3.541 \(\int \frac{1}{4+x+\sqrt{1+x}} \, dx\)

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]

(-2*ArcTan[(1 + 2*Sqrt[1 + x])/Sqrt[11]])/Sqrt[11] + Log[4 + x + Sqrt[1 + x]]

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Rubi [A]  time = 0.0662781, 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]

(-2*ArcTan[(1 + 2*Sqrt[1 + x])/Sqrt[11]])/Sqrt[11] + Log[4 + x + Sqrt[1 + x]]

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Rubi in Sympy [A]  time = 2.86405, 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]

log(x + sqrt(x + 1) + 4) - 2*sqrt(11)*atan(sqrt(11)*(2*sqrt(x + 1)/11 + 1/11))/1
1

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Mathematica [A]  time = 0.0211995, 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]

(-2*ArcTan[(1 + 2*Sqrt[1 + x])/Sqrt[11]])/Sqrt[11] + Log[4 + x + Sqrt[1 + x]]

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Maple [B]  time = 0.018, size = 93, normalized size = 2.5 \[{\frac{1}{2}\ln \left ( 4+x+\sqrt{1+x} \right ) }-{\frac{\sqrt{11}}{11}\arctan \left ({\frac{\sqrt{11}}{11} \left ( 1+2\,\sqrt{1+x} \right ) } \right ) }-{\frac{1}{2}\ln \left ( 4+x-\sqrt{1+x} \right ) }-{\frac{\sqrt{11}}{11}\arctan \left ({\frac{\sqrt{11}}{11} \left ( 2\,\sqrt{1+x}-1 \right ) } \right ) }+{\frac{\sqrt{11}}{11}\arctan \left ({\frac{ \left ( 2\,x+7 \right ) \sqrt{11}}{11}} \right ) }+{\frac{\ln \left ({x}^{2}+7\,x+15 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(4+x+(1+x)^(1/2)),x)

[Out]

1/2*ln(4+x+(1+x)^(1/2))-1/11*arctan(1/11*(1+2*(1+x)^(1/2))*11^(1/2))*11^(1/2)-1/
2*ln(4+x-(1+x)^(1/2))-1/11*11^(1/2)*arctan(1/11*(2*(1+x)^(1/2)-1)*11^(1/2))+1/11
*11^(1/2)*arctan(1/11*(2*x+7)*11^(1/2))+1/2*ln(x^2+7*x+15)

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Maxima [A]  time = 0.802132, 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]

-2/11*sqrt(11)*arctan(1/11*sqrt(11)*(2*sqrt(x + 1) + 1)) + log(x + sqrt(x + 1) +
4)

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Fricas [A]  time = 0.264478, 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]

1/11*sqrt(11)*(sqrt(11)*log(x + sqrt(x + 1) + 4) - 2*arctan(2/11*sqrt(11)*sqrt(x
+ 1) + 1/11*sqrt(11)))

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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]

Integral(1/(x + sqrt(x + 1) + 4), x)

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GIAC/XCAS [A]  time = 0.284902, 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]

-2/11*sqrt(11)*arctan(1/11*sqrt(11)*(2*sqrt(x + 1) + 1)) + ln(x + sqrt(x + 1) +
4)