Integrand size = 73, antiderivative size = 21 \[ \int \frac {72-18 x^2+e^{16-32 x^2+24 x^4-8 x^6+x^8} \left (-4+x^2\right ) \left (774 x^8-1344 x^{10}+864 x^{12}-240 x^{14}+24 x^{16}\right )}{-4 x^7+x^9} \, dx=3 \left (\frac {1}{x^6}+e^{\left (-2+x^2\right )^4} \left (-4+x^2\right )\right ) \] Output:
3*exp(ln(x^2-4)+(x^2-2)^4)+3/x^6
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 4.52 \[ \int \frac {72-18 x^2+e^{16-32 x^2+24 x^4-8 x^6+x^8} \left (-4+x^2\right ) \left (774 x^8-1344 x^{10}+864 x^{12}-240 x^{14}+24 x^{16}\right )}{-4 x^7+x^9} \, dx=\frac {3}{x^6}+3 \left (-2 e^{\left (-2+x^2\right )^4}-\frac {\left (-2+x^2\right ) \Gamma \left (\frac {1}{4},-\left (-2+x^2\right )^4\right )}{4 \sqrt [4]{-\left (-2+x^2\right )^4}}+\frac {\left (2-x^2\right )^5 \Gamma \left (\frac {5}{4},-\left (2-x^2\right )^4\right )}{\left (-\left (2-x^2\right )^4\right )^{5/4}}\right ) \] Input:
Integrate[(72 - 18*x^2 + E^(16 - 32*x^2 + 24*x^4 - 8*x^6 + x^8)*(-4 + x^2) *(774*x^8 - 1344*x^10 + 864*x^12 - 240*x^14 + 24*x^16))/(-4*x^7 + x^9),x]
Output:
3/x^6 + 3*(-2*E^(-2 + x^2)^4 - ((-2 + x^2)*Gamma[1/4, -(-2 + x^2)^4])/(4*( -(-2 + x^2)^4)^(1/4)) + ((2 - x^2)^5*Gamma[5/4, -(2 - x^2)^4])/(-(2 - x^2) ^4)^(5/4))
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.15 (sec) , antiderivative size = 99, normalized size of antiderivative = 4.71, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {2026, 7239, 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.
\(\displaystyle \int \frac {-18 x^2+e^{x^8-8 x^6+24 x^4-32 x^2+16} \left (x^2-4\right ) \left (24 x^{16}-240 x^{14}+864 x^{12}-1344 x^{10}+774 x^8\right )+72}{x^9-4 x^7} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {-18 x^2+e^{x^8-8 x^6+24 x^4-32 x^2+16} \left (x^2-4\right ) \left (24 x^{16}-240 x^{14}+864 x^{12}-1344 x^{10}+774 x^8\right )+72}{x^7 \left (x^2-4\right )}dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \left (6 e^{\left (x^2-2\right )^4} x \left (4 x^8-40 x^6+144 x^4-224 x^2+129\right )-\frac {18}{x^7}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {3}{x^6}-6 e^{\left (x^2-2\right )^4}+\frac {3 \left (2-x^2\right )^5 \Gamma \left (\frac {5}{4},-\left (2-x^2\right )^4\right )}{\left (-\left (2-x^2\right )^4\right )^{5/4}}+\frac {3 \left (2-x^2\right ) \Gamma \left (\frac {1}{4},-\left (2-x^2\right )^4\right )}{4 \sqrt [4]{-\left (2-x^2\right )^4}}\) |
Input:
Int[(72 - 18*x^2 + E^(16 - 32*x^2 + 24*x^4 - 8*x^6 + x^8)*(-4 + x^2)*(774* x^8 - 1344*x^10 + 864*x^12 - 240*x^14 + 24*x^16))/(-4*x^7 + x^9),x]
Output:
-6*E^(-2 + x^2)^4 + 3/x^6 + (3*(2 - x^2)*Gamma[1/4, -(2 - x^2)^4])/(4*(-(2 - x^2)^4)^(1/4)) + (3*(2 - x^2)^5*Gamma[5/4, -(2 - x^2)^4])/(-(2 - x^2)^4 )^(5/4)
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 11.12 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10
method | result | size |
risch | \(\frac {3}{x^{6}}+\left (3 x^{2}-12\right ) {\mathrm e}^{\left (x^{2}-2\right )^{4}}\) | \(23\) |
default | \(\frac {3}{x^{6}}+3 \,{\mathrm e}^{\ln \left (x^{2}-4\right )+x^{8}-8 x^{6}+24 x^{4}-32 x^{2}+16}\) | \(36\) |
parts | \(\frac {3}{x^{6}}+3 \,{\mathrm e}^{\ln \left (x^{2}-4\right )+x^{8}-8 x^{6}+24 x^{4}-32 x^{2}+16}\) | \(36\) |
parallelrisch | \(\frac {24+24 \,{\mathrm e}^{\ln \left (x^{2}-4\right )+x^{8}-8 x^{6}+24 x^{4}-32 x^{2}+16} x^{6}}{8 x^{6}}\) | \(40\) |
orering | \(-\frac {\left (8 x^{18}-128 x^{16}+864 x^{14}-3200 x^{12}+7051 x^{10}-9298 x^{8}+6860 x^{6}-2232 x^{4}+27 x^{2}+21\right ) \left (\left (24 x^{16}-240 x^{14}+864 x^{12}-1344 x^{10}+774 x^{8}\right ) {\mathrm e}^{\ln \left (x^{2}-4\right )+x^{8}-8 x^{6}+24 x^{4}-32 x^{2}+16}-18 x^{2}+72\right )}{12 x \left (4 x^{16}-64 x^{14}+432 x^{12}-1600 x^{10}+3529 x^{8}-4684 x^{6}+3556 x^{4}-1312 x^{2}+129\right ) \left (x^{9}-4 x^{7}\right )}+\frac {\left (x^{10}-10 x^{8}+36 x^{6}-56 x^{4}+33 x^{2}-3\right ) \left (\frac {\left (384 x^{15}-3360 x^{13}+10368 x^{11}-13440 x^{9}+6192 x^{7}\right ) {\mathrm e}^{\ln \left (x^{2}-4\right )+x^{8}-8 x^{6}+24 x^{4}-32 x^{2}+16}+\left (24 x^{16}-240 x^{14}+864 x^{12}-1344 x^{10}+774 x^{8}\right ) \left (\frac {2 x}{x^{2}-4}+8 x^{7}-48 x^{5}+96 x^{3}-64 x \right ) {\mathrm e}^{\ln \left (x^{2}-4\right )+x^{8}-8 x^{6}+24 x^{4}-32 x^{2}+16}-36 x}{x^{9}-4 x^{7}}-\frac {\left (\left (24 x^{16}-240 x^{14}+864 x^{12}-1344 x^{10}+774 x^{8}\right ) {\mathrm e}^{\ln \left (x^{2}-4\right )+x^{8}-8 x^{6}+24 x^{4}-32 x^{2}+16}-18 x^{2}+72\right ) \left (9 x^{8}-28 x^{6}\right )}{\left (x^{9}-4 x^{7}\right )^{2}}\right )}{48 x^{16}-768 x^{14}+5184 x^{12}-19200 x^{10}+42348 x^{8}-56208 x^{6}+42672 x^{4}-15744 x^{2}+1548}\) | \(480\) |
Input:
int(((24*x^16-240*x^14+864*x^12-1344*x^10+774*x^8)*exp(ln(x^2-4)+x^8-8*x^6 +24*x^4-32*x^2+16)-18*x^2+72)/(x^9-4*x^7),x,method=_RETURNVERBOSE)
Output:
3/x^6+(3*x^2-12)*exp((x^2-2)^4)
Time = 0.08 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.81 \[ \int \frac {72-18 x^2+e^{16-32 x^2+24 x^4-8 x^6+x^8} \left (-4+x^2\right ) \left (774 x^8-1344 x^{10}+864 x^{12}-240 x^{14}+24 x^{16}\right )}{-4 x^7+x^9} \, dx=\frac {3 \, {\left (x^{6} e^{\left (x^{8} - 8 \, x^{6} + 24 \, x^{4} - 32 \, x^{2} + \log \left (x^{2} - 4\right ) + 16\right )} + 1\right )}}{x^{6}} \] Input:
integrate(((24*x^16-240*x^14+864*x^12-1344*x^10+774*x^8)*exp(log(x^2-4)+x^ 8-8*x^6+24*x^4-32*x^2+16)-18*x^2+72)/(x^9-4*x^7),x, algorithm="fricas")
Output:
3*(x^6*e^(x^8 - 8*x^6 + 24*x^4 - 32*x^2 + log(x^2 - 4) + 16) + 1)/x^6
Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \frac {72-18 x^2+e^{16-32 x^2+24 x^4-8 x^6+x^8} \left (-4+x^2\right ) \left (774 x^8-1344 x^{10}+864 x^{12}-240 x^{14}+24 x^{16}\right )}{-4 x^7+x^9} \, dx=\left (3 x^{2} - 12\right ) e^{x^{8} - 8 x^{6} + 24 x^{4} - 32 x^{2} + 16} + \frac {3}{x^{6}} \] Input:
integrate(((24*x**16-240*x**14+864*x**12-1344*x**10+774*x**8)*exp(ln(x**2- 4)+x**8-8*x**6+24*x**4-32*x**2+16)-18*x**2+72)/(x**9-4*x**7),x)
Output:
(3*x**2 - 12)*exp(x**8 - 8*x**6 + 24*x**4 - 32*x**2 + 16) + 3/x**6
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (23) = 46\).
Time = 0.15 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.90 \[ \int \frac {72-18 x^2+e^{16-32 x^2+24 x^4-8 x^6+x^8} \left (-4+x^2\right ) \left (774 x^8-1344 x^{10}+864 x^{12}-240 x^{14}+24 x^{16}\right )}{-4 x^7+x^9} \, dx=3 \, {\left (x^{2} e^{16} - 4 \, e^{16}\right )} e^{\left (x^{8} - 8 \, x^{6} + 24 \, x^{4} - 32 \, x^{2}\right )} - \frac {9 \, {\left (x^{2} + 2\right )}}{16 \, x^{4}} + \frac {3 \, {\left (3 \, x^{4} + 6 \, x^{2} + 16\right )}}{16 \, x^{6}} \] Input:
integrate(((24*x^16-240*x^14+864*x^12-1344*x^10+774*x^8)*exp(log(x^2-4)+x^ 8-8*x^6+24*x^4-32*x^2+16)-18*x^2+72)/(x^9-4*x^7),x, algorithm="maxima")
Output:
3*(x^2*e^16 - 4*e^16)*e^(x^8 - 8*x^6 + 24*x^4 - 32*x^2) - 9/16*(x^2 + 2)/x ^4 + 3/16*(3*x^4 + 6*x^2 + 16)/x^6
Leaf count of result is larger than twice the leaf count of optimal. 194 vs. \(2 (23) = 46\).
Time = 0.30 (sec) , antiderivative size = 194, normalized size of antiderivative = 9.24 \[ \int \frac {72-18 x^2+e^{16-32 x^2+24 x^4-8 x^6+x^8} \left (-4+x^2\right ) \left (774 x^8-1344 x^{10}+864 x^{12}-240 x^{14}+24 x^{16}\right )}{-4 x^7+x^9} \, dx=\frac {3 \, {\left ({\left (x^{2} - 4\right )}^{4} e^{\left ({\left (x^{2} - 4\right )}^{4} + 8 \, {\left (x^{2} - 4\right )}^{3} + 24 \, {\left (x^{2} - 4\right )}^{2} + 32 \, x^{2} - 112\right )} + 12 \, {\left (x^{2} - 4\right )}^{3} e^{\left ({\left (x^{2} - 4\right )}^{4} + 8 \, {\left (x^{2} - 4\right )}^{3} + 24 \, {\left (x^{2} - 4\right )}^{2} + 32 \, x^{2} - 112\right )} + 48 \, {\left (x^{2} - 4\right )}^{2} e^{\left ({\left (x^{2} - 4\right )}^{4} + 8 \, {\left (x^{2} - 4\right )}^{3} + 24 \, {\left (x^{2} - 4\right )}^{2} + 32 \, x^{2} - 112\right )} + 64 \, {\left (x^{2} - 4\right )} e^{\left ({\left (x^{2} - 4\right )}^{4} + 8 \, {\left (x^{2} - 4\right )}^{3} + 24 \, {\left (x^{2} - 4\right )}^{2} + 32 \, x^{2} - 112\right )} + 1\right )}}{{\left (x^{2} - 4\right )}^{3} + 12 \, {\left (x^{2} - 4\right )}^{2} + 48 \, x^{2} - 128} \] Input:
integrate(((24*x^16-240*x^14+864*x^12-1344*x^10+774*x^8)*exp(log(x^2-4)+x^ 8-8*x^6+24*x^4-32*x^2+16)-18*x^2+72)/(x^9-4*x^7),x, algorithm="giac")
Output:
3*((x^2 - 4)^4*e^((x^2 - 4)^4 + 8*(x^2 - 4)^3 + 24*(x^2 - 4)^2 + 32*x^2 - 112) + 12*(x^2 - 4)^3*e^((x^2 - 4)^4 + 8*(x^2 - 4)^3 + 24*(x^2 - 4)^2 + 32 *x^2 - 112) + 48*(x^2 - 4)^2*e^((x^2 - 4)^4 + 8*(x^2 - 4)^3 + 24*(x^2 - 4) ^2 + 32*x^2 - 112) + 64*(x^2 - 4)*e^((x^2 - 4)^4 + 8*(x^2 - 4)^3 + 24*(x^2 - 4)^2 + 32*x^2 - 112) + 1)/((x^2 - 4)^3 + 12*(x^2 - 4)^2 + 48*x^2 - 128)
Time = 0.09 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.90 \[ \int \frac {72-18 x^2+e^{16-32 x^2+24 x^4-8 x^6+x^8} \left (-4+x^2\right ) \left (774 x^8-1344 x^{10}+864 x^{12}-240 x^{14}+24 x^{16}\right )}{-4 x^7+x^9} \, dx=\frac {3}{x^6}-12\,{\mathrm {e}}^{x^8}\,{\mathrm {e}}^{16}\,{\mathrm {e}}^{-8\,x^6}\,{\mathrm {e}}^{24\,x^4}\,{\mathrm {e}}^{-32\,x^2}+3\,x^2\,{\mathrm {e}}^{x^8}\,{\mathrm {e}}^{16}\,{\mathrm {e}}^{-8\,x^6}\,{\mathrm {e}}^{24\,x^4}\,{\mathrm {e}}^{-32\,x^2} \] Input:
int(-(exp(log(x^2 - 4) - 32*x^2 + 24*x^4 - 8*x^6 + x^8 + 16)*(774*x^8 - 13 44*x^10 + 864*x^12 - 240*x^14 + 24*x^16) - 18*x^2 + 72)/(4*x^7 - x^9),x)
Output:
3/x^6 - 12*exp(x^8)*exp(16)*exp(-8*x^6)*exp(24*x^4)*exp(-32*x^2) + 3*x^2*e xp(x^8)*exp(16)*exp(-8*x^6)*exp(24*x^4)*exp(-32*x^2)
Time = 0.42 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.38 \[ \int \frac {72-18 x^2+e^{16-32 x^2+24 x^4-8 x^6+x^8} \left (-4+x^2\right ) \left (774 x^8-1344 x^{10}+864 x^{12}-240 x^{14}+24 x^{16}\right )}{-4 x^7+x^9} \, dx=\frac {3 e^{x^{8}+24 x^{4}} e^{16} x^{8}-12 e^{x^{8}+24 x^{4}} e^{16} x^{6}+3 e^{8 x^{6}+32 x^{2}}}{e^{8 x^{6}+32 x^{2}} x^{6}} \] Input:
int(((24*x^16-240*x^14+864*x^12-1344*x^10+774*x^8)*exp(log(x^2-4)+x^8-8*x^ 6+24*x^4-32*x^2+16)-18*x^2+72)/(x^9-4*x^7),x)
Output:
(3*(e**(x**8 + 24*x**4)*e**16*x**8 - 4*e**(x**8 + 24*x**4)*e**16*x**6 + e* *(8*x**6 + 32*x**2)))/(e**(8*x**6 + 32*x**2)*x**6)