3.657 \(\int \frac{1+x}{\left (4+x^2\right ) \sqrt{9+x^2}} \, dx\)

Optimal. Leaf size=53 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{5} x}{2 \sqrt{x^2+9}}\right )}{2 \sqrt{5}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^2+9}}{\sqrt{5}}\right )}{\sqrt{5}} \]

[Out]

ArcTan[(Sqrt[5]*x)/(2*Sqrt[9 + x^2])]/(2*Sqrt[5]) - ArcTanh[Sqrt[9 + x^2]/Sqrt[5
]]/Sqrt[5]

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Rubi [A]  time = 0.0967123, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{5} x}{2 \sqrt{x^2+9}}\right )}{2 \sqrt{5}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{x^2+9}}{\sqrt{5}}\right )}{\sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Int[(1 + x)/((4 + x^2)*Sqrt[9 + x^2]),x]

[Out]

ArcTan[(Sqrt[5]*x)/(2*Sqrt[9 + x^2])]/(2*Sqrt[5]) - ArcTanh[Sqrt[9 + x^2]/Sqrt[5
]]/Sqrt[5]

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Rubi in Sympy [A]  time = 5.92944, size = 48, normalized size = 0.91 \[ \frac{\sqrt{5} \operatorname{atan}{\left (\frac{\sqrt{5} x}{2 \sqrt{x^{2} + 9}} \right )}}{10} - \frac{\sqrt{5} \operatorname{atanh}{\left (\frac{\sqrt{5} \sqrt{x^{2} + 9}}{5} \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((1+x)/(x**2+4)/(x**2+9)**(1/2),x)

[Out]

sqrt(5)*atan(sqrt(5)*x/(2*sqrt(x**2 + 9)))/10 - sqrt(5)*atanh(sqrt(5)*sqrt(x**2
+ 9)/5)/5

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Mathematica [A]  time = 0.0979993, size = 75, normalized size = 1.42 \[ -\frac{-\log \left (x^2+4\right )+\log \left (x^2+2 \sqrt{5} \sqrt{x^2+9}+14\right )+\tan ^{-1}\left (\frac{18-8 x^2}{9 x^2+5 \sqrt{5} \sqrt{x^2+9} x+36}\right )}{2 \sqrt{5}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 + x)/((4 + x^2)*Sqrt[9 + x^2]),x]

[Out]

-(ArcTan[(18 - 8*x^2)/(36 + 9*x^2 + 5*Sqrt[5]*x*Sqrt[9 + x^2])] - Log[4 + x^2] +
Log[14 + x^2 + 2*Sqrt[5]*Sqrt[9 + x^2]])/(2*Sqrt[5])

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Maple [A]  time = 0.02, size = 39, normalized size = 0.7 \[{\frac{\sqrt{5}}{10}\arctan \left ({\frac{x\sqrt{5}}{2}{\frac{1}{\sqrt{{x}^{2}+9}}}} \right ) }-{\frac{\sqrt{5}}{5}{\it Artanh} \left ({\frac{\sqrt{5}}{5}\sqrt{{x}^{2}+9}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((1+x)/(x^2+4)/(x^2+9)^(1/2),x)

[Out]

1/10*arctan(1/2*x*5^(1/2)/(x^2+9)^(1/2))*5^(1/2)-1/5*arctanh(1/5*(x^2+9)^(1/2)*5
^(1/2))*5^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x + 1}{\sqrt{x^{2} + 9}{\left (x^{2} + 4\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(sqrt(x^2 + 9)*(x^2 + 4)),x, algorithm="maxima")

[Out]

integrate((x + 1)/(sqrt(x^2 + 9)*(x^2 + 4)), x)

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Fricas [A]  time = 0.296107, size = 254, normalized size = 4.79 \[ \frac{1}{10} \, \sqrt{5}{\left (2 \, \arctan \left (-\frac{2 \, \sqrt{5}}{\sqrt{5} x - \sqrt{5} \sqrt{x^{2} + 9} - \sqrt{10 \, x^{2} - 10 \, \sqrt{x^{2} + 9}{\left (x + \sqrt{5}\right )} + 10 \, \sqrt{5} x + 90} + 5}\right ) - 2 \, \arctan \left (-\frac{2 \, \sqrt{5}}{\sqrt{5} x - \sqrt{5} \sqrt{x^{2} + 9} - \sqrt{10 \, x^{2} - 10 \, \sqrt{x^{2} + 9}{\left (x - \sqrt{5}\right )} - 10 \, \sqrt{5} x + 90} - 5}\right ) + \log \left (10 \, x^{2} - 10 \, \sqrt{x^{2} + 9}{\left (x + \sqrt{5}\right )} + 10 \, \sqrt{5} x + 90\right ) - \log \left (10 \, x^{2} - 10 \, \sqrt{x^{2} + 9}{\left (x - \sqrt{5}\right )} - 10 \, \sqrt{5} x + 90\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(sqrt(x^2 + 9)*(x^2 + 4)),x, algorithm="fricas")

[Out]

1/10*sqrt(5)*(2*arctan(-2*sqrt(5)/(sqrt(5)*x - sqrt(5)*sqrt(x^2 + 9) - sqrt(10*x
^2 - 10*sqrt(x^2 + 9)*(x + sqrt(5)) + 10*sqrt(5)*x + 90) + 5)) - 2*arctan(-2*sqr
t(5)/(sqrt(5)*x - sqrt(5)*sqrt(x^2 + 9) - sqrt(10*x^2 - 10*sqrt(x^2 + 9)*(x - sq
rt(5)) - 10*sqrt(5)*x + 90) - 5)) + log(10*x^2 - 10*sqrt(x^2 + 9)*(x + sqrt(5))
+ 10*sqrt(5)*x + 90) - log(10*x^2 - 10*sqrt(x^2 + 9)*(x - sqrt(5)) - 10*sqrt(5)*
x + 90))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x + 1}{\left (x^{2} + 4\right ) \sqrt{x^{2} + 9}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((1+x)/(x**2+4)/(x**2+9)**(1/2),x)

[Out]

Integral((x + 1)/((x**2 + 4)*sqrt(x**2 + 9)), x)

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GIAC/XCAS [A]  time = 0.298337, size = 528, normalized size = 9.96 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x + 1)/(sqrt(x^2 + 9)*(x^2 + 4)),x, algorithm="giac")

[Out]

1/40*(9*sqrt(5)*arctan(2/(sqrt(5) + 3)) + 9*sqrt(5)*arctan(2/(sqrt(5) - 3)) + 49
*sqrt(5)*ln(3/2*sqrt(5) + 9/2) - 49*sqrt(5)*ln(-3/2*sqrt(5) + 9/2) - 15*arctan(2
/(sqrt(5) + 3)) + 15*arctan(2/(sqrt(5) - 3)) - 105*ln(3/2*sqrt(5) + 9/2) - 105*l
n(-3/2*sqrt(5) + 9/2))*sign(x) - 1/10*(7*sqrt(5) + 15)*ln((sqrt(9/x^2 + 1) - 3/x
)^2 + 1/2*(3*sqrt(5)*sign(x) + 7*sign(x))/sign(x))*sign(x)/(7*abs(sign(x))*sign(
x) + 3*sqrt(5)) + 1/10*(7*sqrt(5) - 15)*ln((sqrt(9/x^2 + 1) - 3/x)^2 - 1/2*(3*sq
rt(5)*sign(x) - 7*sign(x))/sign(x))*sign(x)/(7*abs(sign(x))*sign(x) - 3*sqrt(5))
- 1/20*(5*(sqrt(5) + 3)*abs(sign(x)) + 3*(3*sqrt(5) + 5)*sign(x))*arctan(2*sqrt
(1/2)*(sqrt(9/x^2 + 1) - 3/x)/sqrt((3*sqrt(5)*sign(x) + 7*sign(x))/sign(x)))/(7*
abs(sign(x))*sign(x) + 3*sqrt(5)) + 1/20*(5*(sqrt(5) - 3)*abs(sign(x)) + 3*(3*sq
rt(5) - 5)*sign(x))*arctan(2*sqrt(1/2)*(sqrt(9/x^2 + 1) - 3/x)/sqrt(-(3*sqrt(5)*
sign(x) - 7*sign(x))/sign(x)))/(7*abs(sign(x))*sign(x) - 3*sqrt(5))