3.2922 \(\int \frac{1}{1+\left (x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=83 \[ -\frac{x \log \left (x^2-\sqrt{x^2}+1\right )}{6 \sqrt{x^2}}+\frac{x \log \left (\sqrt{x^2}+1\right )}{3 \sqrt{x^2}}-\frac{x \tan ^{-1}\left (\frac{1-2 \sqrt{x^2}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt{x^2}} \]

[Out]

-((x*ArcTan[(1 - 2*Sqrt[x^2])/Sqrt[3]])/(Sqrt[3]*Sqrt[x^2])) - (x*Log[1 + x^2 -
Sqrt[x^2]])/(6*Sqrt[x^2]) + (x*Log[1 + Sqrt[x^2]])/(3*Sqrt[x^2])

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Rubi [A]  time = 0.0859858, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.636 \[ -\frac{x \log \left (x^2-\sqrt{x^2}+1\right )}{6 \sqrt{x^2}}+\frac{x \log \left (\sqrt{x^2}+1\right )}{3 \sqrt{x^2}}-\frac{x \tan ^{-1}\left (\frac{1-2 \sqrt{x^2}}{\sqrt{3}}\right )}{\sqrt{3} \sqrt{x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + (x^2)^(3/2))^(-1),x]

[Out]

-((x*ArcTan[(1 - 2*Sqrt[x^2])/Sqrt[3]])/(Sqrt[3]*Sqrt[x^2])) - (x*Log[1 + x^2 -
Sqrt[x^2]])/(6*Sqrt[x^2]) + (x*Log[1 + Sqrt[x^2]])/(3*Sqrt[x^2])

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Rubi in Sympy [A]  time = 7.58725, size = 78, normalized size = 0.94 \[ \frac{x \log{\left (\sqrt{x^{2}} + 1 \right )}}{3 \sqrt{x^{2}}} - \frac{x \log{\left (x^{2} - \sqrt{x^{2}} + 1 \right )}}{6 \sqrt{x^{2}}} + \frac{\sqrt{3} x \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt{x^{2}}}{3} - \frac{1}{3}\right ) \right )}}{3 \sqrt{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0441797, size = 0, normalized size = 0. \[ \int \frac{1}{1+\left (x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

Integrate[(1 + (x^2)^(3/2))^(-1), x]

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Maple [A]  time = 0.011, size = 106, normalized size = 1.3 \[ -{\frac{{x}^{3}}{6} \left ( 2\,\sqrt{3}\arctan \left ( 1/3\,{\sqrt{3} \left ( -2\,x+\sqrt [3]{{\frac{{x}^{3}}{ \left ({x}^{2} \right ) ^{3/2}}}} \right ){\frac{1}{\sqrt [3]{{\frac{{x}^{3}}{ \left ({x}^{2} \right ) ^{3/2}}}}}}} \right ) -2\,\ln \left ( x+\sqrt [3]{{\frac{{x}^{3}}{ \left ({x}^{2} \right ) ^{3/2}}}} \right ) +\ln \left ({x}^{2}-x\sqrt [3]{{{x}^{3} \left ({x}^{2} \right ) ^{-{\frac{3}{2}}}}}+ \left ({{x}^{3} \left ({x}^{2} \right ) ^{-{\frac{3}{2}}}} \right ) ^{{\frac{2}{3}}} \right ) \right ) \left ({x}^{2} \right ) ^{-{\frac{3}{2}}} \left ({{x}^{3} \left ({x}^{2} \right ) ^{-{\frac{3}{2}}}} \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1+(x^2)^(3/2)),x)

[Out]

-1/6*x^3*(2*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(1/(x^2)^(3/2)*x^3)^(1/3))/(1/(x^2)
^(3/2)*x^3)^(1/3))-2*ln(x+(1/(x^2)^(3/2)*x^3)^(1/3))+ln(x^2-x*(1/(x^2)^(3/2)*x^3
)^(1/3)+(1/(x^2)^(3/2)*x^3)^(2/3)))/(x^2)^(3/2)/(1/(x^2)^(3/2)*x^3)^(2/3)

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Maxima [A]  time = 1.62341, size = 46, normalized size = 0.55 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) - \frac{1}{6} \, \log \left (x^{2} - x + 1\right ) + \frac{1}{3} \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.215099, size = 74, normalized size = 0.89 \[ -\frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} \log \left (x^{2} - \sqrt{x^{2}} + 1\right ) - 2 \, \sqrt{3} \log \left (\sqrt{x^{2}} + 1\right ) - 6 \, \arctan \left (\frac{2}{3} \, \sqrt{3} \sqrt{x^{2}} - \frac{1}{3} \, \sqrt{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/18*sqrt(3)*(sqrt(3)*log(x^2 - sqrt(x^2) + 1) - 2*sqrt(3)*log(sqrt(x^2) + 1) -
6*arctan(2/3*sqrt(3)*sqrt(x^2) - 1/3*sqrt(3)))

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Sympy [A]  time = 0.368965, size = 41, normalized size = 0.49 \[ \frac{\log{\left (x + 1 \right )}}{3} - \frac{\log{\left (x^{2} - x + 1 \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x}{3} - \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.237638, size = 146, normalized size = 1.76 \[ -\frac{\sqrt{3}{\left (-i \, \sqrt{3} - 1\right )} \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{1}{{\rm sign}\left (x\right )}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{1}{{\rm sign}\left (x\right )}\right )^{\frac{1}{3}}}\right )}{6 \,{\rm sign}\left (x\right )^{\frac{1}{3}}} - \frac{1}{9} i \, \pi{\rm sign}\left (x\right ) - \frac{{\left (-i \, \sqrt{3} - 1\right )}{\rm ln}\left (x^{2} + x \left (-\frac{1}{{\rm sign}\left (x\right )}\right )^{\frac{1}{3}} + \left (-\frac{1}{{\rm sign}\left (x\right )}\right )^{\frac{2}{3}}\right )}{12 \,{\rm sign}\left (x\right )^{\frac{1}{3}}} - \frac{1}{3} \, \left (-\frac{1}{{\rm sign}\left (x\right )}\right )^{\frac{1}{3}}{\rm ln}\left ({\left | x - \left (-\frac{1}{{\rm sign}\left (x\right )}\right )^{\frac{1}{3}} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*sqrt(3)*(-I*sqrt(3) - 1)*arctan(1/3*sqrt(3)*(2*x + (-1/sign(x))^(1/3))/(-1/
sign(x))^(1/3))/sign(x)^(1/3) - 1/9*I*pi*sign(x) - 1/12*(-I*sqrt(3) - 1)*ln(x^2
+ x*(-1/sign(x))^(1/3) + (-1/sign(x))^(2/3))/sign(x)^(1/3) - 1/3*(-1/sign(x))^(1
/3)*ln(abs(x - (-1/sign(x))^(1/3)))