3.860 \(\int \frac{1+2 x}{\left (1+x^2\right ) \sqrt{2+2 x+x^2}} \, dx\)
Optimal. Leaf size=126 \[ -\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\frac{2 \sqrt{5}-\left (5+\sqrt{5}\right ) x}{\sqrt{10 \left (1+\sqrt{5}\right )} \sqrt{x^2+2 x+2}}\right )-\sqrt{\frac{1}{2} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\frac{\left (5-\sqrt{5}\right ) x+2 \sqrt{5}}{\sqrt{10 \left (\sqrt{5}-1\right )} \sqrt{x^2+2 x+2}}\right ) \]
[Out]
-(Sqrt[(1 + Sqrt[5])/2]*ArcTan[(2*Sqrt[5] - (5 + Sqrt[5])*x)/(Sqrt[10*(1 + Sqrt[
5])]*Sqrt[2 + 2*x + x^2])]) - Sqrt[(-1 + Sqrt[5])/2]*ArcTanh[(2*Sqrt[5] + (5 - S
qrt[5])*x)/(Sqrt[10*(-1 + Sqrt[5])]*Sqrt[2 + 2*x + x^2])]
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Rubi [A] time = 0.354784, antiderivative size = 126, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16
\[ -\sqrt{\frac{1}{2} \left (1+\sqrt{5}\right )} \tan ^{-1}\left (\frac{2 \sqrt{5}-\left (5+\sqrt{5}\right ) x}{\sqrt{10 \left (1+\sqrt{5}\right )} \sqrt{x^2+2 x+2}}\right )-\sqrt{\frac{1}{2} \left (\sqrt{5}-1\right )} \tanh ^{-1}\left (\frac{\left (5-\sqrt{5}\right ) x+2 \sqrt{5}}{\sqrt{10 \left (\sqrt{5}-1\right )} \sqrt{x^2+2 x+2}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 + 2*x)/((1 + x^2)*Sqrt[2 + 2*x + x^2]),x]
[Out]
-(Sqrt[(1 + Sqrt[5])/2]*ArcTan[(2*Sqrt[5] - (5 + Sqrt[5])*x)/(Sqrt[10*(1 + Sqrt[
5])]*Sqrt[2 + 2*x + x^2])]) - Sqrt[(-1 + Sqrt[5])/2]*ArcTanh[(2*Sqrt[5] + (5 - S
qrt[5])*x)/(Sqrt[10*(-1 + Sqrt[5])]*Sqrt[2 + 2*x + x^2])]
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Rubi in Sympy [A] time = 13.0668, size = 141, normalized size = 1.12 \[ \frac{\sqrt{10} \left (2 \sqrt{5} + 10\right ) \operatorname{atan}{\left (\frac{\sqrt{10} \left (x \left (\sqrt{5} + 5\right ) - 2 \sqrt{5}\right )}{10 \sqrt{1 + \sqrt{5}} \sqrt{x^{2} + 2 x + 2}} \right )}}{20 \sqrt{1 + \sqrt{5}}} - \frac{\sqrt{10} \left (- 2 \sqrt{5} + 10\right ) \operatorname{atanh}{\left (\frac{\sqrt{10} \left (x \left (- \sqrt{5} + 5\right ) + 2 \sqrt{5}\right )}{10 \sqrt{-1 + \sqrt{5}} \sqrt{x^{2} + 2 x + 2}} \right )}}{20 \sqrt{-1 + \sqrt{5}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((1+2*x)/(x**2+1)/(x**2+2*x+2)**(1/2),x)
[Out]
sqrt(10)*(2*sqrt(5) + 10)*atan(sqrt(10)*(x*(sqrt(5) + 5) - 2*sqrt(5))/(10*sqrt(1
+ sqrt(5))*sqrt(x**2 + 2*x + 2)))/(20*sqrt(1 + sqrt(5))) - sqrt(10)*(-2*sqrt(5)
+ 10)*atanh(sqrt(10)*(x*(-sqrt(5) + 5) + 2*sqrt(5))/(10*sqrt(-1 + sqrt(5))*sqrt
(x**2 + 2*x + 2)))/(20*sqrt(-1 + sqrt(5)))
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Mathematica [C] time = 0.688957, size = 433, normalized size = 3.44 \[ \frac{1}{4} \left (i \left (\left (\sqrt{1-2 i}-\sqrt{1+2 i}\right ) \log \left (x^2+1\right )-\sqrt{1-2 i} \log \left ((3-2 i) x^2+2 \sqrt{1-2 i} \sqrt{x^2+2 x+2} x+4 \sqrt{1-2 i} \sqrt{x^2+2 x+2}+(8-4 i) x+(7-4 i)\right )+\sqrt{1+2 i} \log \left ((3+2 i) x^2+2 \sqrt{1+2 i} \sqrt{x^2+2 x+2} x+4 \sqrt{1+2 i} \sqrt{x^2+2 x+2}+(8+4 i) x+(7+4 i)\right )\right )+2 \sqrt{1+2 i} \tan ^{-1}\left (\frac{(-1+4 i) x^3+\left (5 \sqrt{1+2 i} \sqrt{x^2+2 x+2}-(2-13 i)\right ) x^2+(1+i) \left (5 \sqrt{1+2 i} \sqrt{x^2+2 x+2}+(9+5 i)\right ) x+5 i \sqrt{1+2 i} \sqrt{x^2+2 x+2}+(8+8 i)}{(-3-8 i) x^3+(4-11 i) x^2+(2+2 i) x+(4+14 i)}\right )+2 i \sqrt{1-2 i} \tanh ^{-1}\left (\frac{(1+4 i) x^3+\left ((2+13 i)-5 \sqrt{1-2 i} \sqrt{x^2+2 x+2}\right ) x^2+(1+i) \left (5 i \sqrt{1-2 i} \sqrt{x^2+2 x+2}+(5+9 i)\right ) x+5 i \sqrt{1-2 i} \sqrt{x^2+2 x+2}-(8-8 i)}{(8+3 i) x^3+(11-4 i) x^2-(2+2 i) x-(14+4 i)}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + 2*x)/((1 + x^2)*Sqrt[2 + 2*x + x^2]),x]
[Out]
(2*Sqrt[1 + 2*I]*ArcTan[((8 + 8*I) - (1 - 4*I)*x^3 + (5*I)*Sqrt[1 + 2*I]*Sqrt[2
+ 2*x + x^2] + x^2*((-2 + 13*I) + 5*Sqrt[1 + 2*I]*Sqrt[2 + 2*x + x^2]) + (1 + I)
*x*((9 + 5*I) + 5*Sqrt[1 + 2*I]*Sqrt[2 + 2*x + x^2]))/((4 + 14*I) + (2 + 2*I)*x
+ (4 - 11*I)*x^2 - (3 + 8*I)*x^3)] + (2*I)*Sqrt[1 - 2*I]*ArcTanh[((-8 + 8*I) + (
1 + 4*I)*x^3 + (5*I)*Sqrt[1 - 2*I]*Sqrt[2 + 2*x + x^2] + x^2*((2 + 13*I) - 5*Sqr
t[1 - 2*I]*Sqrt[2 + 2*x + x^2]) + (1 + I)*x*((5 + 9*I) + (5*I)*Sqrt[1 - 2*I]*Sqr
t[2 + 2*x + x^2]))/((-14 - 4*I) - (2 + 2*I)*x + (11 - 4*I)*x^2 + (8 + 3*I)*x^3)]
+ I*((Sqrt[1 - 2*I] - Sqrt[1 + 2*I])*Log[1 + x^2] - Sqrt[1 - 2*I]*Log[(7 - 4*I)
+ (8 - 4*I)*x + (3 - 2*I)*x^2 + 4*Sqrt[1 - 2*I]*Sqrt[2 + 2*x + x^2] + 2*Sqrt[1
- 2*I]*x*Sqrt[2 + 2*x + x^2]] + Sqrt[1 + 2*I]*Log[(7 + 4*I) + (8 + 4*I)*x + (3 +
2*I)*x^2 + 4*Sqrt[1 + 2*I]*Sqrt[2 + 2*x + x^2] + 2*Sqrt[1 + 2*I]*x*Sqrt[2 + 2*x
+ x^2]]))/4
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Maple [B] time = 0.143, size = 753, normalized size = 6. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((1+2*x)/(x^2+1)/(x^2+2*x+2)^(1/2),x)
[Out]
-1/2*(10*(-1/2*5^(1/2)+1/2+x)^2/(-1/2*5^(1/2)-1/2-x)^2-2*5^(1/2)*(-1/2*5^(1/2)+1
/2+x)^2/(-1/2*5^(1/2)-1/2-x)^2+10+2*5^(1/2))^(1/2)*(5*arctan(1/80*(-1/2*5^(1/2)+
1/2+x)/(-1/2*5^(1/2)-1/2-x)*(-5+5^(1/2))*(-22+10*5^(1/2))^(1/2)*((5-5^(1/2))*(2*
(-1/2*5^(1/2)+1/2+x)^2/(-1/2*5^(1/2)-1/2-x)^2+5^(1/2)+3))^(1/2)*(11*5^(1/2)*(-1/
2*5^(1/2)+1/2+x)^2/(-1/2*5^(1/2)-1/2-x)^2+25*(-1/2*5^(1/2)+1/2+x)^2/(-1/2*5^(1/2
)-1/2-x)^2+4*5^(1/2)+10)/((-1/2*5^(1/2)+1/2+x)^4/(-1/2*5^(1/2)-1/2-x)^4+3*(-1/2*
5^(1/2)+1/2+x)^2/(-1/2*5^(1/2)-1/2-x)^2+1))*(-10+10*5^(1/2))^(1/2)*(-22+10*5^(1/
2))^(1/2)+3*arctan(1/80*(-1/2*5^(1/2)+1/2+x)/(-1/2*5^(1/2)-1/2-x)*(-5+5^(1/2))*(
-22+10*5^(1/2))^(1/2)*((5-5^(1/2))*(2*(-1/2*5^(1/2)+1/2+x)^2/(-1/2*5^(1/2)-1/2-x
)^2+5^(1/2)+3))^(1/2)*(11*5^(1/2)*(-1/2*5^(1/2)+1/2+x)^2/(-1/2*5^(1/2)-1/2-x)^2+
25*(-1/2*5^(1/2)+1/2+x)^2/(-1/2*5^(1/2)-1/2-x)^2+4*5^(1/2)+10)/((-1/2*5^(1/2)+1/
2+x)^4/(-1/2*5^(1/2)-1/2-x)^4+3*(-1/2*5^(1/2)+1/2+x)^2/(-1/2*5^(1/2)-1/2-x)^2+1)
)*5^(1/2)*(-10+10*5^(1/2))^(1/2)*(-22+10*5^(1/2))^(1/2)+20*arctanh((10*(-1/2*5^(
1/2)+1/2+x)^2/(-1/2*5^(1/2)-1/2-x)^2-2*5^(1/2)*(-1/2*5^(1/2)+1/2+x)^2/(-1/2*5^(1
/2)-1/2-x)^2+10+2*5^(1/2))^(1/2)/(-10+10*5^(1/2))^(1/2))*5^(1/2)-60*arctanh((10*
(-1/2*5^(1/2)+1/2+x)^2/(-1/2*5^(1/2)-1/2-x)^2-2*5^(1/2)*(-1/2*5^(1/2)+1/2+x)^2/(
-1/2*5^(1/2)-1/2-x)^2+10+2*5^(1/2))^(1/2)/(-10+10*5^(1/2))^(1/2)))/(-2*(5^(1/2)*
(-1/2*5^(1/2)+1/2+x)^2/(-1/2*5^(1/2)-1/2-x)^2-5*(-1/2*5^(1/2)+1/2+x)^2/(-1/2*5^(
1/2)-1/2-x)^2-5^(1/2)-5)/(1+(-1/2*5^(1/2)+1/2+x)/(-1/2*5^(1/2)-1/2-x))^2)^(1/2)/
(1+(-1/2*5^(1/2)+1/2+x)/(-1/2*5^(1/2)-1/2-x))/(-5+5^(1/2))/(-10+10*5^(1/2))^(1/2
)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x + 1}{\sqrt{x^{2} + 2 \, x + 2}{\left (x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 1)/(sqrt(x^2 + 2*x + 2)*(x^2 + 1)),x, algorithm="maxima")
[Out]
integrate((2*x + 1)/(sqrt(x^2 + 2*x + 2)*(x^2 + 1)), x)
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Fricas [A] time = 0.31703, size = 1114, normalized size = 8.84 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 1)/(sqrt(x^2 + 2*x + 2)*(x^2 + 1)),x, algorithm="fricas")
[Out]
1/2*(5^(1/4)*(sqrt(5) - 1)*log(-1/5*(70*x^2 + 2*5^(1/4)*(sqrt(5)*(7*x + 11) - 15
*x - 25)*sqrt((sqrt(5) - 5)/(sqrt(5) - 3)) - 15*sqrt(5)*(2*x^2 + 2*x + 3) - 2*sq
rt(x^2 + 2*x + 2)*(5^(1/4)*(7*sqrt(5) - 15)*sqrt((sqrt(5) - 5)/(sqrt(5) - 3)) -
15*sqrt(5)*x + 35*x) - 5*sqrt(5)*(3*sqrt(5) - 7) + 70*x + 105)/(3*sqrt(5) - 7))
- 5^(1/4)*(sqrt(5) - 1)*log(-1/5*(70*x^2 - 2*5^(1/4)*(sqrt(5)*(7*x + 11) - 15*x
- 25)*sqrt((sqrt(5) - 5)/(sqrt(5) - 3)) - 15*sqrt(5)*(2*x^2 + 2*x + 3) + 2*sqrt(
x^2 + 2*x + 2)*(5^(1/4)*(7*sqrt(5) - 15)*sqrt((sqrt(5) - 5)/(sqrt(5) - 3)) + 15*
sqrt(5)*x - 35*x) - 5*sqrt(5)*(3*sqrt(5) - 7) + 70*x + 105)/(3*sqrt(5) - 7)) - 8
*5^(1/4)*arctan(((sqrt(5) - 1)*sqrt((sqrt(5) - 5)/(sqrt(5) - 3)) + 5^(1/4)*(sqrt
(5) - 1))/(sqrt(1/5)*(sqrt(5) - 1)*sqrt(-(70*x^2 - 2*5^(1/4)*(sqrt(5)*(7*x + 11)
- 15*x - 25)*sqrt((sqrt(5) - 5)/(sqrt(5) - 3)) - 15*sqrt(5)*(2*x^2 + 2*x + 3) +
2*sqrt(x^2 + 2*x + 2)*(5^(1/4)*(7*sqrt(5) - 15)*sqrt((sqrt(5) - 5)/(sqrt(5) - 3
)) + 15*sqrt(5)*x - 35*x) - 5*sqrt(5)*(3*sqrt(5) - 7) + 70*x + 105)/(3*sqrt(5) -
7))*sqrt((sqrt(5) - 5)/(sqrt(5) - 3)) + sqrt(x^2 + 2*x + 2)*(sqrt(5) - 1)*sqrt(
(sqrt(5) - 5)/(sqrt(5) - 3)) - (sqrt(5)*x - x)*sqrt((sqrt(5) - 5)/(sqrt(5) - 3))
+ 2*5^(1/4))) + 8*5^(1/4)*arctan(((sqrt(5) - 1)*sqrt((sqrt(5) - 5)/(sqrt(5) - 3
)) - 5^(1/4)*(sqrt(5) - 1))/(sqrt(1/5)*(sqrt(5) - 1)*sqrt(-(70*x^2 + 2*5^(1/4)*(
sqrt(5)*(7*x + 11) - 15*x - 25)*sqrt((sqrt(5) - 5)/(sqrt(5) - 3)) - 15*sqrt(5)*(
2*x^2 + 2*x + 3) - 2*sqrt(x^2 + 2*x + 2)*(5^(1/4)*(7*sqrt(5) - 15)*sqrt((sqrt(5)
- 5)/(sqrt(5) - 3)) - 15*sqrt(5)*x + 35*x) - 5*sqrt(5)*(3*sqrt(5) - 7) + 70*x +
105)/(3*sqrt(5) - 7))*sqrt((sqrt(5) - 5)/(sqrt(5) - 3)) + sqrt(x^2 + 2*x + 2)*(
sqrt(5) - 1)*sqrt((sqrt(5) - 5)/(sqrt(5) - 3)) - (sqrt(5)*x - x)*sqrt((sqrt(5) -
5)/(sqrt(5) - 3)) - 2*5^(1/4))))/((sqrt(5) - 1)*sqrt((sqrt(5) - 5)/(sqrt(5) - 3
)))
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{2 x + 1}{\left (x^{2} + 1\right ) \sqrt{x^{2} + 2 x + 2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((1+2*x)/(x**2+1)/(x**2+2*x+2)**(1/2),x)
[Out]
Integral((2*x + 1)/((x**2 + 1)*sqrt(x**2 + 2*x + 2)), x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x + 1}{\sqrt{x^{2} + 2 \, x + 2}{\left (x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2*x + 1)/(sqrt(x^2 + 2*x + 2)*(x^2 + 1)),x, algorithm="giac")
[Out]
integrate((2*x + 1)/(sqrt(x^2 + 2*x + 2)*(x^2 + 1)), x)