3.3.40 \(\int \frac {x}{(3-x^2) \sqrt {5-x^2}} \, dx\) [240]

Optimal. Leaf size=24 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {455, 65, 213} \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/((3 - x^2)*Sqrt[5 - x^2]),x]

[Out]

ArcTanh[Sqrt[5 - x^2]/Sqrt[2]]/Sqrt[2]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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 455

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

Rubi steps

\begin {align*} \int \frac {x}{\left (3-x^2\right ) \sqrt {5-x^2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{(3-x) \sqrt {5-x}} \, dx,x,x^2\right )\\ &=-\text {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {5-x^2}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 24, normalized size = 1.00 \begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {5-x^2}}{\sqrt {2}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/((3 - x^2)*Sqrt[5 - x^2]),x]

[Out]

ArcTanh[Sqrt[5 - x^2]/Sqrt[2]]/Sqrt[2]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs. \(2(20)=40\).
time = 0.12, size = 100, normalized size = 4.17

method	result	size
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}-2\right ) x^{2}-7 \RootOf \left (\textit {\_Z}^{2}-2\right )-4 \sqrt {-x^{2}+5}}{x^{2}-3}\right )}{4}\)	\(48\)
default	\(\frac {\sqrt {2}\, \arctanh \left (\frac {\left (4-2 \sqrt {3}\, \left (x -\sqrt {3}\right )\right ) \sqrt {2}}{4 \sqrt {-\left (x -\sqrt {3}\right )^{2}-2 \sqrt {3}\, \left (x -\sqrt {3}\right )+2}}\right )}{4}+\frac {\sqrt {2}\, \arctanh \left (\frac {\left (4+2 \sqrt {3}\, \left (x +\sqrt {3}\right )\right ) \sqrt {2}}{4 \sqrt {-\left (x +\sqrt {3}\right )^{2}+2 \sqrt {3}\, \left (x +\sqrt {3}\right )+2}}\right )}{4}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(-x^2+3)/(-x^2+5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (20) = 40\).
time = 2.28, size = 112, normalized size = 4.67 \begin {gather*} \frac {1}{12} \, \sqrt {3} {\left (\sqrt {3} \sqrt {2} \log \left (\sqrt {3} + \frac {2 \, \sqrt {2} \sqrt {-x^{2} + 5}}{{\left | 2 \, x + 2 \, \sqrt {3} \right |}} + \frac {4}{{\left | 2 \, x + 2 \, \sqrt {3} \right |}}\right ) + \sqrt {3} \sqrt {2} \log \left (-\sqrt {3} + \frac {2 \, \sqrt {2} \sqrt {-x^{2} + 5}}{{\left | 2 \, x - 2 \, \sqrt {3} \right |}} + \frac {4}{{\left | 2 \, x - 2 \, \sqrt {3} \right |}}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+3)/(-x^2+5)^(1/2),x, algorithm="maxima")

[Out]

1/12*sqrt(3)*(sqrt(3)*sqrt(2)*log(sqrt(3) + 2*sqrt(2)*sqrt(-x^2 + 5)/abs(2*x + 2*sqrt(3)) + 4/abs(2*x + 2*sqrt
(3))) + sqrt(3)*sqrt(2)*log(-sqrt(3) + 2*sqrt(2)*sqrt(-x^2 + 5)/abs(2*x - 2*sqrt(3)) + 4/abs(2*x - 2*sqrt(3)))
)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
time = 0.37, size = 48, normalized size = 2.00 \begin {gather*} \frac {1}{8} \, \sqrt {2} \log \left (\frac {x^{4} - 4 \, \sqrt {2} {\left (x^{2} - 7\right )} \sqrt {-x^{2} + 5} - 22 \, x^{2} + 89}{x^{4} - 6 \, x^{2} + 9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+3)/(-x^2+5)^(1/2),x, algorithm="fricas")

[Out]

1/8*sqrt(2)*log((x^4 - 4*sqrt(2)*(x^2 - 7)*sqrt(-x^2 + 5) - 22*x^2 + 89)/(x^4 - 6*x^2 + 9))

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Sympy [A]
time = 2.33, size = 61, normalized size = 2.54 \begin {gather*} - \begin {cases} - \frac {\sqrt {2} \operatorname {acoth}{\left (\frac {\sqrt {2}}{\sqrt {5 - x^{2}}} \right )}}{2} & \text {for}\: \frac {1}{5 - x^{2}} > \frac {1}{2} \\- \frac {\sqrt {2} \operatorname {atanh}{\left (\frac {\sqrt {2}}{\sqrt {5 - x^{2}}} \right )}}{2} & \text {for}\: \frac {1}{5 - x^{2}} < \frac {1}{2} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x**2+3)/(-x**2+5)**(1/2),x)

[Out]

-Piecewise((-sqrt(2)*acoth(sqrt(2)/sqrt(5 - x**2))/2, 1/(5 - x**2) > 1/2), (-sqrt(2)*atanh(sqrt(2)/sqrt(5 - x*
*2))/2, 1/(5 - x**2) < 1/2))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
time = 0.50, size = 42, normalized size = 1.75 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (\sqrt {2} + \sqrt {-x^{2} + 5}\right ) - \frac {1}{4} \, \sqrt {2} \log \left ({\left | -\sqrt {2} + \sqrt {-x^{2} + 5} \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/(-x^2+3)/(-x^2+5)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 0.79, size = 78, normalized size = 3.25 \begin {gather*} \frac {\sqrt {2}\,\left (\ln \left (\frac {\frac {\sqrt {2}\,\left (\sqrt {3}\,x+5\right )\,1{}\mathrm {i}}{2}+\sqrt {5-x^2}\,1{}\mathrm {i}}{x+\sqrt {3}}\right )+\ln \left (\frac {\frac {\sqrt {2}\,\left (\sqrt {3}\,x-5\right )\,1{}\mathrm {i}}{2}-\sqrt {5-x^2}\,1{}\mathrm {i}}{x-\sqrt {3}}\right )\right )}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x/((x^2 - 3)*(5 - x^2)^(1/2)),x)

[Out]

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

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