Optimal. Leaf size=12 \[ \log \left (a+\sqrt{x^2+1}\right ) \]
[Out]
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Rubi [A] time = 0.0748354, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \log \left (a+\sqrt{x^2+1}\right ) \]
Antiderivative was successfully verified.
[In] Int[x/(1 + x^2 + a*Sqrt[1 + x^2]),x]
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Rubi in Sympy [A] time = 3.57427, size = 10, normalized size = 0.83 \[ \log{\left (a + \sqrt{x^{2} + 1} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x/(1+x**2+a*(x**2+1)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.0124393, size = 12, normalized size = 1. \[ \log \left (a+\sqrt{x^2+1}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x/(1 + x^2 + a*Sqrt[1 + x^2]),x]
[Out]
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Maple [B] time = 0.062, size = 328, normalized size = 27.3 \[{\frac{1}{a}\sqrt{{x}^{2}+1}}-{\frac{1}{2\,a}\sqrt{ \left ( x-\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) ^{2}+2\,\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \left ( x-\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) +{a}^{2}}}+{\frac{a}{2}\ln \left ({1 \left ( 2\,{a}^{2}+2\,\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \left ( x-\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) +2\,\sqrt{{a}^{2}}\sqrt{ \left ( x-\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) ^{2}+2\,\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \left ( x-\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) +{a}^{2}} \right ) \left ( x-\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}-{\frac{1}{2\,a}\sqrt{ \left ( x+\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) ^{2}-2\,\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \left ( x+\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) +{a}^{2}}}+{\frac{a}{2}\ln \left ({1 \left ( 2\,{a}^{2}-2\,\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \left ( x+\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) +2\,\sqrt{{a}^{2}}\sqrt{ \left ( x+\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) ^{2}-2\,\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \left ( x+\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) +{a}^{2}} \right ) \left ( x+\sqrt{ \left ( 1+a \right ) \left ( a-1 \right ) } \right ) ^{-1}} \right ){\frac{1}{\sqrt{{a}^{2}}}}}+{\frac{\ln \left ( -{a}^{2}+{x}^{2}+1 \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x/(1+x^2+a*(x^2+1)^(1/2)),x)
[Out]
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Maxima [A] time = 1.33992, size = 14, normalized size = 1.17 \[ \log \left (a + \sqrt{x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^2 + sqrt(x^2 + 1)*a + 1),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227411, size = 84, normalized size = 7. \[ \frac{1}{2} \, \log \left (-a^{2} + x^{2} + 1\right ) - \frac{1}{2} \, \log \left (a x + x^{2} - \sqrt{x^{2} + 1}{\left (a + x\right )} + 1\right ) + \frac{1}{2} \, \log \left (-a x + x^{2} + \sqrt{x^{2} + 1}{\left (a - x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^2 + sqrt(x^2 + 1)*a + 1),x, algorithm="fricas")
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{a \sqrt{x^{2} + 1} + x^{2} + 1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(1+x**2+a*(x**2+1)**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.217181, size = 15, normalized size = 1.25 \[{\rm ln}\left ({\left | a + \sqrt{x^{2} + 1} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x/(x^2 + sqrt(x^2 + 1)*a + 1),x, algorithm="giac")
[Out]