3.2.35 \(\int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx\) [135]

Optimal. Leaf size=18 \[ \frac {\sqrt {-9+16 x^2}}{9 x} \]

[Out]

1/9*(16*x^2-9)^(1/2)/x

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Rubi [A]
time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \begin {gather*} \frac {\sqrt {16 x^2-9}}{9 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[-9 + 16*x^2]),x]

[Out]

Sqrt[-9 + 16*x^2]/(9*x)

Rule 270

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

Rubi steps

\begin {align*} \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx &=\frac {\sqrt {-9+16 x^2}}{9 x}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {\sqrt {-9+16 x^2}}{9 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[-9 + 16*x^2]),x]

[Out]

Sqrt[-9 + 16*x^2]/(9*x)

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 2 in optimal.
time = 2.18, size = 33, normalized size = 1.83 \begin {gather*} \text {Piecewise}\left [\left \{\left \{\frac {I \sqrt {-16+\frac {9}{x^2}}}{9},\frac {1}{\text {Abs}\left [x^2\right ]}>\frac {16}{9}\right \}\right \},\frac {4 \sqrt {1-\frac {9}{16 x^2}}}{9}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[1/(x^2*Sqrt[16*x^2 - 9]),x]')

[Out]

Piecewise[{{I / 9 Sqrt[-16 + 9 / x ^ 2], 1 / Abs[x ^ 2] > 16 / 9}}, 4 Sqrt[1 - 9 / (16 x ^ 2)] / 9]

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Maple [A]
time = 0.05, size = 15, normalized size = 0.83

method	result	size
default	\(\frac {\sqrt {16 x^{2}-9}}{9 x}\)	\(15\)
trager	\(\frac {\sqrt {16 x^{2}-9}}{9 x}\)	\(15\)
risch	\(\frac {\sqrt {16 x^{2}-9}}{9 x}\)	\(15\)
gosper	\(\frac {\left (4 x -3\right ) \left (3+4 x \right )}{9 x \sqrt {16 x^{2}-9}}\)	\(25\)
meijerg	\(-\frac {\sqrt {-\mathrm {signum}\left (-1+\frac {16 x^{2}}{9}\right )}\, \sqrt {1-\frac {16 x^{2}}{9}}}{3 \sqrt {\mathrm {signum}\left (-1+\frac {16 x^{2}}{9}\right )}\, x}\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^2/(16*x^2-9)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/9*(16*x^2-9)^(1/2)/x

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Maxima [A]
time = 0.35, size = 14, normalized size = 0.78 \begin {gather*} \frac {\sqrt {16 \, x^{2} - 9}}{9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(16*x^2-9)^(1/2),x, algorithm="maxima")

[Out]

1/9*sqrt(16*x^2 - 9)/x

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Fricas [A]
time = 0.32, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 \, x + \sqrt {16 \, x^{2} - 9}}{9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(16*x^2-9)^(1/2),x, algorithm="fricas")

[Out]

1/9*(4*x + sqrt(16*x^2 - 9))/x

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Sympy [A]
time = 0.40, size = 37, normalized size = 2.06 \begin {gather*} \begin {cases} \frac {4 i \sqrt {-1 + \frac {9}{16 x^{2}}}}{9} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > \frac {16}{9} \\\frac {4 \sqrt {1 - \frac {9}{16 x^{2}}}}{9} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(16*x**2-9)**(1/2),x)

[Out]

Piecewise((4*I*sqrt(-1 + 9/(16*x**2))/9, 1/Abs(x**2) > 16/9), (4*sqrt(1 - 9/(16*x**2))/9, True))

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Giac [A]
time = 0.00, size = 23, normalized size = 1.28 \begin {gather*} \frac {8}{\left (\sqrt {16 x^{2}-9}-4 x\right )^{2}+9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(16*x^2-9)^(1/2),x)

[Out]

8/((4*x - sqrt(16*x^2 - 9))^2 + 9)

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Mupad [B]
time = 0.25, size = 14, normalized size = 0.78 \begin {gather*} \frac {\sqrt {16\,x^2-9}}{9\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(16*x^2 - 9)^(1/2)),x)

[Out]

(16*x^2 - 9)^(1/2)/(9*x)

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