3.10.98 \(\int \frac {1+2 x}{(1+x^2) \sqrt {2+2 x+x^2}} \, dx\) [998]

Optimal. Leaf size=126 \[ -\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {2 \sqrt {5}-\left (5+\sqrt {5}\right ) x}{\sqrt {10 \left (1+\sqrt {5}\right )} \sqrt {2+2 x+x^2}}\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2 \sqrt {5}+\left (5-\sqrt {5}\right ) x}{\sqrt {10 \left (-1+\sqrt {5}\right )} \sqrt {2+2 x+x^2}}\right ) \]

[Out]

-1/2*arctanh((x*(5-5^(1/2))+2*5^(1/2))/(x^2+2*x+2)^(1/2)/(-10+10*5^(1/2))^(1/2))*(-2+2*5^(1/2))^(1/2)-1/2*arct
an((2*5^(1/2)-x*(5+5^(1/2)))/(x^2+2*x+2)^(1/2)/(10+10*5^(1/2))^(1/2))*(2+2*5^(1/2))^(1/2)

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Rubi [A]
time = 0.11, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {1050, 1044, 213, 209} \begin {gather*} -\sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \text {ArcTan}\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 ) \end {gather*}

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 - Sqrt[5])*x)/(Sqrt[10*(-1 + Sqrt[5])]*Sqrt[2 + 2*x + x^2])]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 1044

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*
a*g*h, Subst[Int[1/Simp[2*a^2*g*h*c + a*e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; F
reeQ[{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]

Rule 1050

Int[((g_.) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Dist[1/(2*q), Int[Simp[(-a)*h*e - g*(c*d - a*f - q) + (h*(c*d - a*f + q) -
g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[(-a)*h*e - g*(c*d - a*f + q
) + (h*(c*d - a*f - q) - g*c*e)*x, x]/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f, g,
h}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]

Rubi steps

\begin {align*} \int \frac {1+2 x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx &=-\frac {\int \frac {-5-\sqrt {5}-2 \sqrt {5} x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx}{2 \sqrt {5}}+\frac {\int \frac {-5+\sqrt {5}+2 \sqrt {5} x}{\left (1+x^2\right ) \sqrt {2+2 x+x^2}} \, dx}{2 \sqrt {5}}\\ &=\left (2 \left (5-\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{20 \left (1-\sqrt {5}\right )+2 x^2} \, dx,x,\frac {2 \sqrt {5}+\left (5-\sqrt {5}\right ) x}{\sqrt {2+2 x+x^2}}\right )+\left (2 \left (5+\sqrt {5}\right )\right ) \text {Subst}\left (\int \frac {1}{20 \left (1+\sqrt {5}\right )+2 x^2} \, dx,x,\frac {-2 \sqrt {5}+\left (5+\sqrt {5}\right ) x}{\sqrt {2+2 x+x^2}}\right )\\ &=-\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 {2+2 x+x^2}}\right )-\sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {2 \sqrt {5}+\left (5-\sqrt {5}\right ) x}{\sqrt {10 \left (-1+\sqrt {5}\right )} \sqrt {2+2 x+x^2}}\right )\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 0.13, size = 97, normalized size = 0.77 \begin {gather*} \text {RootSum}\left [8-8 \text {$\#$1}+\text {$\#$1}^4\&,\frac {-\log \left (-x+\sqrt {2+2 x+x^2}-\text {$\#$1}\right )-\log \left (-x+\sqrt {2+2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-x+\sqrt {2+2 x+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-2+\text {$\#$1}^3}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

RootSum[8 - 8*#1 + #1^4 & , (-Log[-x + Sqrt[2 + 2*x + x^2] - #1] - Log[-x + Sqrt[2 + 2*x + x^2] - #1]*#1 + Log
[-x + Sqrt[2 + 2*x + x^2] - #1]*#1^2)/(-2 + #1^3) & ]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(752\) vs. \(2(94)=188\).
time = 0.77, size = 753, normalized size = 5.98 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x+1)/(x^2+1)/(x^2+2*x+2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-1/2*(10*(x-1/2*5^(1/2)+1/2)^2/(-1/2*5^(1/2)-1/2-x)^2-2*5^(1/2)*(x-1/2*5^(1/2)+1/2)^2/(-1/2*5^(1/2)-1/2-x)^2+1
0+2*5^(1/2))^(1/2)*(3*(-22+10*5^(1/2))^(1/2)*(-10+10*5^(1/2))^(1/2)*arctan(1/80*(-22+10*5^(1/2))^(1/2)*((5-5^(
1/2))*(2*(x-1/2*5^(1/2)+1/2)^2/(-1/2*5^(1/2)-1/2-x)^2+5^(1/2)+3))^(1/2)*(11*5^(1/2)*(x-1/2*5^(1/2)+1/2)^2/(-1/
2*5^(1/2)-1/2-x)^2+25*(x-1/2*5^(1/2)+1/2)^2/(-1/2*5^(1/2)-1/2-x)^2+4*5^(1/2)+10)*(x-1/2*5^(1/2)+1/2)/(-1/2*5^(
1/2)-1/2-x)*(5^(1/2)-5)/((x-1/2*5^(1/2)+1/2)^4/(-1/2*5^(1/2)-1/2-x)^4+3*(x-1/2*5^(1/2)+1/2)^2/(-1/2*5^(1/2)-1/
2-x)^2+1))*5^(1/2)+5*(-22+10*5^(1/2))^(1/2)*(-10+10*5^(1/2))^(1/2)*arctan(1/80*(-22+10*5^(1/2))^(1/2)*((5-5^(1
/2))*(2*(x-1/2*5^(1/2)+1/2)^2/(-1/2*5^(1/2)-1/2-x)^2+5^(1/2)+3))^(1/2)*(11*5^(1/2)*(x-1/2*5^(1/2)+1/2)^2/(-1/2
*5^(1/2)-1/2-x)^2+25*(x-1/2*5^(1/2)+1/2)^2/(-1/2*5^(1/2)-1/2-x)^2+4*5^(1/2)+10)*(x-1/2*5^(1/2)+1/2)/(-1/2*5^(1
/2)-1/2-x)*(5^(1/2)-5)/((x-1/2*5^(1/2)+1/2)^4/(-1/2*5^(1/2)-1/2-x)^4+3*(x-1/2*5^(1/2)+1/2)^2/(-1/2*5^(1/2)-1/2
-x)^2+1))+20*arctanh((10*(x-1/2*5^(1/2)+1/2)^2/(-1/2*5^(1/2)-1/2-x)^2-2*5^(1/2)*(x-1/2*5^(1/2)+1/2)^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*(x-1/2*5^(1/2)+1/2)^2/(-1/2*
5^(1/2)-1/2-x)^2-2*5^(1/2)*(x-1/2*5^(1/2)+1/2)^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)*(x-1/2*5^(1/2)+1/2)^2/(-1/2*5^(1/2)-1/2-x)^2-5*(x-1/2*5^(1/2)+1/2)^2/(-1/2*5^(1/2)-1/2-x)^
2-5^(1/2)-5)/(1+(x-1/2*5^(1/2)+1/2)/(-1/2*5^(1/2)-1/2-x))^2)^(1/2)/(1+(x-1/2*5^(1/2)+1/2)/(-1/2*5^(1/2)-1/2-x)
)/(5^(1/2)-5)/(-10+10*5^(1/2))^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 770 vs. \(2 (93) = 186\).
time = 0.38, size = 770, normalized size = 6.11 \begin {gather*} \frac {1}{5} \cdot 5^{\frac {3}{4}} \sqrt {2} \sqrt {\sqrt {5} + 5} \arctan \left (\frac {1}{200} \, \sqrt {20 \, x^{2} - 20 \, \sqrt {x^{2} + 2 \, x + 2} x - {\left (2 \cdot 5^{\frac {3}{4}} \sqrt {2} \sqrt {x^{2} + 2 \, x + 2} - 5^{\frac {1}{4}} {\left (\sqrt {5} \sqrt {2} {\left (2 \, x + 1\right )} - 5 \, \sqrt {2}\right )}\right )} \sqrt {\sqrt {5} + 5} + 20 \, x + 10 \, \sqrt {5} + 30} {\left (\sqrt {10} {\left (5^{\frac {3}{4}} {\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} + 2 \cdot 5^{\frac {3}{4}} \sqrt {2}\right )} \sqrt {\sqrt {5} + 5} + 10 \, \sqrt {10} {\left (\sqrt {5} + 3\right )}\right )} + \frac {1}{10} \, \sqrt {5} {\left (\sqrt {5} {\left (2 \, x + 1\right )} + 5\right )} + \frac {1}{2} \, \sqrt {5} x + \frac {1}{20} \, {\left (5^{\frac {3}{4}} {\left (\sqrt {5} \sqrt {2} x - \sqrt {2} {\left (x - 2\right )}\right )} - \sqrt {x^{2} + 2 \, x + 2} {\left (5^{\frac {3}{4}} {\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} + 2 \cdot 5^{\frac {3}{4}} \sqrt {2}\right )} + 5^{\frac {1}{4}} {\left (\sqrt {5} \sqrt {2} {\left (2 \, x + 1\right )} + 5 \, \sqrt {2}\right )}\right )} \sqrt {\sqrt {5} + 5} - \frac {1}{2} \, \sqrt {x^{2} + 2 \, x + 2} {\left (\sqrt {5} + 3\right )} + \frac {1}{2} \, x + 1\right ) + \frac {1}{5} \cdot 5^{\frac {3}{4}} \sqrt {2} \sqrt {\sqrt {5} + 5} \arctan \left (\frac {1}{200} \, \sqrt {20 \, x^{2} - 20 \, \sqrt {x^{2} + 2 \, x + 2} x + {\left (2 \cdot 5^{\frac {3}{4}} \sqrt {2} \sqrt {x^{2} + 2 \, x + 2} - 5^{\frac {1}{4}} {\left (\sqrt {5} \sqrt {2} {\left (2 \, x + 1\right )} - 5 \, \sqrt {2}\right )}\right )} \sqrt {\sqrt {5} + 5} + 20 \, x + 10 \, \sqrt {5} + 30} {\left (\sqrt {10} {\left (5^{\frac {3}{4}} {\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} + 2 \cdot 5^{\frac {3}{4}} \sqrt {2}\right )} \sqrt {\sqrt {5} + 5} - 10 \, \sqrt {10} {\left (\sqrt {5} + 3\right )}\right )} - \frac {1}{10} \, \sqrt {5} {\left (\sqrt {5} {\left (2 \, x + 1\right )} + 5\right )} - \frac {1}{2} \, \sqrt {5} x + \frac {1}{20} \, {\left (5^{\frac {3}{4}} {\left (\sqrt {5} \sqrt {2} x - \sqrt {2} {\left (x - 2\right )}\right )} - \sqrt {x^{2} + 2 \, x + 2} {\left (5^{\frac {3}{4}} {\left (\sqrt {5} \sqrt {2} - \sqrt {2}\right )} + 2 \cdot 5^{\frac {3}{4}} \sqrt {2}\right )} + 5^{\frac {1}{4}} {\left (\sqrt {5} \sqrt {2} {\left (2 \, x + 1\right )} + 5 \, \sqrt {2}\right )}\right )} \sqrt {\sqrt {5} + 5} + \frac {1}{2} \, \sqrt {x^{2} + 2 \, x + 2} {\left (\sqrt {5} + 3\right )} - \frac {1}{2} \, x - 1\right ) + \frac {1}{40} \cdot 5^{\frac {1}{4}} {\left (\sqrt {5} \sqrt {2} - 5 \, \sqrt {2}\right )} \sqrt {\sqrt {5} + 5} \log \left (2 \, x^{2} - 2 \, \sqrt {x^{2} + 2 \, x + 2} x + \frac {1}{10} \, {\left (2 \cdot 5^{\frac {3}{4}} \sqrt {2} \sqrt {x^{2} + 2 \, x + 2} - 5^{\frac {1}{4}} {\left (\sqrt {5} \sqrt {2} {\left (2 \, x + 1\right )} - 5 \, \sqrt {2}\right )}\right )} \sqrt {\sqrt {5} + 5} + 2 \, x + \sqrt {5} + 3\right ) - \frac {1}{40} \cdot 5^{\frac {1}{4}} {\left (\sqrt {5} \sqrt {2} - 5 \, \sqrt {2}\right )} \sqrt {\sqrt {5} + 5} \log \left (2 \, x^{2} - 2 \, \sqrt {x^{2} + 2 \, x + 2} x - \frac {1}{10} \, {\left (2 \cdot 5^{\frac {3}{4}} \sqrt {2} \sqrt {x^{2} + 2 \, x + 2} - 5^{\frac {1}{4}} {\left (\sqrt {5} \sqrt {2} {\left (2 \, x + 1\right )} - 5 \, \sqrt {2}\right )}\right )} \sqrt {\sqrt {5} + 5} + 2 \, x + \sqrt {5} + 3\right ) \end {gather*}

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, algorithm="fricas")

[Out]

1/5*5^(3/4)*sqrt(2)*sqrt(sqrt(5) + 5)*arctan(1/200*sqrt(20*x^2 - 20*sqrt(x^2 + 2*x + 2)*x - (2*5^(3/4)*sqrt(2)
*sqrt(x^2 + 2*x + 2) - 5^(1/4)*(sqrt(5)*sqrt(2)*(2*x + 1) - 5*sqrt(2)))*sqrt(sqrt(5) + 5) + 20*x + 10*sqrt(5)
+ 30)*(sqrt(10)*(5^(3/4)*(sqrt(5)*sqrt(2) - sqrt(2)) + 2*5^(3/4)*sqrt(2))*sqrt(sqrt(5) + 5) + 10*sqrt(10)*(sqr
t(5) + 3)) + 1/10*sqrt(5)*(sqrt(5)*(2*x + 1) + 5) + 1/2*sqrt(5)*x + 1/20*(5^(3/4)*(sqrt(5)*sqrt(2)*x - sqrt(2)
*(x - 2)) - sqrt(x^2 + 2*x + 2)*(5^(3/4)*(sqrt(5)*sqrt(2) - sqrt(2)) + 2*5^(3/4)*sqrt(2)) + 5^(1/4)*(sqrt(5)*s
qrt(2)*(2*x + 1) + 5*sqrt(2)))*sqrt(sqrt(5) + 5) - 1/2*sqrt(x^2 + 2*x + 2)*(sqrt(5) + 3) + 1/2*x + 1) + 1/5*5^
(3/4)*sqrt(2)*sqrt(sqrt(5) + 5)*arctan(1/200*sqrt(20*x^2 - 20*sqrt(x^2 + 2*x + 2)*x + (2*5^(3/4)*sqrt(2)*sqrt(
x^2 + 2*x + 2) - 5^(1/4)*(sqrt(5)*sqrt(2)*(2*x + 1) - 5*sqrt(2)))*sqrt(sqrt(5) + 5) + 20*x + 10*sqrt(5) + 30)*
(sqrt(10)*(5^(3/4)*(sqrt(5)*sqrt(2) - sqrt(2)) + 2*5^(3/4)*sqrt(2))*sqrt(sqrt(5) + 5) - 10*sqrt(10)*(sqrt(5) +
3)) - 1/10*sqrt(5)*(sqrt(5)*(2*x + 1) + 5) - 1/2*sqrt(5)*x + 1/20*(5^(3/4)*(sqrt(5)*sqrt(2)*x - sqrt(2)*(x -
2)) - sqrt(x^2 + 2*x + 2)*(5^(3/4)*(sqrt(5)*sqrt(2) - sqrt(2)) + 2*5^(3/4)*sqrt(2)) + 5^(1/4)*(sqrt(5)*sqrt(2)
*(2*x + 1) + 5*sqrt(2)))*sqrt(sqrt(5) + 5) + 1/2*sqrt(x^2 + 2*x + 2)*(sqrt(5) + 3) - 1/2*x - 1) + 1/40*5^(1/4)
*(sqrt(5)*sqrt(2) - 5*sqrt(2))*sqrt(sqrt(5) + 5)*log(2*x^2 - 2*sqrt(x^2 + 2*x + 2)*x + 1/10*(2*5^(3/4)*sqrt(2)
*sqrt(x^2 + 2*x + 2) - 5^(1/4)*(sqrt(5)*sqrt(2)*(2*x + 1) - 5*sqrt(2)))*sqrt(sqrt(5) + 5) + 2*x + sqrt(5) + 3)
- 1/40*5^(1/4)*(sqrt(5)*sqrt(2) - 5*sqrt(2))*sqrt(sqrt(5) + 5)*log(2*x^2 - 2*sqrt(x^2 + 2*x + 2)*x - 1/10*(2*
5^(3/4)*sqrt(2)*sqrt(x^2 + 2*x + 2) - 5^(1/4)*(sqrt(5)*sqrt(2)*(2*x + 1) - 5*sqrt(2)))*sqrt(sqrt(5) + 5) + 2*x
+ sqrt(5) + 3)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 x + 1}{\left (x^{2} + 1\right ) \sqrt {x^{2} + 2 x + 2}}\, dx \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (93) = 186\).
time = 2.26, size = 444, normalized size = 3.52 \begin {gather*} \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (256 \, {\left (\sqrt {5} {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} - 2 \, x + \sqrt {5} \sqrt {\sqrt {5} - 2} + \sqrt {5} + 2 \, \sqrt {x^{2} + 2 \, x + 2} - 2 \, \sqrt {\sqrt {5} - 2} - 2\right )}^{2} + 256 \, {\left (\sqrt {5} {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} - 2 \, x - \sqrt {5} + 2 \, \sqrt {x^{2} + 2 \, x + 2} + \sqrt {\sqrt {5} - 2} + 2\right )}^{2}\right ) - \frac {1}{4} \, \sqrt {2 \, \sqrt {5} - 2} \log \left (256 \, {\left (\sqrt {5} {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} - 2 \, x - \sqrt {5} \sqrt {\sqrt {5} - 2} + \sqrt {5} + 2 \, \sqrt {x^{2} + 2 \, x + 2} + 2 \, \sqrt {\sqrt {5} - 2} - 2\right )}^{2} + 256 \, {\left (\sqrt {5} {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} - 2 \, x - \sqrt {5} + 2 \, \sqrt {x^{2} + 2 \, x + 2} - \sqrt {\sqrt {5} - 2} + 2\right )}^{2}\right ) + \frac {{\left (\pi + 4 \, \arctan \left (\frac {1}{2} \, {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} {\left (2 \, \sqrt {5} \sqrt {\sqrt {5} - 2} + \sqrt {5} + 4 \, \sqrt {\sqrt {5} - 2} + 3\right )} + \frac {3}{2} \, \sqrt {5} \sqrt {\sqrt {5} - 2} + \frac {1}{2} \, \sqrt {5} + \frac {7}{2} \, \sqrt {\sqrt {5} - 2} + \frac {3}{2}\right )\right )} \sqrt {2 \, \sqrt {5} - 2}}{4 \, {\left (\sqrt {5} - 1\right )}} - \frac {{\left (\pi + 4 \, \arctan \left (-\frac {1}{2} \, {\left (x - \sqrt {x^{2} + 2 \, x + 2}\right )} {\left (2 \, \sqrt {5} \sqrt {\sqrt {5} - 2} - \sqrt {5} + 4 \, \sqrt {\sqrt {5} - 2} - 3\right )} - \frac {3}{2} \, \sqrt {5} \sqrt {\sqrt {5} - 2} + \frac {1}{2} \, \sqrt {5} - \frac {7}{2} \, \sqrt {\sqrt {5} - 2} + \frac {3}{2}\right )\right )} \sqrt {2 \, \sqrt {5} - 2}}{4 \, {\left (\sqrt {5} - 1\right )}} \end {gather*}

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, algorithm="giac")

[Out]

1/4*sqrt(2*sqrt(5) - 2)*log(256*(sqrt(5)*(x - sqrt(x^2 + 2*x + 2)) - 2*x + sqrt(5)*sqrt(sqrt(5) - 2) + sqrt(5)
+ 2*sqrt(x^2 + 2*x + 2) - 2*sqrt(sqrt(5) - 2) - 2)^2 + 256*(sqrt(5)*(x - sqrt(x^2 + 2*x + 2)) - 2*x - sqrt(5)
+ 2*sqrt(x^2 + 2*x + 2) + sqrt(sqrt(5) - 2) + 2)^2) - 1/4*sqrt(2*sqrt(5) - 2)*log(256*(sqrt(5)*(x - sqrt(x^2
+ 2*x + 2)) - 2*x - sqrt(5)*sqrt(sqrt(5) - 2) + sqrt(5) + 2*sqrt(x^2 + 2*x + 2) + 2*sqrt(sqrt(5) - 2) - 2)^2 +
256*(sqrt(5)*(x - sqrt(x^2 + 2*x + 2)) - 2*x - sqrt(5) + 2*sqrt(x^2 + 2*x + 2) - sqrt(sqrt(5) - 2) + 2)^2) +
1/4*(pi + 4*arctan(1/2*(x - sqrt(x^2 + 2*x + 2))*(2*sqrt(5)*sqrt(sqrt(5) - 2) + sqrt(5) + 4*sqrt(sqrt(5) - 2)
+ 3) + 3/2*sqrt(5)*sqrt(sqrt(5) - 2) + 1/2*sqrt(5) + 7/2*sqrt(sqrt(5) - 2) + 3/2))*sqrt(2*sqrt(5) - 2)/(sqrt(5
) - 1) - 1/4*(pi + 4*arctan(-1/2*(x - sqrt(x^2 + 2*x + 2))*(2*sqrt(5)*sqrt(sqrt(5) - 2) - sqrt(5) + 4*sqrt(sqr
t(5) - 2) - 3) - 3/2*sqrt(5)*sqrt(sqrt(5) - 2) + 1/2*sqrt(5) - 7/2*sqrt(sqrt(5) - 2) + 3/2))*sqrt(2*sqrt(5) -
2)/(sqrt(5) - 1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {2\,x+1}{\left (x^2+1\right )\,\sqrt {x^2+2\,x+2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 1)/((x^2 + 1)*(2*x + x^2 + 2)^(1/2)),x)

[Out]

int((2*x + 1)/((x^2 + 1)*(2*x + x^2 + 2)^(1/2)), x)

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