Integrand size = 317, antiderivative size = 36 \[ \int \frac {-9 x^2+6 x^3-16 x^4+8 x^5-8 x^6+2 x^7-x^8+e^x \left (3 x^2-5 x^3+x^4+4 x^5-4 x^6+x^7\right )+\left (-9 x^2+6 x^3+2 x^4-4 x^5+6 x^6-2 x^7+x^8+e^x \left (6 x-10 x^2+2 x^3+8 x^4-8 x^5+2 x^6\right )\right ) \log \left (\frac {-1+x^2}{x}\right )+e^x \left (3-5 x+x^2+4 x^3-4 x^4+x^5\right ) \log ^2\left (\frac {-1+x^2}{x}\right )}{-36 x^4+24 x^5+8 x^6-16 x^7+24 x^8-8 x^9+4 x^{10}+\left (-72 x^3+48 x^4+16 x^5-32 x^6+48 x^7-16 x^8+8 x^9\right ) \log \left (\frac {-1+x^2}{x}\right )+\left (-36 x^2+24 x^3+8 x^4-16 x^5+24 x^6-8 x^7+4 x^8\right ) \log ^2\left (\frac {-1+x^2}{x}\right )} \, dx=\frac {1}{4} \left (\frac {e^x}{x \left (3-x+x^2\right )}+\frac {x}{x+\log \left (-\frac {1}{x}+x\right )}\right ) \]
Time = 0.18 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.03 \[ \int \frac {-9 x^2+6 x^3-16 x^4+8 x^5-8 x^6+2 x^7-x^8+e^x \left (3 x^2-5 x^3+x^4+4 x^5-4 x^6+x^7\right )+\left (-9 x^2+6 x^3+2 x^4-4 x^5+6 x^6-2 x^7+x^8+e^x \left (6 x-10 x^2+2 x^3+8 x^4-8 x^5+2 x^6\right )\right ) \log \left (\frac {-1+x^2}{x}\right )+e^x \left (3-5 x+x^2+4 x^3-4 x^4+x^5\right ) \log ^2\left (\frac {-1+x^2}{x}\right )}{-36 x^4+24 x^5+8 x^6-16 x^7+24 x^8-8 x^9+4 x^{10}+\left (-72 x^3+48 x^4+16 x^5-32 x^6+48 x^7-16 x^8+8 x^9\right ) \log \left (\frac {-1+x^2}{x}\right )+\left (-36 x^2+24 x^3+8 x^4-16 x^5+24 x^6-8 x^7+4 x^8\right ) \log ^2\left (\frac {-1+x^2}{x}\right )} \, dx=\frac {1}{4} \left (\frac {e^x}{3 x-x^2+x^3}+\frac {x}{x+\log \left (-\frac {1}{x}+x\right )}\right ) \]
Integrate[(-9*x^2 + 6*x^3 - 16*x^4 + 8*x^5 - 8*x^6 + 2*x^7 - x^8 + E^x*(3* x^2 - 5*x^3 + x^4 + 4*x^5 - 4*x^6 + x^7) + (-9*x^2 + 6*x^3 + 2*x^4 - 4*x^5 + 6*x^6 - 2*x^7 + x^8 + E^x*(6*x - 10*x^2 + 2*x^3 + 8*x^4 - 8*x^5 + 2*x^6 ))*Log[(-1 + x^2)/x] + E^x*(3 - 5*x + x^2 + 4*x^3 - 4*x^4 + x^5)*Log[(-1 + x^2)/x]^2)/(-36*x^4 + 24*x^5 + 8*x^6 - 16*x^7 + 24*x^8 - 8*x^9 + 4*x^10 + (-72*x^3 + 48*x^4 + 16*x^5 - 32*x^6 + 48*x^7 - 16*x^8 + 8*x^9)*Log[(-1 + x^2)/x] + (-36*x^2 + 24*x^3 + 8*x^4 - 16*x^5 + 24*x^6 - 8*x^7 + 4*x^8)*Log [(-1 + x^2)/x]^2),x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 5.04 (sec) , antiderivative size = 525, normalized size of antiderivative = 14.58, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {7239, 27, 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.
| \(\displaystyle \int \frac {-x^8+2 x^7-8 x^6+8 x^5-16 x^4+6 x^3-9 x^2+e^x \left (x^5-4 x^4+4 x^3+x^2-5 x+3\right ) \log ^2\left (\frac {x^2-1}{x}\right )+e^x \left (x^7-4 x^6+4 x^5+x^4-5 x^3+3 x^2\right )+\left (x^8-2 x^7+6 x^6-4 x^5+2 x^4+6 x^3-9 x^2+e^x \left (2 x^6-8 x^5+8 x^4+2 x^3-10 x^2+6 x\right )\right ) \log \left (\frac {x^2-1}{x}\right )}{4 x^{10}-8 x^9+24 x^8-16 x^7+8 x^6+24 x^5-36 x^4+\left (4 x^8-8 x^7+24 x^6-16 x^5+8 x^4+24 x^3-36 x^2\right ) \log ^2\left (\frac {x^2-1}{x}\right )+\left (8 x^9-16 x^8+48 x^7-32 x^6+16 x^5+48 x^4-72 x^3\right ) \log \left (\frac {x^2-1}{x}\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-\left (x^2-1\right ) \left (x \left (x^2-x+3\right )^2+2 e^x \left (x^3-4 x^2+5 x-3\right )\right ) x \log \left (x-\frac {1}{x}\right )+\left (\left (x^2+1\right ) \left (x^2-x+3\right )^2-e^x \left (x^5-4 x^4+4 x^3+x^2-5 x+3\right )\right ) x^2-e^x \left (x^5-4 x^4+4 x^3+x^2-5 x+3\right ) \log ^2\left (x-\frac {1}{x}\right )}{4 x^2 \left (1-x^2\right ) \left (x^2-x+3\right )^2 \left (x+\log \left (x-\frac {1}{x}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {\left (\left (x^2+1\right ) \left (x^2-x+3\right )^2-e^x \left (x^5-4 x^4+4 x^3+x^2-5 x+3\right )\right ) x^2+\left (1-x^2\right ) \left (x \left (x^2-x+3\right )^2-2 e^x \left (-x^3+4 x^2-5 x+3\right )\right ) \log \left (x-\frac {1}{x}\right ) x-e^x \left (x^5-4 x^4+4 x^3+x^2-5 x+3\right ) \log ^2\left (x-\frac {1}{x}\right )}{x^2 \left (1-x^2\right ) \left (x^2-x+3\right )^2 \left (x+\log \left (x-\frac {1}{x}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \frac {1}{4} \int \left (\frac {e^x \left (x^3-4 x^2+5 x-3\right )}{x^2 \left (x^2-x+3\right )^2}+\frac {\log \left (x-\frac {1}{x}\right ) x^2-x^2-\log \left (x-\frac {1}{x}\right )-1}{\left (x^2-1\right ) \left (x+\log \left (x-\frac {1}{x}\right )\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{4} \left (-\frac {1}{66} \left (11+3 i \sqrt {11}\right ) e^{\frac {1}{2}+\frac {i \sqrt {11}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-i \sqrt {11}-1\right )\right )+\frac {1}{66} \left (1+i \sqrt {11}\right ) e^{\frac {1}{2}+\frac {i \sqrt {11}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-i \sqrt {11}-1\right )\right )+\frac {i e^{\frac {1}{2}+\frac {i \sqrt {11}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-i \sqrt {11}-1\right )\right )}{3 \sqrt {11}}+\frac {5}{33} e^{\frac {1}{2}+\frac {i \sqrt {11}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x-i \sqrt {11}-1\right )\right )-\frac {1}{66} \left (11-3 i \sqrt {11}\right ) e^{\frac {1}{2}-\frac {i \sqrt {11}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+i \sqrt {11}-1\right )\right )+\frac {1}{66} \left (1-i \sqrt {11}\right ) e^{\frac {1}{2}-\frac {i \sqrt {11}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+i \sqrt {11}-1\right )\right )-\frac {i e^{\frac {1}{2}-\frac {i \sqrt {11}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+i \sqrt {11}-1\right )\right )}{3 \sqrt {11}}+\frac {5}{33} e^{\frac {1}{2}-\frac {i \sqrt {11}}{2}} \operatorname {ExpIntegralEi}\left (\frac {1}{2} \left (2 x+i \sqrt {11}-1\right )\right )+\frac {\left (1-i \sqrt {11}\right ) e^x}{33 \left (-2 x-i \sqrt {11}+1\right )}+\frac {10 e^x}{33 \left (-2 x-i \sqrt {11}+1\right )}+\frac {\left (1+i \sqrt {11}\right ) e^x}{33 \left (-2 x+i \sqrt {11}+1\right )}+\frac {10 e^x}{33 \left (-2 x+i \sqrt {11}+1\right )}+\frac {e^x}{3 x}-\frac {1}{\frac {x}{\log \left (x-\frac {1}{x}\right )}+1}\right )\) |
Int[(-9*x^2 + 6*x^3 - 16*x^4 + 8*x^5 - 8*x^6 + 2*x^7 - x^8 + E^x*(3*x^2 - 5*x^3 + x^4 + 4*x^5 - 4*x^6 + x^7) + (-9*x^2 + 6*x^3 + 2*x^4 - 4*x^5 + 6*x ^6 - 2*x^7 + x^8 + E^x*(6*x - 10*x^2 + 2*x^3 + 8*x^4 - 8*x^5 + 2*x^6))*Log [(-1 + x^2)/x] + E^x*(3 - 5*x + x^2 + 4*x^3 - 4*x^4 + x^5)*Log[(-1 + x^2)/ x]^2)/(-36*x^4 + 24*x^5 + 8*x^6 - 16*x^7 + 24*x^8 - 8*x^9 + 4*x^10 + (-72* x^3 + 48*x^4 + 16*x^5 - 32*x^6 + 48*x^7 - 16*x^8 + 8*x^9)*Log[(-1 + x^2)/x ] + (-36*x^2 + 24*x^3 + 8*x^4 - 16*x^5 + 24*x^6 - 8*x^7 + 4*x^8)*Log[(-1 + x^2)/x]^2),x]
((10*E^x)/(33*(1 - I*Sqrt[11] - 2*x)) + ((1 - I*Sqrt[11])*E^x)/(33*(1 - I* Sqrt[11] - 2*x)) + (10*E^x)/(33*(1 + I*Sqrt[11] - 2*x)) + ((1 + I*Sqrt[11] )*E^x)/(33*(1 + I*Sqrt[11] - 2*x)) + E^x/(3*x) + (5*E^(1/2 + (I/2)*Sqrt[11 ])*ExpIntegralEi[(-1 - I*Sqrt[11] + 2*x)/2])/33 + ((I/3)*E^(1/2 + (I/2)*Sq rt[11])*ExpIntegralEi[(-1 - I*Sqrt[11] + 2*x)/2])/Sqrt[11] + ((1 + I*Sqrt[ 11])*E^(1/2 + (I/2)*Sqrt[11])*ExpIntegralEi[(-1 - I*Sqrt[11] + 2*x)/2])/66 - ((11 + (3*I)*Sqrt[11])*E^(1/2 + (I/2)*Sqrt[11])*ExpIntegralEi[(-1 - I*S qrt[11] + 2*x)/2])/66 + (5*E^(1/2 - (I/2)*Sqrt[11])*ExpIntegralEi[(-1 + I* Sqrt[11] + 2*x)/2])/33 - ((I/3)*E^(1/2 - (I/2)*Sqrt[11])*ExpIntegralEi[(-1 + I*Sqrt[11] + 2*x)/2])/Sqrt[11] + ((1 - I*Sqrt[11])*E^(1/2 - (I/2)*Sqrt[ 11])*ExpIntegralEi[(-1 + I*Sqrt[11] + 2*x)/2])/66 - ((11 - (3*I)*Sqrt[11]) *E^(1/2 - (I/2)*Sqrt[11])*ExpIntegralEi[(-1 + I*Sqrt[11] + 2*x)/2])/66 - ( 1 + x/Log[-x^(-1) + x])^(-1))/4
3.26.72.3.1 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]]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(93\) vs. \(2(33)=66\).
Time = 32.37 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.61
| method | result | size |
| parallelrisch | \(-\frac {-9 \,{\mathrm e}^{x} x -9 x^{4}+9 x^{3}-27 x^{2}-9 \,{\mathrm e}^{x} \ln \left (\frac {x^{2}-1}{x}\right )}{36 x \left (\ln \left (\frac {x^{2}-1}{x}\right ) x^{2}+x^{3}-\ln \left (\frac {x^{2}-1}{x}\right ) x -x^{2}+3 \ln \left (\frac {x^{2}-1}{x}\right )+3 x \right )}\) | \(94\) |
| risch | \(\frac {{\mathrm e}^{x}}{4 x \left (x^{2}-x +3\right )}+\frac {x}{-2 i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \left (x^{2}-1\right )\right ) \operatorname {csgn}\left (\frac {i \left (x^{2}-1\right )}{x}\right )+2 i \pi \,\operatorname {csgn}\left (\frac {i}{x}\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-1\right )}{x}\right )}^{2}+2 i \pi \,\operatorname {csgn}\left (i \left (x^{2}-1\right )\right ) {\operatorname {csgn}\left (\frac {i \left (x^{2}-1\right )}{x}\right )}^{2}-2 i \pi {\operatorname {csgn}\left (\frac {i \left (x^{2}-1\right )}{x}\right )}^{3}+4 x -4 \ln \left (x \right )+4 \ln \left (x^{2}-1\right )}\) | \(142\) |
int(((x^5-4*x^4+4*x^3+x^2-5*x+3)*exp(x)*ln((x^2-1)/x)^2+((2*x^6-8*x^5+8*x^ 4+2*x^3-10*x^2+6*x)*exp(x)+x^8-2*x^7+6*x^6-4*x^5+2*x^4+6*x^3-9*x^2)*ln((x^ 2-1)/x)+(x^7-4*x^6+4*x^5+x^4-5*x^3+3*x^2)*exp(x)-x^8+2*x^7-8*x^6+8*x^5-16* x^4+6*x^3-9*x^2)/((4*x^8-8*x^7+24*x^6-16*x^5+8*x^4+24*x^3-36*x^2)*ln((x^2- 1)/x)^2+(8*x^9-16*x^8+48*x^7-32*x^6+16*x^5+48*x^4-72*x^3)*ln((x^2-1)/x)+4* x^10-8*x^9+24*x^8-16*x^7+8*x^6+24*x^5-36*x^4),x,method=_RETURNVERBOSE)
-1/36*(-9*exp(x)*x-9*x^4+9*x^3-27*x^2-9*exp(x)*ln((x^2-1)/x))/x/(ln((x^2-1 )/x)*x^2+x^3-ln((x^2-1)/x)*x-x^2+3*ln((x^2-1)/x)+3*x)
Leaf count of result is larger than twice the leaf count of optimal. 72 vs. \(2 (33) = 66\).
Time = 0.26 (sec) , antiderivative size = 72, normalized size of antiderivative = 2.00 \[ \int \frac {-9 x^2+6 x^3-16 x^4+8 x^5-8 x^6+2 x^7-x^8+e^x \left (3 x^2-5 x^3+x^4+4 x^5-4 x^6+x^7\right )+\left (-9 x^2+6 x^3+2 x^4-4 x^5+6 x^6-2 x^7+x^8+e^x \left (6 x-10 x^2+2 x^3+8 x^4-8 x^5+2 x^6\right )\right ) \log \left (\frac {-1+x^2}{x}\right )+e^x \left (3-5 x+x^2+4 x^3-4 x^4+x^5\right ) \log ^2\left (\frac {-1+x^2}{x}\right )}{-36 x^4+24 x^5+8 x^6-16 x^7+24 x^8-8 x^9+4 x^{10}+\left (-72 x^3+48 x^4+16 x^5-32 x^6+48 x^7-16 x^8+8 x^9\right ) \log \left (\frac {-1+x^2}{x}\right )+\left (-36 x^2+24 x^3+8 x^4-16 x^5+24 x^6-8 x^7+4 x^8\right ) \log ^2\left (\frac {-1+x^2}{x}\right )} \, dx=\frac {x^{4} - x^{3} + 3 \, x^{2} + x e^{x} + e^{x} \log \left (\frac {x^{2} - 1}{x}\right )}{4 \, {\left (x^{4} - x^{3} + 3 \, x^{2} + {\left (x^{3} - x^{2} + 3 \, x\right )} \log \left (\frac {x^{2} - 1}{x}\right )\right )}} \]
integrate(((x^5-4*x^4+4*x^3+x^2-5*x+3)*exp(x)*log((x^2-1)/x)^2+((2*x^6-8*x ^5+8*x^4+2*x^3-10*x^2+6*x)*exp(x)+x^8-2*x^7+6*x^6-4*x^5+2*x^4+6*x^3-9*x^2) *log((x^2-1)/x)+(x^7-4*x^6+4*x^5+x^4-5*x^3+3*x^2)*exp(x)-x^8+2*x^7-8*x^6+8 *x^5-16*x^4+6*x^3-9*x^2)/((4*x^8-8*x^7+24*x^6-16*x^5+8*x^4+24*x^3-36*x^2)* log((x^2-1)/x)^2+(8*x^9-16*x^8+48*x^7-32*x^6+16*x^5+48*x^4-72*x^3)*log((x^ 2-1)/x)+4*x^10-8*x^9+24*x^8-16*x^7+8*x^6+24*x^5-36*x^4),x, algorithm=\
1/4*(x^4 - x^3 + 3*x^2 + x*e^x + e^x*log((x^2 - 1)/x))/(x^4 - x^3 + 3*x^2 + (x^3 - x^2 + 3*x)*log((x^2 - 1)/x))
Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.86 \[ \int \frac {-9 x^2+6 x^3-16 x^4+8 x^5-8 x^6+2 x^7-x^8+e^x \left (3 x^2-5 x^3+x^4+4 x^5-4 x^6+x^7\right )+\left (-9 x^2+6 x^3+2 x^4-4 x^5+6 x^6-2 x^7+x^8+e^x \left (6 x-10 x^2+2 x^3+8 x^4-8 x^5+2 x^6\right )\right ) \log \left (\frac {-1+x^2}{x}\right )+e^x \left (3-5 x+x^2+4 x^3-4 x^4+x^5\right ) \log ^2\left (\frac {-1+x^2}{x}\right )}{-36 x^4+24 x^5+8 x^6-16 x^7+24 x^8-8 x^9+4 x^{10}+\left (-72 x^3+48 x^4+16 x^5-32 x^6+48 x^7-16 x^8+8 x^9\right ) \log \left (\frac {-1+x^2}{x}\right )+\left (-36 x^2+24 x^3+8 x^4-16 x^5+24 x^6-8 x^7+4 x^8\right ) \log ^2\left (\frac {-1+x^2}{x}\right )} \, dx=\frac {x}{4 x + 4 \log {\left (\frac {x^{2} - 1}{x} \right )}} + \frac {e^{x}}{4 x^{3} - 4 x^{2} + 12 x} \]
integrate(((x**5-4*x**4+4*x**3+x**2-5*x+3)*exp(x)*ln((x**2-1)/x)**2+((2*x* *6-8*x**5+8*x**4+2*x**3-10*x**2+6*x)*exp(x)+x**8-2*x**7+6*x**6-4*x**5+2*x* *4+6*x**3-9*x**2)*ln((x**2-1)/x)+(x**7-4*x**6+4*x**5+x**4-5*x**3+3*x**2)*e xp(x)-x**8+2*x**7-8*x**6+8*x**5-16*x**4+6*x**3-9*x**2)/((4*x**8-8*x**7+24* x**6-16*x**5+8*x**4+24*x**3-36*x**2)*ln((x**2-1)/x)**2+(8*x**9-16*x**8+48* x**7-32*x**6+16*x**5+48*x**4-72*x**3)*ln((x**2-1)/x)+4*x**10-8*x**9+24*x** 8-16*x**7+8*x**6+24*x**5-36*x**4),x)
Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (33) = 66\).
Time = 0.27 (sec) , antiderivative size = 105, normalized size of antiderivative = 2.92 \[ \int \frac {-9 x^2+6 x^3-16 x^4+8 x^5-8 x^6+2 x^7-x^8+e^x \left (3 x^2-5 x^3+x^4+4 x^5-4 x^6+x^7\right )+\left (-9 x^2+6 x^3+2 x^4-4 x^5+6 x^6-2 x^7+x^8+e^x \left (6 x-10 x^2+2 x^3+8 x^4-8 x^5+2 x^6\right )\right ) \log \left (\frac {-1+x^2}{x}\right )+e^x \left (3-5 x+x^2+4 x^3-4 x^4+x^5\right ) \log ^2\left (\frac {-1+x^2}{x}\right )}{-36 x^4+24 x^5+8 x^6-16 x^7+24 x^8-8 x^9+4 x^{10}+\left (-72 x^3+48 x^4+16 x^5-32 x^6+48 x^7-16 x^8+8 x^9\right ) \log \left (\frac {-1+x^2}{x}\right )+\left (-36 x^2+24 x^3+8 x^4-16 x^5+24 x^6-8 x^7+4 x^8\right ) \log ^2\left (\frac {-1+x^2}{x}\right )} \, dx=\frac {x^{4} - x^{3} + 3 \, x^{2} + {\left (x - \log \left (x\right )\right )} e^{x} + e^{x} \log \left (x + 1\right ) + e^{x} \log \left (x - 1\right )}{4 \, {\left (x^{4} - x^{3} + 3 \, x^{2} + {\left (x^{3} - x^{2} + 3 \, x\right )} \log \left (x + 1\right ) + {\left (x^{3} - x^{2} + 3 \, x\right )} \log \left (x - 1\right ) - {\left (x^{3} - x^{2} + 3 \, x\right )} \log \left (x\right )\right )}} \]
integrate(((x^5-4*x^4+4*x^3+x^2-5*x+3)*exp(x)*log((x^2-1)/x)^2+((2*x^6-8*x ^5+8*x^4+2*x^3-10*x^2+6*x)*exp(x)+x^8-2*x^7+6*x^6-4*x^5+2*x^4+6*x^3-9*x^2) *log((x^2-1)/x)+(x^7-4*x^6+4*x^5+x^4-5*x^3+3*x^2)*exp(x)-x^8+2*x^7-8*x^6+8 *x^5-16*x^4+6*x^3-9*x^2)/((4*x^8-8*x^7+24*x^6-16*x^5+8*x^4+24*x^3-36*x^2)* log((x^2-1)/x)^2+(8*x^9-16*x^8+48*x^7-32*x^6+16*x^5+48*x^4-72*x^3)*log((x^ 2-1)/x)+4*x^10-8*x^9+24*x^8-16*x^7+8*x^6+24*x^5-36*x^4),x, algorithm=\
1/4*(x^4 - x^3 + 3*x^2 + (x - log(x))*e^x + e^x*log(x + 1) + e^x*log(x - 1 ))/(x^4 - x^3 + 3*x^2 + (x^3 - x^2 + 3*x)*log(x + 1) + (x^3 - x^2 + 3*x)*l og(x - 1) - (x^3 - x^2 + 3*x)*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (33) = 66\).
Time = 0.67 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.53 \[ \int \frac {-9 x^2+6 x^3-16 x^4+8 x^5-8 x^6+2 x^7-x^8+e^x \left (3 x^2-5 x^3+x^4+4 x^5-4 x^6+x^7\right )+\left (-9 x^2+6 x^3+2 x^4-4 x^5+6 x^6-2 x^7+x^8+e^x \left (6 x-10 x^2+2 x^3+8 x^4-8 x^5+2 x^6\right )\right ) \log \left (\frac {-1+x^2}{x}\right )+e^x \left (3-5 x+x^2+4 x^3-4 x^4+x^5\right ) \log ^2\left (\frac {-1+x^2}{x}\right )}{-36 x^4+24 x^5+8 x^6-16 x^7+24 x^8-8 x^9+4 x^{10}+\left (-72 x^3+48 x^4+16 x^5-32 x^6+48 x^7-16 x^8+8 x^9\right ) \log \left (\frac {-1+x^2}{x}\right )+\left (-36 x^2+24 x^3+8 x^4-16 x^5+24 x^6-8 x^7+4 x^8\right ) \log ^2\left (\frac {-1+x^2}{x}\right )} \, dx=\frac {x^{4} - x^{3} + 3 \, x^{2} + x e^{x} + e^{x} \log \left (\frac {x^{2} - 1}{x}\right )}{4 \, {\left (x^{4} + x^{3} \log \left (\frac {x^{2} - 1}{x}\right ) - x^{3} - x^{2} \log \left (\frac {x^{2} - 1}{x}\right ) + 3 \, x^{2} + 3 \, x \log \left (\frac {x^{2} - 1}{x}\right )\right )}} \]
integrate(((x^5-4*x^4+4*x^3+x^2-5*x+3)*exp(x)*log((x^2-1)/x)^2+((2*x^6-8*x ^5+8*x^4+2*x^3-10*x^2+6*x)*exp(x)+x^8-2*x^7+6*x^6-4*x^5+2*x^4+6*x^3-9*x^2) *log((x^2-1)/x)+(x^7-4*x^6+4*x^5+x^4-5*x^3+3*x^2)*exp(x)-x^8+2*x^7-8*x^6+8 *x^5-16*x^4+6*x^3-9*x^2)/((4*x^8-8*x^7+24*x^6-16*x^5+8*x^4+24*x^3-36*x^2)* log((x^2-1)/x)^2+(8*x^9-16*x^8+48*x^7-32*x^6+16*x^5+48*x^4-72*x^3)*log((x^ 2-1)/x)+4*x^10-8*x^9+24*x^8-16*x^7+8*x^6+24*x^5-36*x^4),x, algorithm=\
1/4*(x^4 - x^3 + 3*x^2 + x*e^x + e^x*log((x^2 - 1)/x))/(x^4 + x^3*log((x^2 - 1)/x) - x^3 - x^2*log((x^2 - 1)/x) + 3*x^2 + 3*x*log((x^2 - 1)/x))
Time = 11.43 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.47 \[ \int \frac {-9 x^2+6 x^3-16 x^4+8 x^5-8 x^6+2 x^7-x^8+e^x \left (3 x^2-5 x^3+x^4+4 x^5-4 x^6+x^7\right )+\left (-9 x^2+6 x^3+2 x^4-4 x^5+6 x^6-2 x^7+x^8+e^x \left (6 x-10 x^2+2 x^3+8 x^4-8 x^5+2 x^6\right )\right ) \log \left (\frac {-1+x^2}{x}\right )+e^x \left (3-5 x+x^2+4 x^3-4 x^4+x^5\right ) \log ^2\left (\frac {-1+x^2}{x}\right )}{-36 x^4+24 x^5+8 x^6-16 x^7+24 x^8-8 x^9+4 x^{10}+\left (-72 x^3+48 x^4+16 x^5-32 x^6+48 x^7-16 x^8+8 x^9\right ) \log \left (\frac {-1+x^2}{x}\right )+\left (-36 x^2+24 x^3+8 x^4-16 x^5+24 x^6-8 x^7+4 x^8\right ) \log ^2\left (\frac {-1+x^2}{x}\right )} \, dx=\frac {x^2\,\ln \left (\frac {x^2-1}{x}\right )-x^3\,\ln \left (\frac {x^2-1}{x}\right )+{\mathrm {e}}^x\,\ln \left (\frac {x^2-1}{x}\right )-3\,x\,\ln \left (\frac {x^2-1}{x}\right )+x\,{\mathrm {e}}^x}{4\,x\,\left (x+\ln \left (\frac {x^2-1}{x}\right )\right )\,\left (x^2-x+3\right )} \]
int((exp(x)*(3*x^2 - 5*x^3 + x^4 + 4*x^5 - 4*x^6 + x^7) + log((x^2 - 1)/x) *(exp(x)*(6*x - 10*x^2 + 2*x^3 + 8*x^4 - 8*x^5 + 2*x^6) - 9*x^2 + 6*x^3 + 2*x^4 - 4*x^5 + 6*x^6 - 2*x^7 + x^8) - 9*x^2 + 6*x^3 - 16*x^4 + 8*x^5 - 8* x^6 + 2*x^7 - x^8 + exp(x)*log((x^2 - 1)/x)^2*(x^2 - 5*x + 4*x^3 - 4*x^4 + x^5 + 3))/(log((x^2 - 1)/x)*(48*x^4 - 72*x^3 + 16*x^5 - 32*x^6 + 48*x^7 - 16*x^8 + 8*x^9) + log((x^2 - 1)/x)^2*(24*x^3 - 36*x^2 + 8*x^4 - 16*x^5 + 24*x^6 - 8*x^7 + 4*x^8) - 36*x^4 + 24*x^5 + 8*x^6 - 16*x^7 + 24*x^8 - 8*x^ 9 + 4*x^10),x)