3.13.97 \(\int \frac {-2+x}{\sqrt {-3+x} (-8+x^2)} \, dx\)

Optimal. Leaf size=45 \[ \frac {\tan ^{-1}\left (\left (\sqrt {2}-1\right ) \sqrt {x-3}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\left (1+\sqrt {2}\right ) \sqrt {x-3}\right )}{\sqrt {2}} \]

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Rubi [A]  time = 0.08, antiderivative size = 57, normalized size of antiderivative = 1.27, number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {827, 1163, 203} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {x-3}}{\sqrt {3-2 \sqrt {2}}}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {x-3}}{\sqrt {3+2 \sqrt {2}}}\right )}{\sqrt {2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(-2 + x)/(Sqrt[-3 + x]*(-8 + x^2)),x]

[Out]

ArcTan[Sqrt[-3 + x]/Sqrt[3 - 2*Sqrt[2]]]/Sqrt[2] + ArcTan[Sqrt[-3 + x]/Sqrt[3 + 2*Sqrt[2]]]/Sqrt[2]

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])

Rule 827

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

Rule 1163

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - a*e^2, 0] && GtQ[b^2
- 4*a*c, 0]

Rubi steps

\begin {align*} \int \frac {-2+x}{\sqrt {-3+x} \left (-8+x^2\right )} \, dx &=2 \operatorname {Subst}\left (\int \frac {1+x^2}{1+6 x^2+x^4} \, dx,x,\sqrt {-3+x}\right )\\ &=\frac {1}{2} \left (2-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{3-2 \sqrt {2}+x^2} \, dx,x,\sqrt {-3+x}\right )+\frac {1}{2} \left (2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{3+2 \sqrt {2}+x^2} \, dx,x,\sqrt {-3+x}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt {-3+x}}{\sqrt {3-2 \sqrt {2}}}\right )}{\sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {-3+x}}{\sqrt {3+2 \sqrt {2}}}\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 69, normalized size = 1.53 \begin {gather*} \frac {\left (2+\sqrt {2}\right ) \left (\tan ^{-1}\left (\sqrt {3-2 \sqrt {2}} \sqrt {x-3}\right )+\tan ^{-1}\left (\sqrt {3+2 \sqrt {2}} \sqrt {x-3}\right )\right )}{2 \sqrt {3+2 \sqrt {2}}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[(-2 + x)/(Sqrt[-3 + x]*(-8 + x^2)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.21, size = 37, normalized size = 0.82 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\frac {x-3}{2 \sqrt {2}}-\frac {1}{2 \sqrt {2}}}{\sqrt {x-3}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-2 + x)/(Sqrt[-3 + x]*(-8 + x^2)),x]

[Out]

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

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fricas [A]  time = 0.40, size = 19, normalized size = 0.42 \begin {gather*} \frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x - 4\right )}}{4 \, \sqrt {x - 3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+x)/(x^2-8)/(-3+x)^(1/2),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*arctan(1/4*sqrt(2)*(x - 4)/sqrt(x - 3))

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giac [A]  time = 0.18, size = 23, normalized size = 0.51 \begin {gather*} \frac {1}{4} \, \sqrt {2} {\left (\pi + 2 \, \arctan \left (\frac {\sqrt {2} {\left (x - 4\right )}}{4 \, \sqrt {x - 3}}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+x)/(x^2-8)/(-3+x)^(1/2),x, algorithm="giac")

[Out]

1/4*sqrt(2)*(pi + 2*arctan(1/4*sqrt(2)*(x - 4)/sqrt(x - 3)))

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maple [B]  time = 0.44, size = 119, normalized size = 2.64 \begin {gather*} \frac {2 \arctan \left (\frac {2 \sqrt {x -3}}{-2+2 \sqrt {2}}\right )}{-2+2 \sqrt {2}}-\frac {\sqrt {2}\, \arctan \left (\frac {2 \sqrt {x -3}}{-2+2 \sqrt {2}}\right )}{-2+2 \sqrt {2}}+\frac {\sqrt {2}\, \arctan \left (\frac {2 \sqrt {x -3}}{2+2 \sqrt {2}}\right )}{2+2 \sqrt {2}}+\frac {2 \arctan \left (\frac {2 \sqrt {x -3}}{2+2 \sqrt {2}}\right )}{2+2 \sqrt {2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x-2)/(x^2-8)/(x-3)^(1/2),x)

[Out]

2/(-2+2*2^(1/2))*arctan(2*(x-3)^(1/2)/(-2+2*2^(1/2)))-2^(1/2)/(-2+2*2^(1/2))*arctan(2*(x-3)^(1/2)/(-2+2*2^(1/2
)))+2^(1/2)/(2+2*2^(1/2))*arctan(2*(x-3)^(1/2)/(2+2*2^(1/2)))+2/(2+2*2^(1/2))*arctan(2*(x-3)^(1/2)/(2+2*2^(1/2
)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 2}{{\left (x^{2} - 8\right )} \sqrt {x - 3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+x)/(x^2-8)/(-3+x)^(1/2),x, algorithm="maxima")

[Out]

integrate((x - 2)/((x^2 - 8)*sqrt(x - 3)), x)

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mupad [B]  time = 2.04, size = 39, normalized size = 0.87 \begin {gather*} \frac {\sqrt {2}\,\left (\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {x-3}}{4}\right )+\mathrm {atan}\left (\frac {7\,\sqrt {2}\,\sqrt {x-3}}{4}+\frac {\sqrt {2}\,{\left (x-3\right )}^{3/2}}{4}\right )\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x - 2)/((x^2 - 8)*(x - 3)^(1/2)),x)

[Out]

(2^(1/2)*(atan((2^(1/2)*(x - 3)^(1/2))/4) + atan((7*2^(1/2)*(x - 3)^(1/2))/4 + (2^(1/2)*(x - 3)^(3/2))/4)))/2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x - 2}{\sqrt {x - 3} \left (x^{2} - 8\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2+x)/(x**2-8)/(-3+x)**(1/2),x)

[Out]

Integral((x - 2)/(sqrt(x - 3)*(x**2 - 8)), x)

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