Optimal. Leaf size=12 \[ \tan ^{-1}\left (2 \sqrt{x^2+x}\right ) \]
[Out]
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Rubi [A] time = 0.0241491, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \tan ^{-1}\left (2 \sqrt{x^2+x}\right ) \]
Antiderivative was successfully verified.
[In] Int[1/((1 + 2*x)*Sqrt[x + x^2]),x]
[Out]
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Rubi in Sympy [A] time = 2.24703, size = 10, normalized size = 0.83 \[ \operatorname{atan}{\left (2 \sqrt{x^{2} + x} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(1+2*x)/(x**2+x)**(1/2),x)
[Out]
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Mathematica [B] time = 0.026756, size = 37, normalized size = 3.08 \[ \frac{2 \sqrt{x} \sqrt{x+1} \tan ^{-1}\left (\frac{\sqrt{x}}{\sqrt{x+1}}\right )}{\sqrt{x (x+1)}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((1 + 2*x)*Sqrt[x + x^2]),x]
[Out]
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Maple [A] time = 0.009, size = 15, normalized size = 1.3 \[ -\arctan \left ({\frac{1}{\sqrt{4\, \left ( x+1/2 \right ) ^{2}-1}}} \right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(1+2*x)/(x^2+x)^(1/2),x)
[Out]
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Maxima [A] time = 0.773137, size = 15, normalized size = 1.25 \[ -\arcsin \left (\frac{1}{{\left | 2 \, x + 1 \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^2 + x)*(2*x + 1)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.265038, size = 23, normalized size = 1.92 \[ 2 \, \arctan \left (-2 \, x + 2 \, \sqrt{x^{2} + x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^2 + x)*(2*x + 1)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{x \left (x + 1\right )} \left (2 x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(1+2*x)/(x**2+x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.264439, size = 23, normalized size = 1.92 \[ 2 \, \arctan \left (-2 \, x + 2 \, \sqrt{x^{2} + x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(x^2 + x)*(2*x + 1)),x, algorithm="giac")
[Out]