Optimal. Leaf size=37 \[ \log \left (x+\sqrt{x+1}+4\right )-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt{x+1}+1}{\sqrt{11}}\right )}{\sqrt{11}} \]
[Out]
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Rubi [A] time = 0.0648394, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \log \left (x+\sqrt{x+1}+4\right )-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt{x+1}+1}{\sqrt{11}}\right )}{\sqrt{11}} \]
Antiderivative was successfully verified.
[In] Int[(4 + x + Sqrt[1 + x])^(-1),x]
[Out]
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Rubi in Sympy [A] time = 2.64605, size = 39, normalized size = 1.05 \[ \log{\left (x + \sqrt{x + 1} + 4 \right )} - \frac{2 \sqrt{11} \operatorname{atan}{\left (\sqrt{11} \left (\frac{2 \sqrt{x + 1}}{11} + \frac{1}{11}\right ) \right )}}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(4+x+(1+x)**(1/2)),x)
[Out]
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Mathematica [A] time = 0.0203436, size = 37, normalized size = 1. \[ \log \left (x+\sqrt{x+1}+4\right )-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt{x+1}+1}{\sqrt{11}}\right )}{\sqrt{11}} \]
Antiderivative was successfully verified.
[In] Integrate[(4 + x + Sqrt[1 + x])^(-1),x]
[Out]
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Maple [B] time = 0.022, size = 93, normalized size = 2.5 \[ -{\frac{1}{2}\ln \left ( x+4-\sqrt{1+x} \right ) }-{\frac{\sqrt{11}}{11}\arctan \left ({\frac{\sqrt{11}}{11} \left ( 2\,\sqrt{1+x}-1 \right ) } \right ) }+{\frac{1}{2}\ln \left ( 4+x+\sqrt{1+x} \right ) }-{\frac{\sqrt{11}}{11}\arctan \left ({\frac{\sqrt{11}}{11} \left ( 2\,\sqrt{1+x}+1 \right ) } \right ) }+{\frac{\sqrt{11}}{11}\arctan \left ({\frac{ \left ( 7+2\,x \right ) \sqrt{11}}{11}} \right ) }+{\frac{\ln \left ({x}^{2}+7\,x+15 \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(4+x+(1+x)^(1/2)),x)
[Out]
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Maxima [A] time = 1.58043, size = 41, normalized size = 1.11 \[ -\frac{2}{11} \, \sqrt{11} \arctan \left (\frac{1}{11} \, \sqrt{11}{\left (2 \, \sqrt{x + 1} + 1\right )}\right ) + \log \left (x + \sqrt{x + 1} + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x + sqrt(x + 1) + 4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241737, size = 51, normalized size = 1.38 \[ \frac{1}{11} \, \sqrt{11}{\left (\sqrt{11} \log \left (x + \sqrt{x + 1} + 4\right ) - 2 \, \arctan \left (\frac{2}{11} \, \sqrt{11} \sqrt{x + 1} + \frac{1}{11} \, \sqrt{11}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x + sqrt(x + 1) + 4),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x + \sqrt{x + 1} + 4}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(4+x+(1+x)**(1/2)),x)
[Out]
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GIAC/XCAS [A] time = 0.214459, size = 41, normalized size = 1.11 \[ -\frac{2}{11} \, \sqrt{11} \arctan \left (\frac{1}{11} \, \sqrt{11}{\left (2 \, \sqrt{x + 1} + 1\right )}\right ) +{\rm ln}\left (x + \sqrt{x + 1} + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x + sqrt(x + 1) + 4),x, algorithm="giac")
[Out]