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

Optimal. Leaf size=47 \[ \frac{\sqrt{1-x^2} x}{x^2+1}+2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.020665, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {527, 12, 377, 203} \[ \frac{\sqrt{1-x^2} x}{x^2+1}+2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 527

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> -Simp[
((b*e - a*f)*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*n*(b*c - a*d)*(p + 1)), x] + Dist[1/(a*n*(b*c - a*d
)*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)
*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 203

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

Rubi steps

\begin{align*} \int \frac{5+x^2}{\sqrt{1-x^2} \left (1+x^2\right )^2} \, dx &=\frac{x \sqrt{1-x^2}}{1+x^2}-\frac{1}{4} \int -\frac{16}{\sqrt{1-x^2} \left (1+x^2\right )} \, dx\\ &=\frac{x \sqrt{1-x^2}}{1+x^2}+4 \int \frac{1}{\sqrt{1-x^2} \left (1+x^2\right )} \, dx\\ &=\frac{x \sqrt{1-x^2}}{1+x^2}+4 \operatorname{Subst}\left (\int \frac{1}{1+2 x^2} \, dx,x,\frac{x}{\sqrt{1-x^2}}\right )\\ &=\frac{x \sqrt{1-x^2}}{1+x^2}+2 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.148822, size = 85, normalized size = 1.81 \[ \frac{\sqrt{1-x^2} x}{x^2+1}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} x}{\sqrt{1-x^2}}\right )}{\sqrt{2}}+\frac{3 x \tanh ^{-1}\left (\sqrt{2} \sqrt{\frac{x^2}{x^2-1}}\right )}{\sqrt{2} \sqrt{-x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[1 - x^2])/(1 + x^2) + ArcTan[(Sqrt[2]*x)/Sqrt[1 - x^2]]/Sqrt[2] + (3*x*ArcTanh[Sqrt[2]*Sqrt[x^2/(-1 +
x^2)]])/(Sqrt[2]*Sqrt[-x^2])

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Maple [A]  time = 0.026, size = 70, normalized size = 1.5 \begin{align*} -2\,\sqrt{2}\arctan \left ({\frac{\sqrt{2}\sqrt{-{x}^{2}+1}x}{{x}^{2}-1}} \right ) -{\frac{x}{2\,{x}^{2}-2}\sqrt{-{x}^{2}+1} \left ({\frac{ \left ( -{x}^{2}+1 \right ){x}^{2}}{ \left ({x}^{2}-1 \right ) ^{2}}}+{\frac{1}{2}} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+5)/(x^2+1)^2/(-x^2+1)^(1/2),x)

[Out]

-2*2^(1/2)*arctan(2^(1/2)*(-x^2+1)^(1/2)/(x^2-1)*x)-1/2*(-x^2+1)^(1/2)/(x^2-1)*x/((-x^2+1)/(x^2-1)^2*x^2+1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.85211, size = 122, normalized size = 2.6 \begin{align*} -\frac{2 \, \sqrt{2}{\left (x^{2} + 1\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-x^{2} + 1}}{2 \, x}\right ) - \sqrt{-x^{2} + 1} x}{x^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*sqrt(2)*(x^2 + 1)*arctan(1/2*sqrt(2)*sqrt(-x^2 + 1)/x) - sqrt(-x^2 + 1)*x)/(x^2 + 1)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [B]  time = 1.08738, size = 166, normalized size = 3.53 \begin{align*} \sqrt{2}{\left (\pi \mathrm{sgn}\left (x\right ) + 2 \, \arctan \left (-\frac{\sqrt{2} x{\left (\frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}^{2}}{x^{2}} - 1\right )}}{4 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}}\right )\right )} - \frac{2 \,{\left (\frac{x}{\sqrt{-x^{2} + 1} - 1} - \frac{\sqrt{-x^{2} + 1} - 1}{x}\right )}}{{\left (\frac{x}{\sqrt{-x^{2} + 1} - 1} - \frac{\sqrt{-x^{2} + 1} - 1}{x}\right )}^{2} + 8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*(pi*sgn(x) + 2*arctan(-1/4*sqrt(2)*x*((sqrt(-x^2 + 1) - 1)^2/x^2 - 1)/(sqrt(-x^2 + 1) - 1))) - 2*(x/(s
qrt(-x^2 + 1) - 1) - (sqrt(-x^2 + 1) - 1)/x)/((x/(sqrt(-x^2 + 1) - 1) - (sqrt(-x^2 + 1) - 1)/x)^2 + 8)