + 1)/(9*x**9)

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Mathematica [A]  time = 0.0465588, size = 47, normalized size = 0.66 \[ -\frac{5}{24} \tanh ^{-1}\left (\sqrt{1-x^3}\right )-\frac{\sqrt{1-x^3} \left (15 x^6+10 x^3+8\right )}{72 x^9} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^10*Sqrt[1 - x^3]),x]

[Out]

-(Sqrt[1 - x^3]*(8 + 10*x^3 + 15*x^6))/(72*x^9) - (5*ArcTanh[Sqrt[1 - x^3]])/24

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Maple [A]  time = 0.037, size = 56, normalized size = 0.8 \[ -{\frac{5}{24}{\it Artanh} \left ( \sqrt{-{x}^{3}+1} \right ) }-{\frac{1}{9\,{x}^{9}}\sqrt{-{x}^{3}+1}}-{\frac{5}{36\,{x}^{6}}\sqrt{-{x}^{3}+1}}-{\frac{5}{24\,{x}^{3}}\sqrt{-{x}^{3}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^10/(-x^3+1)^(1/2),x)

[Out]

-5/24*arctanh((-x^3+1)^(1/2))-1/9*(-x^3+1)^(1/2)/x^9-5/36*(-x^3+1)^(1/2)/x^6-5/2
4*(-x^3+1)^(1/2)/x^3

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Maxima [A]  time = 1.44533, size = 122, normalized size = 1.72 \[ -\frac{15 \,{\left (-x^{3} + 1\right )}^{\frac{5}{2}} - 40 \,{\left (-x^{3} + 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{-x^{3} + 1}}{72 \,{\left ({\left (x^{3} - 1\right )}^{3} + 3 \, x^{3} + 3 \,{\left (x^{3} - 1\right )}^{2} - 2\right )}} - \frac{5}{48} \, \log \left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{5}{48} \, \log \left (\sqrt{-x^{3} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^3 + 1)*x^10),x, algorithm="maxima")

[Out]

-1/72*(15*(-x^3 + 1)^(5/2) - 40*(-x^3 + 1)^(3/2) + 33*sqrt(-x^3 + 1))/((x^3 - 1)
^3 + 3*x^3 + 3*(x^3 - 1)^2 - 2) - 5/48*log(sqrt(-x^3 + 1) + 1) + 5/48*log(sqrt(-
x^3 + 1) - 1)

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Fricas [A]  time = 0.228063, size = 85, normalized size = 1.2 \[ -\frac{15 \, x^{9} \log \left (\sqrt{-x^{3} + 1} + 1\right ) - 15 \, x^{9} \log \left (\sqrt{-x^{3} + 1} - 1\right ) + 2 \,{\left (15 \, x^{6} + 10 \, x^{3} + 8\right )} \sqrt{-x^{3} + 1}}{144 \, x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^3 + 1)*x^10),x, algorithm="fricas")

[Out]

-1/144*(15*x^9*log(sqrt(-x^3 + 1) + 1) - 15*x^9*log(sqrt(-x^3 + 1) - 1) + 2*(15*
x^6 + 10*x^3 + 8)*sqrt(-x^3 + 1))/x^9

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Sympy [A]  time = 17.6547, size = 182, normalized size = 2.56 \[ \begin{cases} - \frac{5 \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{24} + \frac{5}{24 x^{\frac{3}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} - \frac{5}{72 x^{\frac{9}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} - \frac{1}{36 x^{\frac{15}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} - \frac{1}{9 x^{\frac{21}{2}} \sqrt{-1 + \frac{1}{x^{3}}}} & \text{for}\: \left |{\frac{1}{x^{3}}}\right | > 1 \\\frac{5 i \operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{24} - \frac{5 i}{24 x^{\frac{3}{2}} \sqrt{1 - \frac{1}{x^{3}}}} + \frac{5 i}{72 x^{\frac{9}{2}} \sqrt{1 - \frac{1}{x^{3}}}} + \frac{i}{36 x^{\frac{15}{2}} \sqrt{1 - \frac{1}{x^{3}}}} + \frac{i}{9 x^{\frac{21}{2}} \sqrt{1 - \frac{1}{x^{3}}}} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**10/(-x**3+1)**(1/2),x)

[Out]

Piecewise((-5*acosh(x**(-3/2))/24 + 5/(24*x**(3/2)*sqrt(-1 + x**(-3))) - 5/(72*x
**(9/2)*sqrt(-1 + x**(-3))) - 1/(36*x**(15/2)*sqrt(-1 + x**(-3))) - 1/(9*x**(21/
2)*sqrt(-1 + x**(-3))), Abs(x**(-3)) > 1), (5*I*asin(x**(-3/2))/24 - 5*I/(24*x**
(3/2)*sqrt(1 - 1/x**3)) + 5*I/(72*x**(9/2)*sqrt(1 - 1/x**3)) + I/(36*x**(15/2)*s
qrt(1 - 1/x**3)) + I/(9*x**(21/2)*sqrt(1 - 1/x**3)), True))

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GIAC/XCAS [A]  time = 0.215833, size = 103, normalized size = 1.45 \[ -\frac{15 \,{\left (x^{3} - 1\right )}^{2} \sqrt{-x^{3} + 1} - 40 \,{\left (-x^{3} + 1\right )}^{\frac{3}{2}} + 33 \, \sqrt{-x^{3} + 1}}{72 \, x^{9}} - \frac{5}{48} \,{\rm ln}\left (\sqrt{-x^{3} + 1} + 1\right ) + \frac{5}{48} \,{\rm ln}\left ({\left | \sqrt{-x^{3} + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-x^3 + 1)*x^10),x, algorithm="giac")

[Out]

-1/72*(15*(x^3 - 1)^2*sqrt(-x^3 + 1) - 40*(-x^3 + 1)^(3/2) + 33*sqrt(-x^3 + 1))/
x^9 - 5/48*ln(sqrt(-x^3 + 1) + 1) + 5/48*ln(abs(sqrt(-x^3 + 1) - 1))