Integrand size = 118, antiderivative size = 24 \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=3-\left (3 e^{x/24}+x-\frac {3 x^2}{25}\right )^4 \]
Time = 0.05 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=-\frac {\left (75 e^{x/24}+(25-3 x) x\right )^4}{390625} \]
Integrate[(-10546875*E^(x/6) - 3125000*x^3 + 1875000*x^4 - 405000*x^5 + 37 800*x^6 - 1296*x^7 + E^(x/8)*(-84375000 + 9703125*x + 1265625*x^2) + E^(x/ 12)*(-84375000*x + 26859375*x^2 - 1586250*x^3 - 50625*x^4) + E^(x/24)*(-28 125000*x^2 + 13109375*x^3 - 1884375*x^4 + 80325*x^5 + 675*x^6))/781250,x]
Leaf count is larger than twice the leaf count of optimal. \(145\) vs. \(2(24)=48\).
Time = 0.59 (sec) , antiderivative size = 145, normalized size of antiderivative = 6.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {27, 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.
\(\displaystyle \int \frac {-1296 x^7+37800 x^6-405000 x^5+1875000 x^4-3125000 x^3+e^{x/8} \left (1265625 x^2+9703125 x-84375000\right )+e^{x/12} \left (-50625 x^4-1586250 x^3+26859375 x^2-84375000 x\right )+e^{x/24} \left (675 x^6+80325 x^5-1884375 x^4+13109375 x^3-28125000 x^2\right )-10546875 e^{x/6}}{781250} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \left (-1296 x^7+37800 x^6-405000 x^5+1875000 x^4-3125000 x^3-10546875 e^{x/6}-421875 e^{x/8} \left (-3 x^2-23 x+200\right )-5625 e^{x/12} \left (9 x^4+282 x^3-4775 x^2+15000 x\right )-25 e^{x/24} \left (-27 x^6-3213 x^5+75375 x^4-524375 x^3+1125000 x^2\right )\right )dx}{781250}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-162 x^8+5400 x^7+16200 e^{x/24} x^6-67500 x^6-405000 e^{x/24} x^5+375000 x^5+3375000 e^{x/24} x^4-607500 e^{x/12} x^4-781250 x^4-9375000 e^{x/24} x^3+10125000 e^{x/12} x^3-42187500 e^{x/12} x^2+10125000 e^{x/8} x^2-84375000 e^{x/8} x-63281250 e^{x/6}}{781250}\) |
Int[(-10546875*E^(x/6) - 3125000*x^3 + 1875000*x^4 - 405000*x^5 + 37800*x^ 6 - 1296*x^7 + E^(x/8)*(-84375000 + 9703125*x + 1265625*x^2) + E^(x/12)*(- 84375000*x + 26859375*x^2 - 1586250*x^3 - 50625*x^4) + E^(x/24)*(-28125000 *x^2 + 13109375*x^3 - 1884375*x^4 + 80325*x^5 + 675*x^6))/781250,x]
(-63281250*E^(x/6) - 84375000*E^(x/8)*x - 42187500*E^(x/12)*x^2 + 10125000 *E^(x/8)*x^2 - 9375000*E^(x/24)*x^3 + 10125000*E^(x/12)*x^3 - 781250*x^4 + 3375000*E^(x/24)*x^4 - 607500*E^(x/12)*x^4 + 375000*x^5 - 405000*E^(x/24) *x^5 - 67500*x^6 + 16200*E^(x/24)*x^6 + 5400*x^7 - 162*x^8)/781250
3.17.81.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Leaf count of result is larger than twice the leaf count of optimal. \(96\) vs. \(2(19)=38\).
Time = 0.17 (sec) , antiderivative size = 97, normalized size of antiderivative = 4.04
method | result | size |
risch | \(-81 \,{\mathrm e}^{\frac {x}{6}}+\frac {\left (10125000 x^{2}-84375000 x \right ) {\mathrm e}^{\frac {x}{8}}}{781250}+\frac {\left (-607500 x^{4}+10125000 x^{3}-42187500 x^{2}\right ) {\mathrm e}^{\frac {x}{12}}}{781250}+\frac {\left (16200 x^{6}-405000 x^{5}+3375000 x^{4}-9375000 x^{3}\right ) {\mathrm e}^{\frac {x}{24}}}{781250}-\frac {81 x^{8}}{390625}+\frac {108 x^{7}}{15625}-\frac {54 x^{6}}{625}+\frac {12 x^{5}}{25}-x^{4}\) | \(97\) |
derivativedivides | \(-81 \,{\mathrm e}^{\frac {x}{6}}+\frac {324 \,{\mathrm e}^{\frac {x}{8}} x^{2}}{25}-108 \,{\mathrm e}^{\frac {x}{8}} x -\frac {486 \,{\mathrm e}^{\frac {x}{12}} x^{4}}{625}+\frac {324 \,{\mathrm e}^{\frac {x}{12}} x^{3}}{25}-54 \,{\mathrm e}^{\frac {x}{12}} x^{2}+\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{6}}{15625}-\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{5}}{625}+\frac {108 \,{\mathrm e}^{\frac {x}{24}} x^{4}}{25}-12 \,{\mathrm e}^{\frac {x}{24}} x^{3}-\frac {81 x^{8}}{390625}+\frac {108 x^{7}}{15625}-\frac {54 x^{6}}{625}+\frac {12 x^{5}}{25}-x^{4}\) | \(124\) |
default | \(-81 \,{\mathrm e}^{\frac {x}{6}}+\frac {324 \,{\mathrm e}^{\frac {x}{8}} x^{2}}{25}-108 \,{\mathrm e}^{\frac {x}{8}} x -\frac {486 \,{\mathrm e}^{\frac {x}{12}} x^{4}}{625}+\frac {324 \,{\mathrm e}^{\frac {x}{12}} x^{3}}{25}-54 \,{\mathrm e}^{\frac {x}{12}} x^{2}+\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{6}}{15625}-\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{5}}{625}+\frac {108 \,{\mathrm e}^{\frac {x}{24}} x^{4}}{25}-12 \,{\mathrm e}^{\frac {x}{24}} x^{3}-\frac {81 x^{8}}{390625}+\frac {108 x^{7}}{15625}-\frac {54 x^{6}}{625}+\frac {12 x^{5}}{25}-x^{4}\) | \(124\) |
parallelrisch | \(-81 \,{\mathrm e}^{\frac {x}{6}}+\frac {324 \,{\mathrm e}^{\frac {x}{8}} x^{2}}{25}-108 \,{\mathrm e}^{\frac {x}{8}} x -\frac {486 \,{\mathrm e}^{\frac {x}{12}} x^{4}}{625}+\frac {324 \,{\mathrm e}^{\frac {x}{12}} x^{3}}{25}-54 \,{\mathrm e}^{\frac {x}{12}} x^{2}+\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{6}}{15625}-\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{5}}{625}+\frac {108 \,{\mathrm e}^{\frac {x}{24}} x^{4}}{25}-12 \,{\mathrm e}^{\frac {x}{24}} x^{3}-\frac {81 x^{8}}{390625}+\frac {108 x^{7}}{15625}-\frac {54 x^{6}}{625}+\frac {12 x^{5}}{25}-x^{4}\) | \(124\) |
parts | \(-81 \,{\mathrm e}^{\frac {x}{6}}+\frac {324 \,{\mathrm e}^{\frac {x}{8}} x^{2}}{25}-108 \,{\mathrm e}^{\frac {x}{8}} x -\frac {486 \,{\mathrm e}^{\frac {x}{12}} x^{4}}{625}+\frac {324 \,{\mathrm e}^{\frac {x}{12}} x^{3}}{25}-54 \,{\mathrm e}^{\frac {x}{12}} x^{2}+\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{6}}{15625}-\frac {324 \,{\mathrm e}^{\frac {x}{24}} x^{5}}{625}+\frac {108 \,{\mathrm e}^{\frac {x}{24}} x^{4}}{25}-12 \,{\mathrm e}^{\frac {x}{24}} x^{3}-\frac {81 x^{8}}{390625}+\frac {108 x^{7}}{15625}-\frac {54 x^{6}}{625}+\frac {12 x^{5}}{25}-x^{4}\) | \(124\) |
int(-27/2*exp(1/24*x)^4+1/781250*(1265625*x^2+9703125*x-84375000)*exp(1/24 *x)^3+1/781250*(-50625*x^4-1586250*x^3+26859375*x^2-84375000*x)*exp(1/24*x )^2+1/781250*(675*x^6+80325*x^5-1884375*x^4+13109375*x^3-28125000*x^2)*exp (1/24*x)-648/390625*x^7+756/15625*x^6-324/625*x^5+12/5*x^4-4*x^3,x,method= _RETURNVERBOSE)
-81*exp(1/6*x)+1/781250*(10125000*x^2-84375000*x)*exp(1/8*x)+1/781250*(-60 7500*x^4+10125000*x^3-42187500*x^2)*exp(1/12*x)+1/781250*(16200*x^6-405000 *x^5+3375000*x^4-9375000*x^3)*exp(1/24*x)-81/390625*x^8+108/15625*x^7-54/6 25*x^6+12/25*x^5-x^4
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (21) = 42\).
Time = 0.24 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.00 \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=-\frac {81}{390625} \, x^{8} + \frac {108}{15625} \, x^{7} - \frac {54}{625} \, x^{6} + \frac {12}{25} \, x^{5} - x^{4} + \frac {108}{25} \, {\left (3 \, x^{2} - 25 \, x\right )} e^{\left (\frac {1}{8} \, x\right )} - \frac {54}{625} \, {\left (9 \, x^{4} - 150 \, x^{3} + 625 \, x^{2}\right )} e^{\left (\frac {1}{12} \, x\right )} + \frac {12}{15625} \, {\left (27 \, x^{6} - 675 \, x^{5} + 5625 \, x^{4} - 15625 \, x^{3}\right )} e^{\left (\frac {1}{24} \, x\right )} - 81 \, e^{\left (\frac {1}{6} \, x\right )} \]
integrate(-27/2*exp(1/24*x)^4+1/781250*(1265625*x^2+9703125*x-84375000)*ex p(1/24*x)^3+1/781250*(-50625*x^4-1586250*x^3+26859375*x^2-84375000*x)*exp( 1/24*x)^2+1/781250*(675*x^6+80325*x^5-1884375*x^4+13109375*x^3-28125000*x^ 2)*exp(1/24*x)-648/390625*x^7+756/15625*x^6-324/625*x^5+12/5*x^4-4*x^3,x, algorithm=\
-81/390625*x^8 + 108/15625*x^7 - 54/625*x^6 + 12/25*x^5 - x^4 + 108/25*(3* x^2 - 25*x)*e^(1/8*x) - 54/625*(9*x^4 - 150*x^3 + 625*x^2)*e^(1/12*x) + 12 /15625*(27*x^6 - 675*x^5 + 5625*x^4 - 15625*x^3)*e^(1/24*x) - 81*e^(1/6*x)
Leaf count of result is larger than twice the leaf count of optimal. 100 vs. \(2 (17) = 34\).
Time = 0.11 (sec) , antiderivative size = 100, normalized size of antiderivative = 4.17 \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=- \frac {81 x^{8}}{390625} + \frac {108 x^{7}}{15625} - \frac {54 x^{6}}{625} + \frac {12 x^{5}}{25} - x^{4} + \frac {\left (3164062500 x^{2} - 26367187500 x\right ) e^{\frac {x}{8}}}{244140625} + \frac {\left (- 189843750 x^{4} + 3164062500 x^{3} - 13183593750 x^{2}\right ) e^{\frac {x}{12}}}{244140625} + \frac {\left (5062500 x^{6} - 126562500 x^{5} + 1054687500 x^{4} - 2929687500 x^{3}\right ) e^{\frac {x}{24}}}{244140625} - 81 e^{\frac {x}{6}} \]
integrate(-27/2*exp(1/24*x)**4+1/781250*(1265625*x**2+9703125*x-84375000)* exp(1/24*x)**3+1/781250*(-50625*x**4-1586250*x**3+26859375*x**2-84375000*x )*exp(1/24*x)**2+1/781250*(675*x**6+80325*x**5-1884375*x**4+13109375*x**3- 28125000*x**2)*exp(1/24*x)-648/390625*x**7+756/15625*x**6-324/625*x**5+12/ 5*x**4-4*x**3,x)
-81*x**8/390625 + 108*x**7/15625 - 54*x**6/625 + 12*x**5/25 - x**4 + (3164 062500*x**2 - 26367187500*x)*exp(x/8)/244140625 + (-189843750*x**4 + 31640 62500*x**3 - 13183593750*x**2)*exp(x/12)/244140625 + (5062500*x**6 - 12656 2500*x**5 + 1054687500*x**4 - 2929687500*x**3)*exp(x/24)/244140625 - 81*ex p(x/6)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (21) = 42\).
Time = 0.19 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.00 \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=-\frac {81}{390625} \, x^{8} + \frac {108}{15625} \, x^{7} - \frac {54}{625} \, x^{6} + \frac {12}{25} \, x^{5} - x^{4} + \frac {108}{25} \, {\left (3 \, x^{2} - 25 \, x\right )} e^{\left (\frac {1}{8} \, x\right )} - \frac {54}{625} \, {\left (9 \, x^{4} - 150 \, x^{3} + 625 \, x^{2}\right )} e^{\left (\frac {1}{12} \, x\right )} + \frac {12}{15625} \, {\left (27 \, x^{6} - 675 \, x^{5} + 5625 \, x^{4} - 15625 \, x^{3}\right )} e^{\left (\frac {1}{24} \, x\right )} - 81 \, e^{\left (\frac {1}{6} \, x\right )} \]
integrate(-27/2*exp(1/24*x)^4+1/781250*(1265625*x^2+9703125*x-84375000)*ex p(1/24*x)^3+1/781250*(-50625*x^4-1586250*x^3+26859375*x^2-84375000*x)*exp( 1/24*x)^2+1/781250*(675*x^6+80325*x^5-1884375*x^4+13109375*x^3-28125000*x^ 2)*exp(1/24*x)-648/390625*x^7+756/15625*x^6-324/625*x^5+12/5*x^4-4*x^3,x, algorithm=\
-81/390625*x^8 + 108/15625*x^7 - 54/625*x^6 + 12/25*x^5 - x^4 + 108/25*(3* x^2 - 25*x)*e^(1/8*x) - 54/625*(9*x^4 - 150*x^3 + 625*x^2)*e^(1/12*x) + 12 /15625*(27*x^6 - 675*x^5 + 5625*x^4 - 15625*x^3)*e^(1/24*x) - 81*e^(1/6*x)
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (21) = 42\).
Time = 0.26 (sec) , antiderivative size = 96, normalized size of antiderivative = 4.00 \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=-\frac {81}{390625} \, x^{8} + \frac {108}{15625} \, x^{7} - \frac {54}{625} \, x^{6} + \frac {12}{25} \, x^{5} - x^{4} + \frac {108}{25} \, {\left (3 \, x^{2} - 25 \, x\right )} e^{\left (\frac {1}{8} \, x\right )} - \frac {54}{625} \, {\left (9 \, x^{4} - 150 \, x^{3} + 625 \, x^{2}\right )} e^{\left (\frac {1}{12} \, x\right )} + \frac {12}{15625} \, {\left (27 \, x^{6} - 675 \, x^{5} + 5625 \, x^{4} - 15625 \, x^{3}\right )} e^{\left (\frac {1}{24} \, x\right )} - 81 \, e^{\left (\frac {1}{6} \, x\right )} \]
integrate(-27/2*exp(1/24*x)^4+1/781250*(1265625*x^2+9703125*x-84375000)*ex p(1/24*x)^3+1/781250*(-50625*x^4-1586250*x^3+26859375*x^2-84375000*x)*exp( 1/24*x)^2+1/781250*(675*x^6+80325*x^5-1884375*x^4+13109375*x^3-28125000*x^ 2)*exp(1/24*x)-648/390625*x^7+756/15625*x^6-324/625*x^5+12/5*x^4-4*x^3,x, algorithm=\
-81/390625*x^8 + 108/15625*x^7 - 54/625*x^6 + 12/25*x^5 - x^4 + 108/25*(3* x^2 - 25*x)*e^(1/8*x) - 54/625*(9*x^4 - 150*x^3 + 625*x^2)*e^(1/12*x) + 12 /15625*(27*x^6 - 675*x^5 + 5625*x^4 - 15625*x^3)*e^(1/24*x) - 81*e^(1/6*x)
Time = 8.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {-10546875 e^{x/6}-3125000 x^3+1875000 x^4-405000 x^5+37800 x^6-1296 x^7+e^{x/8} \left (-84375000+9703125 x+1265625 x^2\right )+e^{x/12} \left (-84375000 x+26859375 x^2-1586250 x^3-50625 x^4\right )+e^{x/24} \left (-28125000 x^2+13109375 x^3-1884375 x^4+80325 x^5+675 x^6\right )}{781250} \, dx=-\frac {{\left (25\,x+75\,{\mathrm {e}}^{x/24}-3\,x^2\right )}^4}{390625} \]
int((exp(x/8)*(9703125*x + 1265625*x^2 - 84375000))/781250 - (27*exp(x/6)) /2 + (exp(x/24)*(13109375*x^3 - 28125000*x^2 - 1884375*x^4 + 80325*x^5 + 6 75*x^6))/781250 - (exp(x/12)*(84375000*x - 26859375*x^2 + 1586250*x^3 + 50 625*x^4))/781250 - 4*x^3 + (12*x^4)/5 - (324*x^5)/625 + (756*x^6)/15625 - (648*x^7)/390625,x)