3.279 \(\int \frac{a+b x^3}{1+x^2} \, dx\)

Optimal. Leaf size=24 \[ a \tan ^{-1}(x)+\frac{b x^2}{2}-\frac{1}{2} b \log \left (x^2+1\right ) \]

[Out]

(b*x^2)/2 + a*ArcTan[x] - (b*Log[1 + x^2])/2

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Rubi [A]  time = 0.0419264, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ a \tan ^{-1}(x)+\frac{b x^2}{2}-\frac{1}{2} b \log \left (x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)/(1 + x^2),x]

[Out]

(b*x^2)/2 + a*ArcTan[x] - (b*Log[1 + x^2])/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ a \operatorname{atan}{\left (x \right )} - \frac{b \log{\left (x^{2} + 1 \right )}}{2} + b \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*atan(x) - b*log(x**2 + 1)/2 + b*Integral(x, x)

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Mathematica [A]  time = 0.0148715, size = 22, normalized size = 0.92 \[ a \tan ^{-1}(x)+\frac{1}{2} b \left (x^2-\log \left (x^2+1\right )\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)/(1 + x^2),x]

[Out]

a*ArcTan[x] + (b*(x^2 - Log[1 + x^2]))/2

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Maple [A]  time = 0.003, size = 21, normalized size = 0.9 \[{\frac{b{x}^{2}}{2}}+a\arctan \left ( x \right ) -{\frac{b\ln \left ({x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)/(x^2+1),x)

[Out]

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

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Maxima [A]  time = 0.87947, size = 27, normalized size = 1.12 \[ \frac{1}{2} \, b x^{2} + a \arctan \left (x\right ) - \frac{1}{2} \, b \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b*x^2 + a*arctan(x) - 1/2*b*log(x^2 + 1)

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Fricas [A]  time = 0.257545, size = 27, normalized size = 1.12 \[ \frac{1}{2} \, b x^{2} + a \arctan \left (x\right ) - \frac{1}{2} \, b \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b*x^2 + a*arctan(x) - 1/2*b*log(x^2 + 1)

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Sympy [A]  time = 1.29482, size = 34, normalized size = 1.42 \[ \frac{b x^{2}}{2} + \left (- \frac{i a}{2} - \frac{b}{2}\right ) \log{\left (x - i \right )} + \left (\frac{i a}{2} - \frac{b}{2}\right ) \log{\left (x + i \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b*x**2/2 + (-I*a/2 - b/2)*log(x - I) + (I*a/2 - b/2)*log(x + I)

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GIAC/XCAS [A]  time = 0.259862, size = 27, normalized size = 1.12 \[ \frac{1}{2} \, b x^{2} + a \arctan \left (x\right ) - \frac{1}{2} \, b{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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