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