3.31 \(\int \frac{1+x}{(4+2 x+x^2) \sqrt{5+2 x+x^2}} \, dx\)

Optimal. Leaf size=15 \[ -\tanh ^{-1}\left (\sqrt{x^2+2 x+5}\right ) \]

[Out]

-ArcTanh[Sqrt[5 + 2*x + x^2]]

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Rubi [A]  time = 0.016535, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {1024, 206} \[ -\tanh ^{-1}\left (\sqrt{x^2+2 x+5}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 + x)/((4 + 2*x + x^2)*Sqrt[5 + 2*x + x^2]),x]

[Out]

-ArcTanh[Sqrt[5 + 2*x + x^2]]

Rule 1024

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

Rule 206

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

Rubi steps

\begin{align*} \int \frac{1+x}{\left (4+2 x+x^2\right ) \sqrt{5+2 x+x^2}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{2-2 x^2} \, dx,x,\sqrt{5+2 x+x^2}\right )\right )\\ &=-\tanh ^{-1}\left (\sqrt{5+2 x+x^2}\right )\\ \end{align*}

Mathematica [C]  time = 0.0441422, size = 79, normalized size = 5.27 \[ \frac{1}{2} \left (-\tanh ^{-1}\left (\frac{-i \sqrt{3} x-i \sqrt{3}+4}{\sqrt{x^2+2 x+5}}\right )-\tanh ^{-1}\left (\frac{i \sqrt{3} x+i \sqrt{3}+4}{\sqrt{x^2+2 x+5}}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/((4 + 2*x + x^2)*Sqrt[5 + 2*x + x^2]),x]

[Out]

(-ArcTanh[(4 - I*Sqrt[3] - I*Sqrt[3]*x)/Sqrt[5 + 2*x + x^2]] - ArcTanh[(4 + I*Sqrt[3] + I*Sqrt[3]*x)/Sqrt[5 +
2*x + x^2]])/2

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Maple [A]  time = 0.046, size = 14, normalized size = 0.9 \begin{align*} -{\it Artanh} \left ( \sqrt{{x}^{2}+2\,x+5} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)/(x^2+2*x+4)/(x^2+2*x+5)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x + 1}{\sqrt{x^{2} + 2 \, x + 5}{\left (x^{2} + 2 \, x + 4\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^2+2*x+4)/(x^2+2*x+5)^(1/2),x, algorithm="maxima")

[Out]

integrate((x + 1)/(sqrt(x^2 + 2*x + 5)*(x^2 + 2*x + 4)), x)

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Fricas [B]  time = 1.32252, size = 136, normalized size = 9.07 \begin{align*} \frac{1}{2} \, \log \left (x^{2} - \sqrt{x^{2} + 2 \, x + 5}{\left (x + 2\right )} + 3 \, x + 6\right ) - \frac{1}{2} \, \log \left (x^{2} - \sqrt{x^{2} + 2 \, x + 5} x + x + 4\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^2+2*x+4)/(x^2+2*x+5)^(1/2),x, algorithm="fricas")

[Out]

1/2*log(x^2 - sqrt(x^2 + 2*x + 5)*(x + 2) + 3*x + 6) - 1/2*log(x^2 - sqrt(x^2 + 2*x + 5)*x + x + 4)

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Sympy [B]  time = 3.44338, size = 36, normalized size = 2.4 \begin{align*} \frac{\log{\left (-1 + \frac{1}{\sqrt{x^{2} + 2 x + 5}} \right )}}{2} - \frac{\log{\left (1 + \frac{1}{\sqrt{x^{2} + 2 x + 5}} \right )}}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x**2+2*x+4)/(x**2+2*x+5)**(1/2),x)

[Out]

log(-1 + 1/sqrt(x**2 + 2*x + 5))/2 - log(1 + 1/sqrt(x**2 + 2*x + 5))/2

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Giac [B]  time = 1.15753, size = 78, normalized size = 5.2 \begin{align*} \frac{1}{2} \, \log \left ({\left (x - \sqrt{x^{2} + 2 \, x + 5}\right )}^{2} + 4 \, x - 4 \, \sqrt{x^{2} + 2 \, x + 5} + 7\right ) - \frac{1}{2} \, \log \left ({\left (x - \sqrt{x^{2} + 2 \, x + 5}\right )}^{2} + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^2+2*x+4)/(x^2+2*x+5)^(1/2),x, algorithm="giac")

[Out]

1/2*log((x - sqrt(x^2 + 2*x + 5))^2 + 4*x - 4*sqrt(x^2 + 2*x + 5) + 7) - 1/2*log((x - sqrt(x^2 + 2*x + 5))^2 +
3)