Integrand size = 15, antiderivative size = 57 \[ \int \frac {1}{x^5 \sqrt {9-4 x^2}} \, dx=-\frac {\sqrt {9-4 x^2}}{36 x^4}-\frac {\sqrt {9-4 x^2}}{54 x^2}-\frac {2}{81} \text {arctanh}\left (\frac {1}{3} \sqrt {9-4 x^2}\right ) \] Output:
-1/36*(-4*x^2+9)^(1/2)/x^4-1/54*(-4*x^2+9)^(1/2)/x^2-2/81*arctanh(1/3*(-4* x^2+9)^(1/2))
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x^5 \sqrt {9-4 x^2}} \, dx=\frac {\sqrt {9-4 x^2} \left (-3-2 x^2\right )}{108 x^4}-\frac {2}{81} \text {arctanh}\left (\frac {1}{3} \sqrt {9-4 x^2}\right ) \] Input:
Integrate[1/(x^5*Sqrt[9 - 4*x^2]),x]
Output:
(Sqrt[9 - 4*x^2]*(-3 - 2*x^2))/(108*x^4) - (2*ArcTanh[Sqrt[9 - 4*x^2]/3])/ 81
Time = 0.15 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {243, 52, 52, 73, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{x^5 \sqrt {9-4 x^2}} \, dx\) |
\(\Big \downarrow \) 243 |
\(\displaystyle \frac {1}{2} \int \frac {1}{x^6 \sqrt {9-4 x^2}}dx^2\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \int \frac {1}{x^4 \sqrt {9-4 x^2}}dx^2-\frac {\sqrt {9-4 x^2}}{18 x^4}\right )\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (\frac {2}{9} \int \frac {1}{x^2 \sqrt {9-4 x^2}}dx^2-\frac {\sqrt {9-4 x^2}}{9 x^2}\right )-\frac {\sqrt {9-4 x^2}}{18 x^4}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (-\frac {1}{9} \int \frac {1}{\frac {9}{4}-\frac {x^4}{4}}d\sqrt {9-4 x^2}-\frac {\sqrt {9-4 x^2}}{9 x^2}\right )-\frac {\sqrt {9-4 x^2}}{18 x^4}\right )\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{3} \left (-\frac {4}{27} \text {arctanh}\left (\frac {1}{3} \sqrt {9-4 x^2}\right )-\frac {\sqrt {9-4 x^2}}{9 x^2}\right )-\frac {\sqrt {9-4 x^2}}{18 x^4}\right )\) |
Input:
Int[1/(x^5*Sqrt[9 - 4*x^2]),x]
Output:
(-1/18*Sqrt[9 - 4*x^2]/x^4 + (-1/9*Sqrt[9 - 4*x^2]/x^2 - (4*ArcTanh[Sqrt[9 - 4*x^2]/3])/27)/3)/2
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Time = 0.34 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.72
method | result | size |
trager | \(-\frac {\left (2 x^{2}+3\right ) \sqrt {-4 x^{2}+9}}{108 x^{4}}+\frac {2 \ln \left (\frac {-3+\sqrt {-4 x^{2}+9}}{x}\right )}{81}\) | \(41\) |
risch | \(\frac {8 x^{4}-6 x^{2}-27}{108 x^{4} \sqrt {-4 x^{2}+9}}-\frac {2 \,\operatorname {arctanh}\left (\frac {3}{\sqrt {-4 x^{2}+9}}\right )}{81}\) | \(42\) |
default | \(-\frac {\sqrt {-4 x^{2}+9}}{36 x^{4}}-\frac {\sqrt {-4 x^{2}+9}}{54 x^{2}}-\frac {2 \,\operatorname {arctanh}\left (\frac {3}{\sqrt {-4 x^{2}+9}}\right )}{81}\) | \(44\) |
pseudoelliptic | \(\frac {-\frac {16 \ln \left (\sqrt {-4 x^{2}+9}+3\right ) x^{4}}{81}+\frac {16 \ln \left (-3+\sqrt {-4 x^{2}+9}\right ) x^{4}}{81}-\frac {8 x^{2} \sqrt {-4 x^{2}+9}}{27}-\frac {4 \sqrt {-4 x^{2}+9}}{9}}{\left (\sqrt {-4 x^{2}+9}+3\right )^{2} \left (-3+\sqrt {-4 x^{2}+9}\right )^{2}}\) | \(89\) |
meijerg | \(\frac {-\frac {\sqrt {\pi }}{12 x^{4}}-\frac {\sqrt {\pi }}{27 x^{2}}+\frac {\left (\frac {7}{6}+2 \ln \left (x \right )-2 \ln \left (3\right )+i \pi \right ) \sqrt {\pi }}{81}+\frac {\sqrt {\pi }\, \left (-\frac {112}{81} x^{4}+\frac {32}{9} x^{2}+8\right )}{96 x^{4}}-\frac {\sqrt {\pi }\, \left (\frac {16 x^{2}}{3}+8\right ) \sqrt {-\frac {4 x^{2}}{9}+1}}{96 x^{4}}-\frac {2 \sqrt {\pi }\, \ln \left (\frac {1}{2}+\frac {\sqrt {-\frac {4 x^{2}}{9}+1}}{2}\right )}{81}}{\sqrt {\pi }}\) | \(105\) |
Input:
int(1/x^5/(-4*x^2+9)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/108*(2*x^2+3)/x^4*(-4*x^2+9)^(1/2)+2/81*ln((-3+(-4*x^2+9)^(1/2))/x)
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.79 \[ \int \frac {1}{x^5 \sqrt {9-4 x^2}} \, dx=\frac {8 \, x^{4} \log \left (\frac {\sqrt {-4 \, x^{2} + 9} - 3}{x}\right ) - 3 \, {\left (2 \, x^{2} + 3\right )} \sqrt {-4 \, x^{2} + 9}}{324 \, x^{4}} \] Input:
integrate(1/x^5/(-4*x^2+9)^(1/2),x, algorithm="fricas")
Output:
1/324*(8*x^4*log((sqrt(-4*x^2 + 9) - 3)/x) - 3*(2*x^2 + 3)*sqrt(-4*x^2 + 9 ))/x^4
Result contains complex when optimal does not.
Time = 2.75 (sec) , antiderivative size = 136, normalized size of antiderivative = 2.39 \[ \int \frac {1}{x^5 \sqrt {9-4 x^2}} \, dx=\begin {cases} - \frac {2 \operatorname {acosh}{\left (\frac {3}{2 x} \right )}}{81} + \frac {1}{27 x \sqrt {-1 + \frac {9}{4 x^{2}}}} - \frac {1}{36 x^{3} \sqrt {-1 + \frac {9}{4 x^{2}}}} - \frac {1}{8 x^{5} \sqrt {-1 + \frac {9}{4 x^{2}}}} & \text {for}\: \frac {1}{\left |{x^{2}}\right |} > \frac {4}{9} \\\frac {2 i \operatorname {asin}{\left (\frac {3}{2 x} \right )}}{81} - \frac {i}{27 x \sqrt {1 - \frac {9}{4 x^{2}}}} + \frac {i}{36 x^{3} \sqrt {1 - \frac {9}{4 x^{2}}}} + \frac {i}{8 x^{5} \sqrt {1 - \frac {9}{4 x^{2}}}} & \text {otherwise} \end {cases} \] Input:
integrate(1/x**5/(-4*x**2+9)**(1/2),x)
Output:
Piecewise((-2*acosh(3/(2*x))/81 + 1/(27*x*sqrt(-1 + 9/(4*x**2))) - 1/(36*x **3*sqrt(-1 + 9/(4*x**2))) - 1/(8*x**5*sqrt(-1 + 9/(4*x**2))), 1/Abs(x**2) > 4/9), (2*I*asin(3/(2*x))/81 - I/(27*x*sqrt(1 - 9/(4*x**2))) + I/(36*x** 3*sqrt(1 - 9/(4*x**2))) + I/(8*x**5*sqrt(1 - 9/(4*x**2))), True))
Time = 0.10 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \frac {1}{x^5 \sqrt {9-4 x^2}} \, dx=-\frac {\sqrt {-4 \, x^{2} + 9}}{54 \, x^{2}} - \frac {\sqrt {-4 \, x^{2} + 9}}{36 \, x^{4}} - \frac {2}{81} \, \log \left (\frac {6 \, \sqrt {-4 \, x^{2} + 9}}{{\left | x \right |}} + \frac {18}{{\left | x \right |}}\right ) \] Input:
integrate(1/x^5/(-4*x^2+9)^(1/2),x, algorithm="maxima")
Output:
-1/54*sqrt(-4*x^2 + 9)/x^2 - 1/36*sqrt(-4*x^2 + 9)/x^4 - 2/81*log(6*sqrt(- 4*x^2 + 9)/abs(x) + 18/abs(x))
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^5 \sqrt {9-4 x^2}} \, dx=\frac {{\left (-4 \, x^{2} + 9\right )}^{\frac {3}{2}} - 15 \, \sqrt {-4 \, x^{2} + 9}}{216 \, x^{4}} - \frac {1}{81} \, \log \left (\sqrt {-4 \, x^{2} + 9} + 3\right ) + \frac {1}{81} \, \log \left (-\sqrt {-4 \, x^{2} + 9} + 3\right ) \] Input:
integrate(1/x^5/(-4*x^2+9)^(1/2),x, algorithm="giac")
Output:
1/216*((-4*x^2 + 9)^(3/2) - 15*sqrt(-4*x^2 + 9))/x^4 - 1/81*log(sqrt(-4*x^ 2 + 9) + 3) + 1/81*log(-sqrt(-4*x^2 + 9) + 3)
Time = 0.18 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^5 \sqrt {9-4 x^2}} \, dx=\frac {2\,\ln \left (\sqrt {\frac {9}{4\,x^2}-1}-\sqrt {\frac {9}{4\,x^2}}\right )}{81}-\frac {\sqrt {\frac {9}{4}-x^2}\,\left (\frac {2}{27\,x^2}+\frac {1}{9\,x^4}\right )}{2} \] Input:
int(1/(x^5*(9 - 4*x^2)^(1/2)),x)
Output:
(2*log((9/(4*x^2) - 1)^(1/2) - (9/(4*x^2))^(1/2)))/81 - ((9/4 - x^2)^(1/2) *(2/(27*x^2) + 1/(9*x^4)))/2
Time = 0.24 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^5 \sqrt {9-4 x^2}} \, dx=\frac {-6 \sqrt {-4 x^{2}+9}\, x^{2}-9 \sqrt {-4 x^{2}+9}+8 \,\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (\frac {2 x}{3}\right )}{2}\right )\right ) x^{4}}{324 x^{4}} \] Input:
int(1/x^5/(-4*x^2+9)^(1/2),x)
Output:
( - 6*sqrt( - 4*x**2 + 9)*x**2 - 9*sqrt( - 4*x**2 + 9) + 8*log(tan(asin((2 *x)/3)/2))*x**4)/(324*x**4)