3.2.34 \(\int \frac {\sqrt {-4+9 x^2}}{x} \, dx\) [134]

Optimal. Leaf size=30 \[ \sqrt {-4+9 x^2}-2 \tan ^{-1}\left (\frac {1}{2} \sqrt {-4+9 x^2}\right ) \]

[Out]

-2*arctan(1/2*(9*x^2-4)^(1/2))+(9*x^2-4)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {272, 52, 65, 209} \begin {gather*} \sqrt {9 x^2-4}-2 \tan ^{-1}\left (\frac {1}{2} \sqrt {9 x^2-4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[-4 + 9*x^2]/x,x]

[Out]

Sqrt[-4 + 9*x^2] - 2*ArcTan[Sqrt[-4 + 9*x^2]/2]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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 209

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

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\sqrt {-4+9 x^2}}{x} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {-4+9 x}}{x} \, dx,x,x^2\right )\\ &=\sqrt {-4+9 x^2}-2 \text {Subst}\left (\int \frac {1}{x \sqrt {-4+9 x}} \, dx,x,x^2\right )\\ &=\sqrt {-4+9 x^2}-\frac {4}{9} \text {Subst}\left (\int \frac {1}{\frac {4}{9}+\frac {x^2}{9}} \, dx,x,\sqrt {-4+9 x^2}\right )\\ &=\sqrt {-4+9 x^2}-2 \tan ^{-1}\left (\frac {1}{2} \sqrt {-4+9 x^2}\right )\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 30, normalized size = 1.00 \begin {gather*} \sqrt {-4+9 x^2}-2 \tan ^{-1}\left (\frac {1}{2} \sqrt {-4+9 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[-4 + 9*x^2]/x,x]

[Out]

Sqrt[-4 + 9*x^2] - 2*ArcTan[Sqrt[-4 + 9*x^2]/2]

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 2.78, size = 68, normalized size = 2.27 \begin {gather*} \text {Piecewise}\left [\left \{\left \{I \left (x \sqrt {-9+\frac {4}{x^2}}-2 \text {ArcCosh}\left [\frac {2}{3 x}\right ]\right ),\frac {1}{\text {Abs}\left [x^2\right ]}>\frac {9}{4}\right \}\right \},\frac {-4}{3 x \sqrt {1-\frac {4}{9 x^2}}}+\frac {3 x}{\sqrt {1-\frac {4}{9 x^2}}}+2 \text {ArcSin}\left [\frac {2}{3 x}\right ]\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[Sqrt[9*x^2 - 4]/x,x]')

[Out]

Piecewise[{{I (x Sqrt[-9 + 4 / x ^ 2] - 2 ArcCosh[2 / (3 x)]), 1 / Abs[x ^ 2] > 9 / 4}}, -4 / (3 x Sqrt[1 - 4
/ (9 x ^ 2)]) + 3 x / Sqrt[1 - 4 / (9 x ^ 2)] + 2 ArcSin[2 / (3 x)]]

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Maple [A]
time = 0.10, size = 25, normalized size = 0.83

method	result	size
default	\(\sqrt {9 x^{2}-4}+2 \arctan \left (\frac {2}{\sqrt {9 x^{2}-4}}\right )\)	\(25\)
trager	\(\sqrt {9 x^{2}-4}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\sqrt {9 x^{2}-4}-2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x}\right )\)	\(42\)
meijerg	\(-\frac {\sqrt {\mathrm {signum}\left (-1+\frac {9 x^{2}}{4}\right )}\, \left (-2 \left (2-4 \ln \left (2\right )+2 \ln \left (x \right )+2 \ln \left (3\right )+i \pi \right ) \sqrt {\pi }+4 \sqrt {\pi }-4 \sqrt {\pi }\, \sqrt {1-\frac {9 x^{2}}{4}}+4 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {1-\frac {9 x^{2}}{4}}}{2}\right )\right )}{2 \sqrt {\pi }\, \sqrt {-\mathrm {signum}\left (-1+\frac {9 x^{2}}{4}\right )}}\)	\(90\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x^2-4)^(1/2)/x,x,method=_RETURNVERBOSE)

[Out]

(9*x^2-4)^(1/2)+2*arctan(2/(9*x^2-4)^(1/2))

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Maxima [A]
time = 0.35, size = 19, normalized size = 0.63 \begin {gather*} \sqrt {9 \, x^{2} - 4} + 2 \, \arcsin \left (\frac {2}{3 \, {\left | x \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2-4)^(1/2)/x,x, algorithm="maxima")

[Out]

sqrt(9*x^2 - 4) + 2*arcsin(2/3/abs(x))

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Fricas [A]
time = 0.34, size = 28, normalized size = 0.93 \begin {gather*} \sqrt {9 \, x^{2} - 4} - 4 \, \arctan \left (-\frac {3}{2} \, x + \frac {1}{2} \, \sqrt {9 \, x^{2} - 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2-4)^(1/2)/x,x, algorithm="fricas")

[Out]

sqrt(9*x^2 - 4) - 4*arctan(-3/2*x + 1/2*sqrt(9*x^2 - 4))

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Sympy [A]
time = 0.67, size = 92, normalized size = 3.07 \begin {gather*} \begin {cases} - \frac {3 i x}{\sqrt {-1 + \frac {4}{9 x^{2}}}} - 2 i \operatorname {acosh}{\left (\frac {2}{3 x} \right )} + \frac {4 i}{3 x \sqrt {-1 + \frac {4}{9 x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > \frac {9}{4} \\\frac {3 x}{\sqrt {1 - \frac {4}{9 x^{2}}}} + 2 \operatorname {asin}{\left (\frac {2}{3 x} \right )} - \frac {4}{3 x \sqrt {1 - \frac {4}{9 x^{2}}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x**2-4)**(1/2)/x,x)

[Out]

Piecewise((-3*I*x/sqrt(-1 + 4/(9*x**2)) - 2*I*acosh(2/(3*x)) + 4*I/(3*x*sqrt(-1 + 4/(9*x**2))), 1/Abs(x**2) >
9/4), (3*x/sqrt(1 - 4/(9*x**2)) + 2*asin(2/(3*x)) - 4/(3*x*sqrt(1 - 4/(9*x**2))), True))

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Giac [A]
time = 0.00, size = 30, normalized size = 1.00 \begin {gather*} \sqrt {9 x^{2}-4}-2 \arctan \left (\frac {\sqrt {9 x^{2}-4}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((9*x^2-4)^(1/2)/x,x)

[Out]

sqrt(9*x^2 - 4) - 2*arctan(1/2*sqrt(9*x^2 - 4))

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Mupad [B]
time = 0.31, size = 24, normalized size = 0.80 \begin {gather*} \sqrt {9\,x^2-4}-2\,\mathrm {atan}\left (\frac {\sqrt {9\,x^2-4}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((9*x^2 - 4)^(1/2)/x,x)

[Out]

(9*x^2 - 4)^(1/2) - 2*atan((9*x^2 - 4)^(1/2)/2)

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