Integrand size = 94, antiderivative size = 28 \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=\frac {x^2}{\left (5-x+8 x \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2\right )^2} \]
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(93\) vs. \(2(28)=56\).
Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.32 \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=-\frac {5 \left (5-2 x \left (-7+8 \pi ^2+16 i \pi (-1+\log (4))+16 \log (4)-8 \log ^2(4)\right )\right )}{\left (-7+8 \pi ^2+16 i \pi (-1+\log (4))+16 \log (4)-8 \log ^2(4)\right )^2 \left (-5+x \left (-7+8 \pi ^2+16 i \pi (-1+\log (4))+16 \log (4)-8 \log ^2(4)\right )\right )^2} \]
Integrate[(10*x)/(125 - 75*x + 15*x^2 - x^3 + (600*x - 240*x^2 + 24*x^3)*( I*Pi + Log[E/4])^2 + (960*x^2 - 192*x^3)*(I*Pi + Log[E/4])^4 + 512*x^3*(I* Pi + Log[E/4])^6),x]
(-5*(5 - 2*x*(-7 + 8*Pi^2 + (16*I)*Pi*(-1 + Log[4]) + 16*Log[4] - 8*Log[4] ^2)))/((-7 + 8*Pi^2 + (16*I)*Pi*(-1 + Log[4]) + 16*Log[4] - 8*Log[4]^2)^2* (-5 + x*(-7 + 8*Pi^2 + (16*I)*Pi*(-1 + Log[4]) + 16*Log[4] - 8*Log[4]^2))^ 2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(154\) vs. \(2(28)=56\).
Time = 0.84 (sec) , antiderivative size = 154, normalized size of antiderivative = 5.50, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {6, 27, 2007, 48}
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.
| \(\displaystyle \int \frac {10 x}{-x^3+512 x^3 \left (\log \left (\frac {e}{4}\right )+i \pi \right )^6+15 x^2+\left (960 x^2-192 x^3\right ) \left (\log \left (\frac {e}{4}\right )+i \pi \right )^4+\left (24 x^3-240 x^2+600 x\right ) \left (\log \left (\frac {e}{4}\right )+i \pi \right )^2-75 x+125} \, dx\) |
| \(\Big \downarrow \) 6 |
| \(\displaystyle \int \frac {10 x}{x^3 \left (-1+512 \left (\log \left (\frac {e}{4}\right )+i \pi \right )^6\right )+15 x^2+\left (960 x^2-192 x^3\right ) \left (\log \left (\frac {e}{4}\right )+i \pi \right )^4+\left (24 x^3-240 x^2+600 x\right ) \left (\log \left (\frac {e}{4}\right )+i \pi \right )^2-75 x+125}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 10 \int \frac {x}{-\left (\left (1-512 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6\right ) x^3\right )+15 x^2-75 x+24 \left (x^3-10 x^2+25 x\right ) (1+i \pi -\log (4))^2+192 \left (5 x^2-x^3\right ) (\pi -i (1-\log (4)))^4+125}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle 10 \int \frac {x}{\left (5-x \sqrt [3]{-343+720 i \pi +6552 \pi ^2+9472 i \pi ^3-7488 \pi ^4+512 \pi ^6+2352 \log (4)-2256 i \pi \log (4)-28416 \pi ^2 \log (4)-29952 i \pi ^3 \log (4)+15360 \pi ^4 \log (4)-6552 \log ^2(4)+2304 i \pi \log ^2(4)+44928 \pi ^2 \log ^2(4)+30720 i \pi ^3 \log ^2(4)-7680 \pi ^4 \log ^2(4)+9472 \log ^3(4)-768 i \pi \log ^3(4)-30720 \pi ^2 \log ^3(4)-10240 i \pi ^3 \log ^3(4)-7488 \log ^4(4)+7680 \pi ^2 \log ^4(4)+3072 \log ^5(4)-512 \log ^6(4)-3072 i \pi ^5 \log \left (\frac {e}{4}\right )-3072 i \pi \log ^5\left (\frac {e}{4}\right )}\right )^3}dx\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {x^2}{\left (5-x \sqrt [3]{512 \pi ^6-\left (7+8 \log ^2(4)-16 \log (4)\right )^3-192 \pi ^4 \left (39+40 \log ^2(4)-80 \log (4)\right )+256 i \pi ^3 \left (37-40 \log ^3(4)+120 \log ^2(4)-117 \log (4)\right )+48 i \pi \left (15-64 \log ^5\left (\frac {e}{4}\right )-16 \log ^3(4)+48 \log ^2(4)-47 \log (4)\right )+24 \pi ^2 \left (273+320 \log ^4(4)-1280 \log ^3(4)+1872 \log ^2(4)-1184 \log (4)\right )-3072 i \pi ^5 (1-\log (4))}\right )^2}\) |
Int[(10*x)/(125 - 75*x + 15*x^2 - x^3 + (600*x - 240*x^2 + 24*x^3)*(I*Pi + Log[E/4])^2 + (960*x^2 - 192*x^3)*(I*Pi + Log[E/4])^4 + 512*x^3*(I*Pi + L og[E/4])^6),x]
x^2/(5 - x*(512*Pi^6 - (3072*I)*Pi^5*(1 - Log[4]) - (7 - 16*Log[4] + 8*Log [4]^2)^3 - 192*Pi^4*(39 - 80*Log[4] + 40*Log[4]^2) + (256*I)*Pi^3*(37 - 11 7*Log[4] + 120*Log[4]^2 - 40*Log[4]^3) + 24*Pi^2*(273 - 1184*Log[4] + 1872 *Log[4]^2 - 1280*Log[4]^3 + 320*Log[4]^4) + (48*I)*Pi*(15 - 47*Log[4] + 48 *Log[4]^2 - 16*Log[4]^3 - 64*Log[E/4]^5))^(1/3))^2
3.28.95.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(21)=42\).
Time = 2.84 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75
| method | result | size |
| parallelrisch | \(\frac {x^{2}}{64 \ln \left (-\frac {{\mathrm e}}{4}\right )^{4} x^{2}-16 \ln \left (-\frac {{\mathrm e}}{4}\right )^{2} x^{2}+80 x \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}+x^{2}-10 x +25}\) | \(49\) |
| default | \(-\frac {10}{\left (8 \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}-1\right )^{2} \left (8 x \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}+5-x \right )}+\frac {25}{\left (8 \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}-1\right )^{2} \left (8 x \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}+5-x \right )^{2}}\) | \(66\) |
| gosper | \(-\frac {5 \left (16 x \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}-2 x +5\right )}{\left (64 \ln \left (-\frac {{\mathrm e}}{4}\right )^{4} x^{2}-16 \ln \left (-\frac {{\mathrm e}}{4}\right )^{2} x^{2}+80 x \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}+x^{2}-10 x +25\right ) \left (8 \ln \left (-\frac {{\mathrm e}}{4}\right )^{2}-1\right )^{2}}\) | \(75\) |
| risch | \(\frac {\frac {10 x}{-448+2048 \ln \left (2\right )-1024 i \pi -2048 \ln \left (2\right )^{2}+2048 i \ln \left (2\right ) \pi +512 \pi ^{2}}-\frac {25}{64 \left (-7+32 \ln \left (2\right )-16 i \pi -32 \ln \left (2\right )^{2}+32 i \ln \left (2\right ) \pi +8 \pi ^{2}\right )^{2}}}{-23 i \pi \ln \left (2\right ) x^{2}+\frac {7 i \pi \,x^{2}}{2}+\pi ^{4} x^{2}+\frac {5 i \pi x}{2}-24 \pi ^{2} \ln \left (2\right )^{2} x^{2}+8 i \pi ^{3} \ln \left (2\right ) x^{2}+16 x^{2} \ln \left (2\right )^{4}+24 \pi ^{2} \ln \left (2\right ) x^{2}-5 i \pi \ln \left (2\right ) x -32 x^{2} \ln \left (2\right )^{3}-\frac {23 \pi ^{2} x^{2}}{4}-32 i \pi \ln \left (2\right )^{3} x^{2}-4 i \pi ^{3} x^{2}+23 x^{2} \ln \left (2\right )^{2}-\frac {5 x \,\pi ^{2}}{4}+48 i \pi \ln \left (2\right )^{2} x^{2}+5 x \ln \left (2\right )^{2}-7 x^{2} \ln \left (2\right )-5 x \ln \left (2\right )+\frac {49 x^{2}}{64}+\frac {35 x}{32}+\frac {25}{64}}\) | \(238\) |
int(10*x/(512*x^3*ln(-1/4*exp(1))^6+(-192*x^3+960*x^2)*ln(-1/4*exp(1))^4+( 24*x^3-240*x^2+600*x)*ln(-1/4*exp(1))^2-x^3+15*x^2-75*x+125),x,method=_RET URNVERBOSE)
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 199, normalized size of antiderivative = 7.11 \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=-\frac {5 \, {\left (16 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{2} x - 32 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )} x + 14 \, x + 5\right )}}{4096 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{8} x^{2} - 32768 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{7} x^{2} + 5120 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{6} {\left (22 \, x^{2} + x\right )} - 2048 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{5} {\left (106 \, x^{2} + 15 \, x\right )} + 64 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{4} {\left (4006 \, x^{2} + 1170 \, x + 25\right )} - 256 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{3} {\left (742 \, x^{2} + 370 \, x + 25\right )} + 80 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )}^{2} {\left (1078 \, x^{2} + 819 \, x + 115\right )} - 224 \, {\left (i \, \pi + 2 \, \log \left (2\right )\right )} {\left (98 \, x^{2} + 105 \, x + 25\right )} + 2401 \, x^{2} + 3430 \, x + 1225} \]
integrate(10*x/(512*x^3*log(-1/4*exp(1))^6+(-192*x^3+960*x^2)*log(-1/4*exp (1))^4+(24*x^3-240*x^2+600*x)*log(-1/4*exp(1))^2-x^3+15*x^2-75*x+125),x, a lgorithm=\
-5*(16*(I*pi + 2*log(2))^2*x - 32*(I*pi + 2*log(2))*x + 14*x + 5)/(4096*(I *pi + 2*log(2))^8*x^2 - 32768*(I*pi + 2*log(2))^7*x^2 + 5120*(I*pi + 2*log (2))^6*(22*x^2 + x) - 2048*(I*pi + 2*log(2))^5*(106*x^2 + 15*x) + 64*(I*pi + 2*log(2))^4*(4006*x^2 + 1170*x + 25) - 256*(I*pi + 2*log(2))^3*(742*x^2 + 370*x + 25) + 80*(I*pi + 2*log(2))^2*(1078*x^2 + 819*x + 115) - 224*(I* pi + 2*log(2))*(98*x^2 + 105*x + 25) + 2401*x^2 + 3430*x + 1225)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 765 vs. \(2 (22) = 44\).
Time = 4.15 (sec) , antiderivative size = 765, normalized size of antiderivative = 27.32 \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=\text {Too large to display} \]
integrate(10*x/(512*x**3*ln(-1/4*exp(1))**6+(-192*x**3+960*x**2)*ln(-1/4*e xp(1))**4+(24*x**3-240*x**2+600*x)*ln(-1/4*exp(1))**2-x**3+15*x**2-75*x+12 5),x)
-10*(x*(-16*pi**2 - 64*log(2) + 14 + 64*log(2)**2 - 64*I*pi*log(2) + 32*I* pi) + 5)/(x**2*(-18350080*pi**4*log(2)**3 - 917504*pi**6*log(2)**2 - 43417 60*pi**4*log(2) - 225280*pi**6 - 54067200*pi**2*log(2)**4 - 12306432*pi**2 *log(2)**2 - 14680064*pi**2*log(2)**6 - 13893632*log(2)**5 - 172480*pi**2 - 3039232*log(2)**3 - 8388608*log(2)**7 - 87808*log(2) + 4802 + 2097152*lo g(2)**8 + 689920*log(2)**2 + 14417920*log(2)**6 + 8204288*log(2)**4 + 2279 424*pi**2*log(2) + 512768*pi**4 + 44040192*pi**2*log(2)**5 + 8192*pi**8 + 34734080*pi**2*log(2)**3 + 9175040*pi**4*log(2)**4 + 917504*pi**6*log(2) + 13516800*pi**4*log(2)**2 - 2703360*I*pi**5*log(2) - 3670016*I*pi**5*log(2 )**3 - 36700160*I*pi**3*log(2)**4 - 17367040*I*pi**3*log(2)**2 - 65536*I*p i**7 - 43253760*I*pi*log(2)**5 - 16408576*I*pi*log(2)**3 - 379904*I*pi**3 - 8388608*I*pi*log(2)**7 - 689920*I*pi*log(2) + 43904*I*pi + 4558848*I*pi* log(2)**2 + 29360128*I*pi*log(2)**6 + 34734080*I*pi*log(2)**4 + 14680064*I *pi**3*log(2)**5 + 4102144*I*pi**3*log(2) + 434176*I*pi**5 + 131072*I*pi** 7*log(2) + 36044800*I*pi**3*log(2)**3 + 5505024*I*pi**5*log(2)**2) + x*(-6 14400*pi**4*log(2) - 3594240*pi**2*log(2)**2 - 10240*pi**6 - 2457600*pi**2 *log(2)**4 - 131040*pi**2 - 1515520*log(2)**3 - 1966080*log(2)**5 - 94080* log(2) + 6860 + 655360*log(2)**6 + 524160*log(2)**2 + 2396160*log(2)**4 + 1136640*pi**2*log(2) + 149760*pi**4 + 4915200*pi**2*log(2)**3 + 614400*pi* *4*log(2)**2 - 2457600*I*pi**3*log(2)**2 - 122880*I*pi**5*log(2) - 1894...
Leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (21) = 42\).
Time = 0.20 (sec) , antiderivative size = 114, normalized size of antiderivative = 4.07 \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=-\frac {5 \, {\left (2 \, {\left (8 \, \log \left (-\frac {1}{4} \, e\right )^{2} - 1\right )} x + 5\right )}}{1600 \, \log \left (-\frac {1}{4} \, e\right )^{4} + {\left (4096 \, \log \left (-\frac {1}{4} \, e\right )^{8} - 2048 \, \log \left (-\frac {1}{4} \, e\right )^{6} + 384 \, \log \left (-\frac {1}{4} \, e\right )^{4} - 32 \, \log \left (-\frac {1}{4} \, e\right )^{2} + 1\right )} x^{2} + 10 \, {\left (512 \, \log \left (-\frac {1}{4} \, e\right )^{6} - 192 \, \log \left (-\frac {1}{4} \, e\right )^{4} + 24 \, \log \left (-\frac {1}{4} \, e\right )^{2} - 1\right )} x - 400 \, \log \left (-\frac {1}{4} \, e\right )^{2} + 25} \]
integrate(10*x/(512*x^3*log(-1/4*exp(1))^6+(-192*x^3+960*x^2)*log(-1/4*exp (1))^4+(24*x^3-240*x^2+600*x)*log(-1/4*exp(1))^2-x^3+15*x^2-75*x+125),x, a lgorithm=\
-5*(2*(8*log(-1/4*e)^2 - 1)*x + 5)/(1600*log(-1/4*e)^4 + (4096*log(-1/4*e) ^8 - 2048*log(-1/4*e)^6 + 384*log(-1/4*e)^4 - 32*log(-1/4*e)^2 + 1)*x^2 + 10*(512*log(-1/4*e)^6 - 192*log(-1/4*e)^4 + 24*log(-1/4*e)^2 - 1)*x - 400* log(-1/4*e)^2 + 25)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (21) = 42\).
Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00 \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=-\frac {5 \, {\left (16 \, x \log \left (-\frac {1}{4} \, e\right )^{2} - 2 \, x + 5\right )}}{{\left (64 \, \log \left (-\frac {1}{4} \, e\right )^{4} - 16 \, \log \left (-\frac {1}{4} \, e\right )^{2} + 1\right )} {\left (8 \, x \log \left (-\frac {1}{4} \, e\right )^{2} - x + 5\right )}^{2}} \]
integrate(10*x/(512*x^3*log(-1/4*exp(1))^6+(-192*x^3+960*x^2)*log(-1/4*exp (1))^4+(24*x^3-240*x^2+600*x)*log(-1/4*exp(1))^2-x^3+15*x^2-75*x+125),x, a lgorithm=\
-5*(16*x*log(-1/4*e)^2 - 2*x + 5)/((64*log(-1/4*e)^4 - 16*log(-1/4*e)^2 + 1)*(8*x*log(-1/4*e)^2 - x + 5)^2)
Time = 0.22 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {10 x}{125-75 x+15 x^2-x^3+\left (600 x-240 x^2+24 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^2+\left (960 x^2-192 x^3\right ) \left (i \pi +\log \left (\frac {e}{4}\right )\right )^4+512 x^3 \left (i \pi +\log \left (\frac {e}{4}\right )\right )^6} \, dx=-\frac {5\,\left (16\,x\,{\ln \left (-\frac {\mathrm {e}}{4}\right )}^2-2\,x+5\right )}{{\left (8\,{\ln \left (-\frac {\mathrm {e}}{4}\right )}^2-1\right )}^2\,{\left (8\,x\,{\ln \left (-\frac {\mathrm {e}}{4}\right )}^2-x+5\right )}^2} \]
int((10*x)/(log(-exp(1)/4)^2*(600*x - 240*x^2 + 24*x^3) - 75*x + log(-exp( 1)/4)^4*(960*x^2 - 192*x^3) + 512*x^3*log(-exp(1)/4)^6 + 15*x^2 - x^3 + 12 5),x)