3.3.35 \(\int \frac {1}{\sqrt {4+3 x-2 x^2}} \, dx\) [235]

Optimal. Leaf size=19 \[ -\frac {\sin ^{-1}\left (\frac {3-4 x}{\sqrt {41}}\right )}{\sqrt {2}} \]

[Out]

-1/2*arcsin(1/41*(3-4*x)*41^(1/2))*2^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {633, 222} \begin {gather*} -\frac {\text {ArcSin}\left (\frac {3-4 x}{\sqrt {41}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[4 + 3*x - 2*x^2],x]

[Out]

-(ArcSin[(3 - 4*x)/Sqrt[41]]/Sqrt[2])

Rule 222

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

Rule 633

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*(-4*(c/(b^2 - 4*a*c)))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {4+3 x-2 x^2}} \, dx &=-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{41}}} \, dx,x,3-4 x\right )}{\sqrt {82}}\\ &=-\frac {\sin ^{-1}\left (\frac {3-4 x}{\sqrt {41}}\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 32, normalized size = 1.68 \begin {gather*} \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} x}{-2+\sqrt {4+3 x-2 x^2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[4 + 3*x - 2*x^2],x]

[Out]

Sqrt[2]*ArcTan[(Sqrt[2]*x)/(-2 + Sqrt[4 + 3*x - 2*x^2])]

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Maple [A]
time = 0.14, size = 15, normalized size = 0.79

method	result	size
default	\(\frac {\sqrt {2}\, \arcsin \left (\frac {4 \sqrt {41}\, \left (x -\frac {3}{4}\right )}{41}\right )}{2}\)	\(15\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+2\right ) \ln \left (-4 \RootOf \left (\textit {\_Z}^{2}+2\right ) x +4 \sqrt {-2 x^{2}+3 x +4}+3 \RootOf \left (\textit {\_Z}^{2}+2\right )\right )}{2}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-2*x^2+3*x+4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*2^(1/2)*arcsin(4/41*41^(1/2)*(x-3/4))

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Maxima [A]
time = 4.39, size = 16, normalized size = 0.84 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arcsin \left (-\frac {1}{41} \, \sqrt {41} {\left (4 \, x - 3\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^2+3*x+4)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(2)*arcsin(-1/41*sqrt(41)*(4*x - 3))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (16) = 32\).
time = 0.39, size = 33, normalized size = 1.74 \begin {gather*} -\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-2 \, x^{2} + 3 \, x + 4} - 2 \, \sqrt {2}}{2 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^2+3*x+4)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- 2 x^{2} + 3 x + 4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x**2+3*x+4)**(1/2),x)

[Out]

Integral(1/sqrt(-2*x**2 + 3*x + 4), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (16) = 32\).
time = 0.46, size = 36, normalized size = 1.89 \begin {gather*} \frac {1}{8} \, \sqrt {-2 \, x^{2} + 3 \, x + 4} {\left (4 \, x - 3\right )} + \frac {41}{32} \, \sqrt {2} \arcsin \left (\frac {1}{41} \, \sqrt {41} {\left (4 \, x - 3\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-2*x^2+3*x+4)^(1/2),x, algorithm="giac")

[Out]

1/8*sqrt(-2*x^2 + 3*x + 4)*(4*x - 3) + 41/32*sqrt(2)*arcsin(1/41*sqrt(41)*(4*x - 3))

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Mupad [B]
time = 0.20, size = 16, normalized size = 0.84 \begin {gather*} \frac {\sqrt {2}\,\mathrm {asin}\left (\frac {\sqrt {41}\,\left (4\,x-3\right )}{41}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x - 2*x^2 + 4)^(1/2),x)

[Out]

(2^(1/2)*asin((41^(1/2)*(4*x - 3))/41))/2

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