Integrand size = 29, antiderivative size = 56 \[ \int \frac {-3 \left (-3 x+x^3\right )+\left (-3 x+x^3\right )^3}{\sqrt {-4+x^2}} \, dx=\sqrt {-4+x^2}+\frac {10}{3} \left (-4+x^2\right )^{3/2}+3 \left (-4+x^2\right )^{5/2}+\left (-4+x^2\right )^{7/2}+\frac {1}{9} \left (-4+x^2\right )^{9/2} \] Output:
(x^2-4)^(1/2)+10/3*(x^2-4)^(3/2)+3*(x^2-4)^(5/2)+(x^2-4)^(7/2)+1/9*(x^2-4) ^(9/2)
Time = 0.03 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.59 \[ \int \frac {-3 \left (-3 x+x^3\right )+\left (-3 x+x^3\right )^3}{\sqrt {-4+x^2}} \, dx=\frac {1}{9} \sqrt {-4+x^2} \left (1-10 x^2+15 x^4-7 x^6+x^8\right ) \] Input:
Integrate[(-3*(-3*x + x^3) + (-3*x + x^3)^3)/Sqrt[-4 + x^2],x]
Output:
(Sqrt[-4 + x^2]*(1 - 10*x^2 + 15*x^4 - 7*x^6 + x^8))/9
Time = 0.29 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2342, 2331, 2389, 2009}
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.
\(\displaystyle \int \frac {\left (x^3-3 x\right )^3-3 \left (x^3-3 x\right )}{\sqrt {x^2-4}} \, dx\) |
\(\Big \downarrow \) 2342 |
\(\displaystyle \int \frac {x \left (x^8-9 x^6+27 x^4-30 x^2+9\right )}{\sqrt {x^2-4}}dx\) |
\(\Big \downarrow \) 2331 |
\(\displaystyle \frac {1}{2} \int \frac {x^8-9 x^6+27 x^4-30 x^2+9}{\sqrt {x^2-4}}dx^2\) |
\(\Big \downarrow \) 2389 |
\(\displaystyle \frac {1}{2} \int \left (\left (x^2-4\right )^{7/2}+7 \left (x^2-4\right )^{5/2}+15 \left (x^2-4\right )^{3/2}+10 \sqrt {x^2-4}+\frac {1}{\sqrt {x^2-4}}\right )dx^2\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{9} \left (x^2-4\right )^{9/2}+2 \left (x^2-4\right )^{7/2}+6 \left (x^2-4\right )^{5/2}+\frac {20}{3} \left (x^2-4\right )^{3/2}+2 \sqrt {x^2-4}\right )\) |
Input:
Int[(-3*(-3*x + x^3) + (-3*x + x^3)^3)/Sqrt[-4 + x^2],x]
Output:
(2*Sqrt[-4 + x^2] + (20*(-4 + x^2)^(3/2))/3 + 6*(-4 + x^2)^(5/2) + 2*(-4 + x^2)^(7/2) + (2*(-4 + x^2)^(9/2))/9)/2
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[1/2 S ubst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[x*PolynomialQuotient [Pq, x, x]*(a + b*x^2)^p, x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && EqQ[ Coeff[Pq, x, 0], 0] && !MatchQ[Pq, x^(m_.)*(u_.) /; IntegerQ[m]]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand [Pq*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p , 0] || EqQ[n, 1])
Time = 0.15 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.54
method | result | size |
risch | \(\frac {\left (x^{8}-7 x^{6}+15 x^{4}-10 x^{2}+1\right ) \sqrt {x^{2}-4}}{9}\) | \(30\) |
pseudoelliptic | \(\frac {\left (x^{8}-7 x^{6}+15 x^{4}-10 x^{2}+1\right ) \sqrt {x^{2}-4}}{9}\) | \(30\) |
trager | \(\left (\frac {1}{9} x^{8}-\frac {7}{9} x^{6}+\frac {5}{3} x^{4}-\frac {10}{9} x^{2}+\frac {1}{9}\right ) \sqrt {x^{2}-4}\) | \(31\) |
gosper | \(\frac {\left (x^{8}-7 x^{6}+15 x^{4}-10 x^{2}+1\right ) \left (-2+x \right ) \left (2+x \right )}{9 \sqrt {x^{2}-4}}\) | \(36\) |
default | \(\frac {x^{8} \sqrt {x^{2}-4}}{9}-\frac {7 x^{6} \sqrt {x^{2}-4}}{9}+\frac {5 x^{4} \sqrt {x^{2}-4}}{3}-\frac {10 x^{2} \sqrt {x^{2}-4}}{9}+\frac {\sqrt {x^{2}-4}}{9}\) | \(59\) |
orering | \(\frac {\left (x^{8}-7 x^{6}+15 x^{4}-10 x^{2}+1\right ) \left (-2+x \right ) \left (2+x \right ) \left (-3 x^{3}+9 x +\left (x^{3}-3 x \right )^{3}\right )}{9 x \left (x^{2}-3\right ) \left (x^{6}-6 x^{4}+9 x^{2}-3\right ) \sqrt {x^{2}-4}}\) | \(81\) |
meijerg | \(-\frac {120 \sqrt {-\operatorname {signum}\left (-1+\frac {x^{2}}{4}\right )}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (x^{2}+8\right ) \sqrt {1-\frac {x^{2}}{4}}}{6}\right )}{\sqrt {\pi }\, \sqrt {\operatorname {signum}\left (-1+\frac {x^{2}}{4}\right )}}-\frac {9 \sqrt {-\operatorname {signum}\left (-1+\frac {x^{2}}{4}\right )}\, \left (-2 \sqrt {\pi }+2 \sqrt {\pi }\, \sqrt {1-\frac {x^{2}}{4}}\right )}{\sqrt {\pi }\, \sqrt {\operatorname {signum}\left (-1+\frac {x^{2}}{4}\right )}}-\frac {256 \sqrt {-\operatorname {signum}\left (-1+\frac {x^{2}}{4}\right )}\, \left (-\frac {256 \sqrt {\pi }}{315}+\frac {\sqrt {\pi }\, \left (\frac {35}{128} x^{8}+\frac {5}{4} x^{6}+6 x^{4}+32 x^{2}+256\right ) \sqrt {1-\frac {x^{2}}{4}}}{315}\right )}{\sqrt {\pi }\, \sqrt {\operatorname {signum}\left (-1+\frac {x^{2}}{4}\right )}}-\frac {576 \sqrt {-\operatorname {signum}\left (-1+\frac {x^{2}}{4}\right )}\, \left (\frac {32 \sqrt {\pi }}{35}-\frac {\sqrt {\pi }\, \left (\frac {5}{8} x^{6}+3 x^{4}+16 x^{2}+128\right ) \sqrt {1-\frac {x^{2}}{4}}}{140}\right )}{\sqrt {\pi }\, \sqrt {\operatorname {signum}\left (-1+\frac {x^{2}}{4}\right )}}-\frac {432 \sqrt {-\operatorname {signum}\left (-1+\frac {x^{2}}{4}\right )}\, \left (-\frac {16 \sqrt {\pi }}{15}+\frac {\sqrt {\pi }\, \left (\frac {3}{8} x^{4}+2 x^{2}+16\right ) \sqrt {1-\frac {x^{2}}{4}}}{15}\right )}{\sqrt {\pi }\, \sqrt {\operatorname {signum}\left (-1+\frac {x^{2}}{4}\right )}}\) | \(293\) |
Input:
int((-3*x^3+9*x+(x^3-3*x)^3)/(x^2-4)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/9*(x^8-7*x^6+15*x^4-10*x^2+1)*(x^2-4)^(1/2)
Time = 0.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.52 \[ \int \frac {-3 \left (-3 x+x^3\right )+\left (-3 x+x^3\right )^3}{\sqrt {-4+x^2}} \, dx=\frac {1}{9} \, {\left (x^{8} - 7 \, x^{6} + 15 \, x^{4} - 10 \, x^{2} + 1\right )} \sqrt {x^{2} - 4} \] Input:
integrate((-3*x^3+9*x+(x^3-3*x)^3)/(x^2-4)^(1/2),x, algorithm="fricas")
Output:
1/9*(x^8 - 7*x^6 + 15*x^4 - 10*x^2 + 1)*sqrt(x^2 - 4)
Time = 0.81 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.21 \[ \int \frac {-3 \left (-3 x+x^3\right )+\left (-3 x+x^3\right )^3}{\sqrt {-4+x^2}} \, dx=\frac {x^{8} \sqrt {x^{2} - 4}}{9} - \frac {7 x^{6} \sqrt {x^{2} - 4}}{9} + \frac {5 x^{4} \sqrt {x^{2} - 4}}{3} - \frac {10 x^{2} \sqrt {x^{2} - 4}}{9} + \frac {\sqrt {x^{2} - 4}}{9} \] Input:
integrate((-3*x**3+9*x+(x**3-3*x)**3)/(x**2-4)**(1/2),x)
Output:
x**8*sqrt(x**2 - 4)/9 - 7*x**6*sqrt(x**2 - 4)/9 + 5*x**4*sqrt(x**2 - 4)/3 - 10*x**2*sqrt(x**2 - 4)/9 + sqrt(x**2 - 4)/9
Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.04 \[ \int \frac {-3 \left (-3 x+x^3\right )+\left (-3 x+x^3\right )^3}{\sqrt {-4+x^2}} \, dx=\frac {1}{9} \, \sqrt {x^{2} - 4} x^{8} - \frac {7}{9} \, \sqrt {x^{2} - 4} x^{6} + \frac {5}{3} \, \sqrt {x^{2} - 4} x^{4} - \frac {10}{9} \, \sqrt {x^{2} - 4} x^{2} + \frac {1}{9} \, \sqrt {x^{2} - 4} \] Input:
integrate((-3*x^3+9*x+(x^3-3*x)^3)/(x^2-4)^(1/2),x, algorithm="maxima")
Output:
1/9*sqrt(x^2 - 4)*x^8 - 7/9*sqrt(x^2 - 4)*x^6 + 5/3*sqrt(x^2 - 4)*x^4 - 10 /9*sqrt(x^2 - 4)*x^2 + 1/9*sqrt(x^2 - 4)
Time = 0.11 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.75 \[ \int \frac {-3 \left (-3 x+x^3\right )+\left (-3 x+x^3\right )^3}{\sqrt {-4+x^2}} \, dx=\frac {1}{9} \, {\left (x^{2} - 4\right )}^{\frac {9}{2}} + {\left (x^{2} - 4\right )}^{\frac {7}{2}} + 3 \, {\left (x^{2} - 4\right )}^{\frac {5}{2}} + \frac {10}{3} \, {\left (x^{2} - 4\right )}^{\frac {3}{2}} + \sqrt {x^{2} - 4} \] Input:
integrate((-3*x^3+9*x+(x^3-3*x)^3)/(x^2-4)^(1/2),x, algorithm="giac")
Output:
1/9*(x^2 - 4)^(9/2) + (x^2 - 4)^(7/2) + 3*(x^2 - 4)^(5/2) + 10/3*(x^2 - 4) ^(3/2) + sqrt(x^2 - 4)
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.54 \[ \int \frac {-3 \left (-3 x+x^3\right )+\left (-3 x+x^3\right )^3}{\sqrt {-4+x^2}} \, dx=\sqrt {x^2-4}\,\left (\frac {x^8}{9}-\frac {7\,x^6}{9}+\frac {5\,x^4}{3}-\frac {10\,x^2}{9}+\frac {1}{9}\right ) \] Input:
int(-((3*x - x^3)^3 - 9*x + 3*x^3)/(x^2 - 4)^(1/2),x)
Output:
(x^2 - 4)^(1/2)*((5*x^4)/3 - (10*x^2)/9 - (7*x^6)/9 + x^8/9 + 1/9)
Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.50 \[ \int \frac {-3 \left (-3 x+x^3\right )+\left (-3 x+x^3\right )^3}{\sqrt {-4+x^2}} \, dx=\frac {\sqrt {x^{2}-4}\, \left (x^{8}-7 x^{6}+15 x^{4}-10 x^{2}+1\right )}{9} \] Input:
int((-3*x^3+9*x+(x^3-3*x)^3)/(x^2-4)^(1/2),x)
Output:
(sqrt(x**2 - 4)*(x**8 - 7*x**6 + 15*x**4 - 10*x**2 + 1))/9