3.444 \(\int \frac{1}{x^7 \sqrt{1+x^3}} \, dx\)

Optimal. Leaf size=47 \[ \frac{\sqrt{x^3+1}}{4 x^3}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{\sqrt{x^3+1}}{6 x^6} \]

[Out]

-Sqrt[1 + x^3]/(6*x^6) + Sqrt[1 + x^3]/(4*x^3) - ArcTanh[Sqrt[1 + x^3]]/4

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Rubi [A]  time = 0.0495388, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\sqrt{x^3+1}}{4 x^3}-\frac{1}{4} \tanh ^{-1}\left (\sqrt{x^3+1}\right )-\frac{\sqrt{x^3+1}}{6 x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*Sqrt[1 + x^3]),x]

[Out]

-Sqrt[1 + x^3]/(6*x^6) + Sqrt[1 + x^3]/(4*x^3) - ArcTanh[Sqrt[1 + x^3]]/4

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Rubi in Sympy [A]  time = 4.8524, size = 37, normalized size = 0.79 \[ - \frac{\operatorname{atanh}{\left (\sqrt{x^{3} + 1} \right )}}{4} + \frac{\sqrt{x^{3} + 1}}{4 x^{3}} - \frac{\sqrt{x^{3} + 1}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(x**3+1)**(1/2),x)

[Out]

-atanh(sqrt(x**3 + 1))/4 + sqrt(x**3 + 1)/(4*x**3) - sqrt(x**3 + 1)/(6*x**6)

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Mathematica [A]  time = 0.0419965, size = 37, normalized size = 0.79 \[ \frac{1}{12} \left (\frac{\sqrt{x^3+1} \left (3 x^3-2\right )}{x^6}-3 \tanh ^{-1}\left (\sqrt{x^3+1}\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*Sqrt[1 + x^3]),x]

[Out]

((Sqrt[1 + x^3]*(-2 + 3*x^3))/x^6 - 3*ArcTanh[Sqrt[1 + x^3]])/12

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Maple [A]  time = 0.03, size = 36, normalized size = 0.8 \[ -{\frac{1}{4}{\it Artanh} \left ( \sqrt{{x}^{3}+1} \right ) }-{\frac{1}{6\,{x}^{6}}\sqrt{{x}^{3}+1}}+{\frac{1}{4\,{x}^{3}}\sqrt{{x}^{3}+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(x^3+1)^(1/2),x)

[Out]

-1/4*arctanh((x^3+1)^(1/2))-1/6*(x^3+1)^(1/2)/x^6+1/4*(x^3+1)^(1/2)/x^3

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Maxima [A]  time = 1.43456, size = 86, normalized size = 1.83 \[ -\frac{3 \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{x^{3} + 1}}{12 \,{\left (2 \, x^{3} -{\left (x^{3} + 1\right )}^{2} + 1\right )}} - \frac{1}{8} \, \log \left (\sqrt{x^{3} + 1} + 1\right ) + \frac{1}{8} \, \log \left (\sqrt{x^{3} + 1} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^3 + 1)*x^7),x, algorithm="maxima")

[Out]

-1/12*(3*(x^3 + 1)^(3/2) - 5*sqrt(x^3 + 1))/(2*x^3 - (x^3 + 1)^2 + 1) - 1/8*log(
sqrt(x^3 + 1) + 1) + 1/8*log(sqrt(x^3 + 1) - 1)

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Fricas [A]  time = 0.228907, size = 70, normalized size = 1.49 \[ -\frac{3 \, x^{6} \log \left (\sqrt{x^{3} + 1} + 1\right ) - 3 \, x^{6} \log \left (\sqrt{x^{3} + 1} - 1\right ) - 2 \,{\left (3 \, x^{3} - 2\right )} \sqrt{x^{3} + 1}}{24 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^3 + 1)*x^7),x, algorithm="fricas")

[Out]

-1/24*(3*x^6*log(sqrt(x^3 + 1) + 1) - 3*x^6*log(sqrt(x^3 + 1) - 1) - 2*(3*x^3 -
2)*sqrt(x^3 + 1))/x^6

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Sympy [A]  time = 10.5292, size = 65, normalized size = 1.38 \[ - \frac{\operatorname{asinh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{4} + \frac{1}{4 x^{\frac{3}{2}} \sqrt{1 + \frac{1}{x^{3}}}} + \frac{1}{12 x^{\frac{9}{2}} \sqrt{1 + \frac{1}{x^{3}}}} - \frac{1}{6 x^{\frac{15}{2}} \sqrt{1 + \frac{1}{x^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(x**3+1)**(1/2),x)

[Out]

-asinh(x**(-3/2))/4 + 1/(4*x**(3/2)*sqrt(1 + x**(-3))) + 1/(12*x**(9/2)*sqrt(1 +
x**(-3))) - 1/(6*x**(15/2)*sqrt(1 + x**(-3)))

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GIAC/XCAS [A]  time = 0.246308, size = 68, normalized size = 1.45 \[ \frac{3 \,{\left (x^{3} + 1\right )}^{\frac{3}{2}} - 5 \, \sqrt{x^{3} + 1}}{12 \, x^{6}} - \frac{1}{8} \,{\rm ln}\left (\sqrt{x^{3} + 1} + 1\right ) + \frac{1}{8} \,{\rm ln}\left ({\left | \sqrt{x^{3} + 1} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(x^3 + 1)*x^7),x, algorithm="giac")

[Out]

1/12*(3*(x^3 + 1)^(3/2) - 5*sqrt(x^3 + 1))/x^6 - 1/8*ln(sqrt(x^3 + 1) + 1) + 1/8
*ln(abs(sqrt(x^3 + 1) - 1))