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

Optimal result

Integrand size = 15, antiderivative size = 18 \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\frac {\sqrt {-9+16 x^2}}{9 x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {270} \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\frac {\sqrt {16 x^2-9}}{9 x} \]

[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*} \text {integral}& = \frac {\sqrt {-9+16 x^2}}{9 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\frac {\sqrt {-9+16 x^2}}{9 x} \]

[In]

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

[Out]

Sqrt[-9 + 16*x^2]/(9*x)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 15, normalized size of antiderivative = 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	\(\frac {\sqrt {16 x^{2}-9}}{9 x}\)	\(15\)
pseudoelliptic	\(\frac {\sqrt {16 x^{2}-9}}{9 x}\)	\(15\)
gosper	\(\frac {\left (4 x -3\right ) \left (4 x +3\right )}{9 x \sqrt {16 x^{2}-9}}\)	\(25\)
meijerg	\(-\frac {\sqrt {-\operatorname {signum}\left (-1+\frac {16 x^{2}}{9}\right )}\, \sqrt {1-\frac {16 x^{2}}{9}}}{3 \sqrt {\operatorname {signum}\left (-1+\frac {16 x^{2}}{9}\right )}\, x}\)	\(37\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\frac {4 \, x + \sqrt {16 \, x^{2} - 9}}{9 \, x} \]

[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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.45 (sec) , antiderivative size = 37, normalized size of antiderivative = 2.06 \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\begin {cases} \frac {4 i \sqrt {-1 + \frac {9}{16 x^{2}}}}{9} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > \frac {16}{9} \\\frac {4 \sqrt {1 - \frac {9}{16 x^{2}}}}{9} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\frac {\sqrt {16 \, x^{2} - 9}}{9 \, x} \]

[In]

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

[Out]

1/9*sqrt(16*x^2 - 9)/x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28 \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\frac {8}{{\left (4 \, x - \sqrt {16 \, x^{2} - 9}\right )}^{2} + 9} \]

[In]

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

[Out]

8/((4*x - sqrt(16*x^2 - 9))^2 + 9)

Mupad [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \sqrt {-9+16 x^2}} \, dx=\frac {\sqrt {16\,x^2-9}}{9\,x} \]

[In]

int(1/(x^2*(16*x^2 - 9)^(1/2)),x)

[Out]

(16*x^2 - 9)^(1/2)/(9*x)