3.36 \(\int \frac{1}{x+b x^3} \, dx\)

Optimal. Leaf size=15 \[ \log (x)-\frac{1}{2} \log \left (b x^2+1\right ) \]

[Out]

Log[x] - Log[1 + b*x^2]/2

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Rubi [A]  time = 0.0254655, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556 \[ \log (x)-\frac{1}{2} \log \left (b x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(x + b*x^3)^(-1),x]

[Out]

Log[x] - Log[1 + b*x^2]/2

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Rubi in Sympy [A]  time = 25.0264, size = 15, normalized size = 1. \[ \frac{\log{\left (x^{2} \right )}}{2} - \frac{\log{\left (b x^{2} + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**3+x),x)

[Out]

log(x**2)/2 - log(b*x**2 + 1)/2

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Mathematica [A]  time = 0.00553795, size = 15, normalized size = 1. \[ \log (x)-\frac{1}{2} \log \left (b x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(x + b*x^3)^(-1),x]

[Out]

Log[x] - Log[1 + b*x^2]/2

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Maple [A]  time = 0.006, size = 14, normalized size = 0.9 \[ \ln \left ( x \right ) -{\frac{\ln \left ( b{x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^3+x),x)

[Out]

ln(x)-1/2*ln(b*x^2+1)

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Maxima [A]  time = 1.37166, size = 18, normalized size = 1.2 \[ -\frac{1}{2} \, \log \left (b x^{2} + 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*log(b*x^2 + 1) + log(x)

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Fricas [A]  time = 0.200676, size = 18, normalized size = 1.2 \[ -\frac{1}{2} \, \log \left (b x^{2} + 1\right ) + \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*log(b*x^2 + 1) + log(x)

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Sympy [A]  time = 0.28776, size = 12, normalized size = 0.8 \[ \log{\left (x \right )} - \frac{\log{\left (x^{2} + \frac{1}{b} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**3+x),x)

[Out]

log(x) - log(x**2 + 1/b)/2

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GIAC/XCAS [A]  time = 0.216922, size = 24, normalized size = 1.6 \[ \frac{1}{2} \,{\rm ln}\left (x^{2}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | b x^{2} + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*ln(x^2) - 1/2*ln(abs(b*x^2 + 1))