Integrand size = 101, antiderivative size = 28 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=e^x \left (x^2-\log \left (\frac {2}{5 \left (8+\frac {x}{-2+x^2}\right )}\right )\right ) \]
Time = 0.07 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=e^x \left (x^2-\log \left (\frac {2 \left (-2+x^2\right )}{5 \left (-16+x+8 x^2\right )}\right )\right ) \]
Integrate[(E^x*(-2 + 64*x + 27*x^2 - 66*x^3 - 30*x^4 + 17*x^5 + 8*x^6) + E ^x*(-32 + 2*x + 32*x^2 - x^3 - 8*x^4)*Log[(-4 + 2*x^2)/(-80 + 5*x + 40*x^2 )])/(32 - 2*x - 32*x^2 + x^3 + 8*x^4),x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 14.20 (sec) , antiderivative size = 723, normalized size of antiderivative = 25.82, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2463, 7239, 7276, 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.
| \(\displaystyle \int \frac {e^x \left (-8 x^4-x^3+32 x^2+2 x-32\right ) \log \left (\frac {2 x^2-4}{40 x^2+5 x-80}\right )+e^x \left (8 x^6+17 x^5-30 x^4-66 x^3+27 x^2+64 x-2\right )}{8 x^4+x^3-32 x^2-2 x+32} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (\frac {x \left (e^x \left (-8 x^4-x^3+32 x^2+2 x-32\right ) \log \left (\frac {2 x^2-4}{40 x^2+5 x-80}\right )+e^x \left (8 x^6+17 x^5-30 x^4-66 x^3+27 x^2+64 x-2\right )\right )}{2 \left (x^2-2\right )}+\frac {(-8 x-1) \left (e^x \left (-8 x^4-x^3+32 x^2+2 x-32\right ) \log \left (\frac {2 x^2-4}{40 x^2+5 x-80}\right )+e^x \left (8 x^6+17 x^5-30 x^4-66 x^3+27 x^2+64 x-2\right )\right )}{2 \left (8 x^2+x-16\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^x \left (8 x^6+17 x^5-30 x^4-66 x^3+27 x^2-\left (8 x^4+x^3-32 x^2-2 x+32\right ) \log \left (\frac {2 \left (x^2-2\right )}{5 \left (8 x^2+x-16\right )}\right )+64 x-2\right )}{\left (-8 x^2-x+16\right ) \left (2-x^2\right )}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {27 e^x x^2}{\left (x^2-2\right ) \left (8 x^2+x-16\right )}+\frac {64 e^x x}{\left (x^2-2\right ) \left (8 x^2+x-16\right )}-\frac {2 e^x}{\left (x^2-2\right ) \left (8 x^2+x-16\right )}-e^x \log \left (\frac {2 \left (x^2-2\right )}{5 \left (8 x^2+x-16\right )}\right )+\frac {8 e^x x^6}{\left (x^2-2\right ) \left (8 x^2+x-16\right )}+\frac {17 e^x x^5}{\left (x^2-2\right ) \left (8 x^2+x-16\right )}-\frac {30 e^x x^4}{\left (x^2-2\right ) \left (8 x^2+x-16\right )}-\frac {66 e^x x^3}{\left (x^2-2\right ) \left (8 x^2+x-16\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{342} \left (171+\sqrt {57}\right ) e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )-\frac {3}{38} \left (171-\sqrt {57}\right ) e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )-\frac {11}{456} \left (171-257 \sqrt {57}\right ) e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )+\frac {5 \left (22059-385 \sqrt {57}\right ) e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )}{3648}+\frac {17 \left (43947-33281 \sqrt {57}\right ) e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )}{175104}-\frac {\left (2867499-82561 \sqrt {57}\right ) e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )}{175104}-\frac {512 e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )}{3 \sqrt {57}}-e^{\frac {1}{16} \left (3 \sqrt {57}-1\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x-3 \sqrt {57}+1\right )\right )-\frac {\left (2867499+82561 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )}{175104}+\frac {17 \left (43947+33281 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )}{175104}+\frac {5 \left (22059+385 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )}{3648}-\frac {11}{456} \left (171+257 \sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )-\frac {3}{38} \left (171+\sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )+\frac {1}{342} \left (171-\sqrt {57}\right ) e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )+\frac {512 e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )}{3 \sqrt {57}}-e^{\frac {1}{16} \left (-1-3 \sqrt {57}\right )} \operatorname {ExpIntegralEi}\left (\frac {1}{16} \left (16 x+3 \sqrt {57}+1\right )\right )+e^x x^2-e^x \log \left (\frac {2 \left (2-x^2\right )}{5 \left (-8 x^2-x+16\right )}\right )\) |
Int[(E^x*(-2 + 64*x + 27*x^2 - 66*x^3 - 30*x^4 + 17*x^5 + 8*x^6) + E^x*(-3 2 + 2*x + 32*x^2 - x^3 - 8*x^4)*Log[(-4 + 2*x^2)/(-80 + 5*x + 40*x^2)])/(3 2 - 2*x - 32*x^2 + x^3 + 8*x^4),x]
E^x*x^2 - E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/1 6] - (512*E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/1 6])/(3*Sqrt[57]) - ((2867499 - 82561*Sqrt[57])*E^((-1 + 3*Sqrt[57])/16)*Ex pIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16])/175104 + (17*(43947 - 33281*Sqrt[ 57])*E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16])/1 75104 + (5*(22059 - 385*Sqrt[57])*E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[( 1 - 3*Sqrt[57] + 16*x)/16])/3648 - (11*(171 - 257*Sqrt[57])*E^((-1 + 3*Sqr t[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16])/456 - (3*(171 - Sqrt [57])*E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[57] + 16*x)/16])/ 38 + ((171 + Sqrt[57])*E^((-1 + 3*Sqrt[57])/16)*ExpIntegralEi[(1 - 3*Sqrt[ 57] + 16*x)/16])/342 - E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[ 57] + 16*x)/16] + (512*E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[ 57] + 16*x)/16])/(3*Sqrt[57]) + ((171 - Sqrt[57])*E^((-1 - 3*Sqrt[57])/16) *ExpIntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/342 - (3*(171 + Sqrt[57])*E^(( -1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/38 - (11*( 171 + 257*Sqrt[57])*E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/456 + (5*(22059 + 385*Sqrt[57])*E^((-1 - 3*Sqrt[57])/16)*Exp IntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/3648 + (17*(43947 + 33281*Sqrt[57] )*E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi[(1 + 3*Sqrt[57] + 16*x)/16])/1751 04 - ((2867499 + 82561*Sqrt[57])*E^((-1 - 3*Sqrt[57])/16)*ExpIntegralEi...
3.9.20.3.1 Defintions of rubi rules used
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 4.53 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07
| method | result | size |
| parallelrisch | \({\mathrm e}^{x} x^{2}-{\mathrm e}^{x} \ln \left (\frac {\frac {2 x^{2}}{5}-\frac {4}{5}}{8 x^{2}+x -16}\right )\) | \(30\) |
| risch | \({\mathrm e}^{x} \ln \left (x^{2}+\frac {1}{8} x -2\right )-\ln \left (x^{2}-2\right ) {\mathrm e}^{x}+\frac {i {\mathrm e}^{x} \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}+\frac {1}{8} x -2}\right ) \operatorname {csgn}\left (i \left (x^{2}-2\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )}{x^{2}+\frac {1}{8} x -2}\right )}{2}-\frac {i {\mathrm e}^{x} \pi \,\operatorname {csgn}\left (\frac {i}{x^{2}+\frac {1}{8} x -2}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )}{x^{2}+\frac {1}{8} x -2}\right )}^{2}}{2}-\frac {i {\mathrm e}^{x} \pi \,\operatorname {csgn}\left (i \left (x^{2}-2\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )}{x^{2}+\frac {1}{8} x -2}\right )}^{2}}{2}+\frac {i {\mathrm e}^{x} \pi {\operatorname {csgn}\left (\frac {i \left (x^{2}-2\right )}{x^{2}+\frac {1}{8} x -2}\right )}^{3}}{2}+{\mathrm e}^{x} x^{2}+2 \,{\mathrm e}^{x} \ln \left (2\right )+{\mathrm e}^{x} \ln \left (5\right )\) | \(193\) |
int(((-8*x^4-x^3+32*x^2+2*x-32)*exp(x)*ln((2*x^2-4)/(40*x^2+5*x-80))+(8*x^ 6+17*x^5-30*x^4-66*x^3+27*x^2+64*x-2)*exp(x))/(8*x^4+x^3-32*x^2-2*x+32),x, method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=x^{2} e^{x} - e^{x} \log \left (\frac {2 \, {\left (x^{2} - 2\right )}}{5 \, {\left (8 \, x^{2} + x - 16\right )}}\right ) \]
integrate(((-8*x^4-x^3+32*x^2+2*x-32)*exp(x)*log((2*x^2-4)/(40*x^2+5*x-80) )+(8*x^6+17*x^5-30*x^4-66*x^3+27*x^2+64*x-2)*exp(x))/(8*x^4+x^3-32*x^2-2*x +32),x, algorithm=\
Time = 0.45 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=\left (x^{2} - \log {\left (\frac {2 x^{2} - 4}{40 x^{2} + 5 x - 80} \right )}\right ) e^{x} \]
integrate(((-8*x**4-x**3+32*x**2+2*x-32)*exp(x)*ln((2*x**2-4)/(40*x**2+5*x -80))+(8*x**6+17*x**5-30*x**4-66*x**3+27*x**2+64*x-2)*exp(x))/(8*x**4+x**3 -32*x**2-2*x+32),x)
Time = 0.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx={\left (x^{2} + \log \left (5\right ) - \log \left (2\right )\right )} e^{x} + e^{x} \log \left (8 \, x^{2} + x - 16\right ) - e^{x} \log \left (x^{2} - 2\right ) \]
integrate(((-8*x^4-x^3+32*x^2+2*x-32)*exp(x)*log((2*x^2-4)/(40*x^2+5*x-80) )+(8*x^6+17*x^5-30*x^4-66*x^3+27*x^2+64*x-2)*exp(x))/(8*x^4+x^3-32*x^2-2*x +32),x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=x^{2} e^{x} - e^{x} \log \left (\frac {2 \, {\left (x^{2} - 2\right )}}{5 \, {\left (8 \, x^{2} + x - 16\right )}}\right ) \]
integrate(((-8*x^4-x^3+32*x^2+2*x-32)*exp(x)*log((2*x^2-4)/(40*x^2+5*x-80) )+(8*x^6+17*x^5-30*x^4-66*x^3+27*x^2+64*x-2)*exp(x))/(8*x^4+x^3-32*x^2-2*x +32),x, algorithm=\
Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {e^x \left (-2+64 x+27 x^2-66 x^3-30 x^4+17 x^5+8 x^6\right )+e^x \left (-32+2 x+32 x^2-x^3-8 x^4\right ) \log \left (\frac {-4+2 x^2}{-80+5 x+40 x^2}\right )}{32-2 x-32 x^2+x^3+8 x^4} \, dx=-{\mathrm {e}}^x\,\left (\ln \left (\frac {2\,x^2-4}{40\,x^2+5\,x-80}\right )-x^2\right ) \]