Integrand size = 120, antiderivative size = 31 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=-x+\frac {3}{5} x^2 (4+x) \left (-e^x+x\right ) \log \left (\left (\frac {4}{x}+x\right )^2\right ) \] Output:
3/5*ln((x+4/x)^2)*x^2*(x-exp(x))*(4+x)-x
Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.13 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=\frac {1}{5} \left (-5 x+3 x^2 (4+x) \left (-e^x+x\right ) \log \left (\frac {\left (4+x^2\right )^2}{x^2}\right )\right ) \] Input:
Integrate[(-20 - 101*x^2 - 24*x^3 + 24*x^4 + 6*x^5 + E^x*(96*x + 24*x^2 - 24*x^3 - 6*x^4) + (144*x^2 + 48*x^3 + 36*x^4 + 12*x^5 + E^x*(-96*x - 84*x^ 2 - 36*x^3 - 21*x^4 - 3*x^5))*Log[(16 + 8*x^2 + x^4)/x^2])/(20 + 5*x^2),x]
Output:
(-5*x + 3*x^2*(4 + x)*(-E^x + x)*Log[(4 + x^2)^2/x^2])/5
Leaf count is larger than twice the leaf count of optimal. \(86\) vs. \(2(31)=62\).
Time = 3.89 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.77, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.017, Rules used = {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.
\(\displaystyle \int \frac {6 x^5+24 x^4-24 x^3-101 x^2+e^x \left (-6 x^4-24 x^3+24 x^2+96 x\right )+\left (12 x^5+36 x^4+48 x^3+144 x^2+e^x \left (-3 x^5-21 x^4-36 x^3-84 x^2-96 x\right )\right ) \log \left (\frac {x^4+8 x^2+16}{x^2}\right )-20}{5 x^2+20} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (-\frac {101 x^2}{5 \left (x^2+4\right )}-\frac {4}{x^2+4}+\frac {144 x^2 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )}{5 \left (x^2+4\right )}+\frac {6 x^5}{5 \left (x^2+4\right )}+\frac {12 x^5 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )}{5 \left (x^2+4\right )}+\frac {24 x^4}{5 \left (x^2+4\right )}+\frac {36 x^4 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )}{5 \left (x^2+4\right )}-\frac {24 x^3}{5 \left (x^2+4\right )}+\frac {48 x^3 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )}{5 \left (x^2+4\right )}-\frac {3 e^x x \left (2 x^3+8 x^2+12 x^2 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )+28 x \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )+32 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )+x^4 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )+7 x^3 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )-8 x-32\right )}{5 \left (x^2+4\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {12}{5} e^x x^2 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )+\frac {3}{5} x^4 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )-\frac {3}{5} e^x x^3 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )+\frac {12}{5} x^3 \log \left (\frac {\left (x^2+4\right )^2}{x^2}\right )-x\) |
Input:
Int[(-20 - 101*x^2 - 24*x^3 + 24*x^4 + 6*x^5 + E^x*(96*x + 24*x^2 - 24*x^3 - 6*x^4) + (144*x^2 + 48*x^3 + 36*x^4 + 12*x^5 + E^x*(-96*x - 84*x^2 - 36 *x^3 - 21*x^4 - 3*x^5))*Log[(16 + 8*x^2 + x^4)/x^2])/(20 + 5*x^2),x]
Output:
-x - (12*E^x*x^2*Log[(4 + x^2)^2/x^2])/5 + (12*x^3*Log[(4 + x^2)^2/x^2])/5 - (3*E^x*x^3*Log[(4 + x^2)^2/x^2])/5 + (3*x^4*Log[(4 + x^2)^2/x^2])/5
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.15 (sec) , antiderivative size = 1157, normalized size of antiderivative = 37.32
\[\text {Expression too large to display}\]
Input:
int((((-3*x^5-21*x^4-36*x^3-84*x^2-96*x)*exp(x)+12*x^5+36*x^4+48*x^3+144*x ^2)*ln((x^4+8*x^2+16)/x^2)+(-6*x^4-24*x^3+24*x^2+96*x)*exp(x)+6*x^5+24*x^4 -24*x^3-101*x^2-20)/(5*x^2+20),x)
Output:
-x-6/5*x^4*ln(x)-24/5*x^3*ln(x)+6/5*x^3*exp(x)*ln(x)+24/5*x^2*exp(x)*ln(x) +3/10*I*Pi*x^3*csgn(I*(x^2+4))^2*csgn(I*(x^2+4)^2)*exp(x)+6/5*I*Pi*x^2*csg n(I*(x^2+4))^2*csgn(I*(x^2+4)^2)*exp(x)-3/5*I*Pi*x^3*csgn(I*(x^2+4))*csgn( I*(x^2+4)^2)^2*exp(x)-12/5*I*Pi*x^2*csgn(I*(x^2+4))*csgn(I*(x^2+4)^2)^2*ex p(x)+3/5*I*Pi*x^3*csgn(I*x^2)^2*csgn(I*x)*exp(x)-3/10*I*Pi*x^4*csgn(I*(x^2 +4)^2)*csgn(I/x^2*(x^2+4)^2)*csgn(I/x^2)-6/5*I*Pi*x^3*csgn(I*(x^2+4)^2)*cs gn(I/x^2*(x^2+4)^2)*csgn(I/x^2)+12/5*I*Pi*x^2*csgn(I*x^2)^2*csgn(I*x)*exp( x)-3/10*I*Pi*x^3*csgn(I*x^2)*csgn(I*x)^2*exp(x)-6/5*I*Pi*x^2*csgn(I*x^2)*c sgn(I*x)^2*exp(x)-3/10*I*Pi*x^3*csgn(I*(x^2+4)^2)*csgn(I/x^2*(x^2+4)^2)^2* exp(x)+6/5*I*Pi*x^3*csgn(I*(x^2+4)^2)*csgn(I/x^2*(x^2+4)^2)^2+3/10*I*Pi*x^ 4*csgn(I/x^2*(x^2+4)^2)^2*csgn(I/x^2)+6/5*I*Pi*x^3*csgn(I/x^2*(x^2+4)^2)^2 *csgn(I/x^2)-3/5*I*Pi*x^4*csgn(I*x^2)^2*csgn(I*x)-12/5*I*Pi*x^3*csgn(I*x^2 )^2*csgn(I*x)-3/10*I*Pi*x^4*csgn(I*(x^2+4))^2*csgn(I*(x^2+4)^2)-6/5*I*Pi*x ^3*csgn(I*(x^2+4))^2*csgn(I*(x^2+4)^2)+3/5*I*Pi*x^4*csgn(I*(x^2+4))*csgn(I *(x^2+4)^2)^2+12/5*I*Pi*x^3*csgn(I*(x^2+4))*csgn(I*(x^2+4)^2)^2+3/10*I*Pi* x^3*csgn(I/x^2*(x^2+4)^2)^3*exp(x)+6/5*I*Pi*x^2*csgn(I/x^2*(x^2+4)^2)^3*ex p(x)+3/10*I*Pi*x^3*csgn(I*(x^2+4)^2)^3*exp(x)+6/5*I*Pi*x^2*csgn(I*(x^2+4)^ 2)^3*exp(x)-3/10*I*Pi*x^3*csgn(I*x^2)^3*exp(x)-6/5*I*Pi*x^2*csgn(I*x^2)^3* exp(x)+3/10*I*Pi*x^4*csgn(I*x^2)*csgn(I*x)^2+6/5*I*Pi*x^3*csgn(I*x^2)*csgn (I*x)^2+3/10*I*Pi*x^4*csgn(I*(x^2+4)^2)*csgn(I/x^2*(x^2+4)^2)^2-6/5*I*P...
Time = 0.09 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.39 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=\frac {3}{5} \, {\left (x^{4} + 4 \, x^{3} - {\left (x^{3} + 4 \, x^{2}\right )} e^{x}\right )} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) - x \] Input:
integrate((((-3*x^5-21*x^4-36*x^3-84*x^2-96*x)*exp(x)+12*x^5+36*x^4+48*x^3 +144*x^2)*log((x^4+8*x^2+16)/x^2)+(-6*x^4-24*x^3+24*x^2+96*x)*exp(x)+6*x^5 +24*x^4-24*x^3-101*x^2-20)/(5*x^2+20),x, algorithm="fricas")
Output:
3/5*(x^4 + 4*x^3 - (x^3 + 4*x^2)*e^x)*log((x^4 + 8*x^2 + 16)/x^2) - x
Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (24) = 48\).
Time = 0.31 (sec) , antiderivative size = 76, normalized size of antiderivative = 2.45 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=- x + \left (\frac {3 x^{4}}{5} + \frac {12 x^{3}}{5}\right ) \log {\left (\frac {x^{4} + 8 x^{2} + 16}{x^{2}} \right )} + \frac {\left (- 3 x^{3} \log {\left (\frac {x^{4} + 8 x^{2} + 16}{x^{2}} \right )} - 12 x^{2} \log {\left (\frac {x^{4} + 8 x^{2} + 16}{x^{2}} \right )}\right ) e^{x}}{5} \] Input:
integrate((((-3*x**5-21*x**4-36*x**3-84*x**2-96*x)*exp(x)+12*x**5+36*x**4+ 48*x**3+144*x**2)*ln((x**4+8*x**2+16)/x**2)+(-6*x**4-24*x**3+24*x**2+96*x) *exp(x)+6*x**5+24*x**4-24*x**3-101*x**2-20)/(5*x**2+20),x)
Output:
-x + (3*x**4/5 + 12*x**3/5)*log((x**4 + 8*x**2 + 16)/x**2) + (-3*x**3*log( (x**4 + 8*x**2 + 16)/x**2) - 12*x**2*log((x**4 + 8*x**2 + 16)/x**2))*exp(x )/5
Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (28) = 56\).
Time = 0.15 (sec) , antiderivative size = 71, normalized size of antiderivative = 2.29 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=\frac {6}{5} \, {\left (x^{3} + 4 \, x^{2}\right )} e^{x} \log \left (x\right ) + \frac {6}{5} \, {\left (x^{4} + 4 \, x^{3} - {\left (x^{3} + 4 \, x^{2}\right )} e^{x} - 16\right )} \log \left (x^{2} + 4\right ) - \frac {6}{5} \, {\left (x^{4} + 4 \, x^{3}\right )} \log \left (x\right ) - x + \frac {96}{5} \, \log \left (x^{2} + 4\right ) \] Input:
integrate((((-3*x^5-21*x^4-36*x^3-84*x^2-96*x)*exp(x)+12*x^5+36*x^4+48*x^3 +144*x^2)*log((x^4+8*x^2+16)/x^2)+(-6*x^4-24*x^3+24*x^2+96*x)*exp(x)+6*x^5 +24*x^4-24*x^3-101*x^2-20)/(5*x^2+20),x, algorithm="maxima")
Output:
6/5*(x^3 + 4*x^2)*e^x*log(x) + 6/5*(x^4 + 4*x^3 - (x^3 + 4*x^2)*e^x - 16)* log(x^2 + 4) - 6/5*(x^4 + 4*x^3)*log(x) - x + 96/5*log(x^2 + 4)
Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (28) = 56\).
Time = 0.15 (sec) , antiderivative size = 88, normalized size of antiderivative = 2.84 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=\frac {3}{5} \, x^{4} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) - \frac {3}{5} \, x^{3} e^{x} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) + \frac {12}{5} \, x^{3} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) - \frac {12}{5} \, x^{2} e^{x} \log \left (\frac {x^{4} + 8 \, x^{2} + 16}{x^{2}}\right ) - x \] Input:
integrate((((-3*x^5-21*x^4-36*x^3-84*x^2-96*x)*exp(x)+12*x^5+36*x^4+48*x^3 +144*x^2)*log((x^4+8*x^2+16)/x^2)+(-6*x^4-24*x^3+24*x^2+96*x)*exp(x)+6*x^5 +24*x^4-24*x^3-101*x^2-20)/(5*x^2+20),x, algorithm="giac")
Output:
3/5*x^4*log((x^4 + 8*x^2 + 16)/x^2) - 3/5*x^3*e^x*log((x^4 + 8*x^2 + 16)/x ^2) + 12/5*x^3*log((x^4 + 8*x^2 + 16)/x^2) - 12/5*x^2*e^x*log((x^4 + 8*x^2 + 16)/x^2) - x
Time = 2.59 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.48 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=\ln \left (\frac {x^4+8\,x^2+16}{x^2}\right )\,\left (\frac {12\,x^3}{5}-{\mathrm {e}}^x\,\left (\frac {3\,x^3}{5}+\frac {12\,x^2}{5}\right )+\frac {3\,x^4}{5}\right )-x \] Input:
int((exp(x)*(96*x + 24*x^2 - 24*x^3 - 6*x^4) - 101*x^2 - 24*x^3 + 24*x^4 + 6*x^5 + log((8*x^2 + x^4 + 16)/x^2)*(144*x^2 - exp(x)*(96*x + 84*x^2 + 36 *x^3 + 21*x^4 + 3*x^5) + 48*x^3 + 36*x^4 + 12*x^5) - 20)/(5*x^2 + 20),x)
Output:
log((8*x^2 + x^4 + 16)/x^2)*((12*x^3)/5 - exp(x)*((12*x^2)/5 + (3*x^3)/5) + (3*x^4)/5) - x
Time = 0.18 (sec) , antiderivative size = 119, normalized size of antiderivative = 3.84 \[ \int \frac {-20-101 x^2-24 x^3+24 x^4+6 x^5+e^x \left (96 x+24 x^2-24 x^3-6 x^4\right )+\left (144 x^2+48 x^3+36 x^4+12 x^5+e^x \left (-96 x-84 x^2-36 x^3-21 x^4-3 x^5\right )\right ) \log \left (\frac {16+8 x^2+x^4}{x^2}\right )}{20+5 x^2} \, dx=-\frac {3 e^{x} \mathrm {log}\left (\frac {x^{4}+8 x^{2}+16}{x^{2}}\right ) x^{3}}{5}-\frac {12 e^{x} \mathrm {log}\left (\frac {x^{4}+8 x^{2}+16}{x^{2}}\right ) x^{2}}{5}+\frac {96 \,\mathrm {log}\left (x^{2}+4\right )}{5}+\frac {3 \,\mathrm {log}\left (\frac {x^{4}+8 x^{2}+16}{x^{2}}\right ) x^{4}}{5}+\frac {12 \,\mathrm {log}\left (\frac {x^{4}+8 x^{2}+16}{x^{2}}\right ) x^{3}}{5}-\frac {48 \,\mathrm {log}\left (\frac {x^{4}+8 x^{2}+16}{x^{2}}\right )}{5}-\frac {96 \,\mathrm {log}\left (x \right )}{5}-x \] Input:
int((((-3*x^5-21*x^4-36*x^3-84*x^2-96*x)*exp(x)+12*x^5+36*x^4+48*x^3+144*x ^2)*log((x^4+8*x^2+16)/x^2)+(-6*x^4-24*x^3+24*x^2+96*x)*exp(x)+6*x^5+24*x^ 4-24*x^3-101*x^2-20)/(5*x^2+20),x)
Output:
( - 3*e**x*log((x**4 + 8*x**2 + 16)/x**2)*x**3 - 12*e**x*log((x**4 + 8*x** 2 + 16)/x**2)*x**2 + 96*log(x**2 + 4) + 3*log((x**4 + 8*x**2 + 16)/x**2)*x **4 + 12*log((x**4 + 8*x**2 + 16)/x**2)*x**3 - 48*log((x**4 + 8*x**2 + 16) /x**2) - 96*log(x) - 5*x)/5