3.253 \(\int \frac{x}{1+x^2+a \sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=12 \[ \log \left (a+\sqrt{x^2+1}\right ) \]

[Out]

Log[a + Sqrt[1 + x^2]]

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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]

[Out]

Log[a + Sqrt[1 + x^2]]

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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]

log(a + sqrt(x**2 + 1))

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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]

Log[a + Sqrt[1 + x^2]]

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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]

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

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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]

log(a + sqrt(x^2 + 1))

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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")

[Out]

1/2*log(-a^2 + x^2 + 1) - 1/2*log(a*x + x^2 - sqrt(x^2 + 1)*(a + x) + 1) + 1/2*l
og(-a*x + x^2 + sqrt(x^2 + 1)*(a - x) + 1)

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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]

Integral(x/(a*sqrt(x**2 + 1) + x**2 + 1), x)

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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]

ln(abs(a + sqrt(x^2 + 1)))