Optimal. Leaf size=26 \[ \sqrt{x^2+1} \log \left (\sqrt{x^2+1}+x\right )-x \]
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Rubi [A] time = 0.0533101, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \sqrt{x^2+1} \log \left (\sqrt{x^2+1}+x\right )-x \]
Antiderivative was successfully verified.
[In] Int[(x*Log[x + Sqrt[1 + x^2]])/Sqrt[1 + x^2],x]
[Out]
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Rubi in Sympy [A] time = 4.1515, size = 20, normalized size = 0.77 \[ - x + \sqrt{x^{2} + 1} \log{\left (x + \sqrt{x^{2} + 1} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x*ln(x+(x**2+1)**(1/2))/(x**2+1)**(1/2),x)
[Out]
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Mathematica [A] time = 0.017903, size = 26, normalized size = 1. \[ \sqrt{x^2+1} \log \left (\sqrt{x^2+1}+x\right )-x \]
Antiderivative was successfully verified.
[In] Integrate[(x*Log[x + Sqrt[1 + x^2]])/Sqrt[1 + x^2],x]
[Out]
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Maple [F] time = 0.022, size = 0, normalized size = 0. \[ \int{x\ln \left ( x+\sqrt{{x}^{2}+1} \right ){\frac{1}{\sqrt{{x}^{2}+1}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x*ln(x+(x^2+1)^(1/2))/(x^2+1)^(1/2),x)
[Out]
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Maxima [A] time = 1.64804, size = 30, normalized size = 1.15 \[ \sqrt{x^{2} + 1} \log \left (x + \sqrt{x^{2} + 1}\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*log(x + sqrt(x^2 + 1))/sqrt(x^2 + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220853, size = 30, normalized size = 1.15 \[ \sqrt{x^{2} + 1} \log \left (x + \sqrt{x^{2} + 1}\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*log(x + sqrt(x^2 + 1))/sqrt(x^2 + 1),x, algorithm="fricas")
[Out]
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Sympy [A] time = 59.3081, size = 20, normalized size = 0.77 \[ - x + \sqrt{x^{2} + 1} \log{\left (x + \sqrt{x^{2} + 1} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*ln(x+(x**2+1)**(1/2))/(x**2+1)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.202008, size = 30, normalized size = 1.15 \[ \sqrt{x^{2} + 1}{\rm ln}\left (x + \sqrt{x^{2} + 1}\right ) - x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x*log(x + sqrt(x^2 + 1))/sqrt(x^2 + 1),x, algorithm="giac")
[Out]