3.72.19 \(\int \frac {e (-1250+2000 x-1250 x^2+390 x^3-52 x^4)}{-5625 x+14625 x^2-14550 x^3+7110 x^4-1729 x^5+169 x^6+(3750 x-9750 x^2+9650 x^3-4650 x^4+1104 x^5-104 x^6) \log (\frac {1-2 x+x^2}{x^2})+(-625 x+1625 x^2-1600 x^3+760 x^4-176 x^5+16 x^6) \log ^2(\frac {1-2 x+x^2}{x^2})} \, dx\)

Optimal. Leaf size=32 \[ 5+\frac {e}{-3-\frac {x^2}{(-5+2 x)^2}+\log \left (\frac {(1-x)^2}{x^2}\right )} \]

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Rubi [F]  time = 1.37, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e \left (-1250+2000 x-1250 x^2+390 x^3-52 x^4\right )}{-5625 x+14625 x^2-14550 x^3+7110 x^4-1729 x^5+169 x^6+\left (3750 x-9750 x^2+9650 x^3-4650 x^4+1104 x^5-104 x^6\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )+\left (-625 x+1625 x^2-1600 x^3+760 x^4-176 x^5+16 x^6\right ) \log ^2\left (\frac {1-2 x+x^2}{x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E*(-1250 + 2000*x - 1250*x^2 + 390*x^3 - 52*x^4))/(-5625*x + 14625*x^2 - 14550*x^3 + 7110*x^4 - 1729*x^5
+ 169*x^6 + (3750*x - 9750*x^2 + 9650*x^3 - 4650*x^4 + 1104*x^5 - 104*x^6)*Log[(1 - 2*x + x^2)/x^2] + (-625*x
+ 1625*x^2 - 1600*x^3 + 760*x^4 - 176*x^5 + 16*x^6)*Log[(1 - 2*x + x^2)/x^2]^2),x]

[Out]

-912*E*Defer[Int][(75 - 60*x + 13*x^2 - (5 - 2*x)^2*Log[(-1 + x)^2/x^2])^(-2), x] - 162*E*Defer[Int][1/((-1 +
x)*(75 - 60*x + 13*x^2 - (5 - 2*x)^2*Log[(-1 + x)^2/x^2])^2), x] + 1250*E*Defer[Int][1/(x*(75 - 60*x + 13*x^2
- (5 - 2*x)^2*Log[(-1 + x)^2/x^2])^2), x] + 338*E*Defer[Int][x/(75 - 60*x + 13*x^2 - (5 - 2*x)^2*Log[(-1 + x)^
2/x^2])^2, x] - 52*E*Defer[Int][x^2/(75 - 60*x + 13*x^2 - (5 - 2*x)^2*Log[(-1 + x)^2/x^2])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e \int \frac {-1250+2000 x-1250 x^2+390 x^3-52 x^4}{-5625 x+14625 x^2-14550 x^3+7110 x^4-1729 x^5+169 x^6+\left (3750 x-9750 x^2+9650 x^3-4650 x^4+1104 x^5-104 x^6\right ) \log \left (\frac {1-2 x+x^2}{x^2}\right )+\left (-625 x+1625 x^2-1600 x^3+760 x^4-176 x^5+16 x^6\right ) \log ^2\left (\frac {1-2 x+x^2}{x^2}\right )} \, dx\\ &=e \int \frac {2 \left (625-1000 x+625 x^2-195 x^3+26 x^4\right )}{(1-x) x \left (75-60 x+13 x^2-(5-2 x)^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2} \, dx\\ &=(2 e) \int \frac {625-1000 x+625 x^2-195 x^3+26 x^4}{(1-x) x \left (75-60 x+13 x^2-(5-2 x)^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2} \, dx\\ &=(2 e) \int \left (-\frac {456}{\left (-75+60 x-13 x^2+25 \log \left (\frac {(-1+x)^2}{x^2}\right )-20 x \log \left (\frac {(-1+x)^2}{x^2}\right )+4 x^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2}-\frac {81}{(-1+x) \left (-75+60 x-13 x^2+25 \log \left (\frac {(-1+x)^2}{x^2}\right )-20 x \log \left (\frac {(-1+x)^2}{x^2}\right )+4 x^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2}+\frac {625}{x \left (-75+60 x-13 x^2+25 \log \left (\frac {(-1+x)^2}{x^2}\right )-20 x \log \left (\frac {(-1+x)^2}{x^2}\right )+4 x^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2}+\frac {169 x}{\left (-75+60 x-13 x^2+25 \log \left (\frac {(-1+x)^2}{x^2}\right )-20 x \log \left (\frac {(-1+x)^2}{x^2}\right )+4 x^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2}-\frac {26 x^2}{\left (-75+60 x-13 x^2+25 \log \left (\frac {(-1+x)^2}{x^2}\right )-20 x \log \left (\frac {(-1+x)^2}{x^2}\right )+4 x^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2}\right ) \, dx\\ &=-\left ((52 e) \int \frac {x^2}{\left (-75+60 x-13 x^2+25 \log \left (\frac {(-1+x)^2}{x^2}\right )-20 x \log \left (\frac {(-1+x)^2}{x^2}\right )+4 x^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2} \, dx\right )-(162 e) \int \frac {1}{(-1+x) \left (-75+60 x-13 x^2+25 \log \left (\frac {(-1+x)^2}{x^2}\right )-20 x \log \left (\frac {(-1+x)^2}{x^2}\right )+4 x^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2} \, dx+(338 e) \int \frac {x}{\left (-75+60 x-13 x^2+25 \log \left (\frac {(-1+x)^2}{x^2}\right )-20 x \log \left (\frac {(-1+x)^2}{x^2}\right )+4 x^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2} \, dx-(912 e) \int \frac {1}{\left (-75+60 x-13 x^2+25 \log \left (\frac {(-1+x)^2}{x^2}\right )-20 x \log \left (\frac {(-1+x)^2}{x^2}\right )+4 x^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2} \, dx+(1250 e) \int \frac {1}{x \left (-75+60 x-13 x^2+25 \log \left (\frac {(-1+x)^2}{x^2}\right )-20 x \log \left (\frac {(-1+x)^2}{x^2}\right )+4 x^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2} \, dx\\ &=-\left ((52 e) \int \frac {x^2}{\left (75-60 x+13 x^2-(5-2 x)^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2} \, dx\right )-(162 e) \int \frac {1}{(-1+x) \left (75-60 x+13 x^2-(5-2 x)^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2} \, dx+(338 e) \int \frac {x}{\left (75-60 x+13 x^2-(5-2 x)^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2} \, dx-(912 e) \int \frac {1}{\left (75-60 x+13 x^2-(5-2 x)^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2} \, dx+(1250 e) \int \frac {1}{x \left (75-60 x+13 x^2-(5-2 x)^2 \log \left (\frac {(-1+x)^2}{x^2}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.97, size = 39, normalized size = 1.22 \begin {gather*} \frac {e (5-2 x)^2}{-75+60 x-13 x^2+(5-2 x)^2 \log \left (\frac {(-1+x)^2}{x^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E*(-1250 + 2000*x - 1250*x^2 + 390*x^3 - 52*x^4))/(-5625*x + 14625*x^2 - 14550*x^3 + 7110*x^4 - 172
9*x^5 + 169*x^6 + (3750*x - 9750*x^2 + 9650*x^3 - 4650*x^4 + 1104*x^5 - 104*x^6)*Log[(1 - 2*x + x^2)/x^2] + (-
625*x + 1625*x^2 - 1600*x^3 + 760*x^4 - 176*x^5 + 16*x^6)*Log[(1 - 2*x + x^2)/x^2]^2),x]

[Out]

(E*(5 - 2*x)^2)/(-75 + 60*x - 13*x^2 + (5 - 2*x)^2*Log[(-1 + x)^2/x^2])

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fricas [A]  time = 1.01, size = 51, normalized size = 1.59 \begin {gather*} -\frac {{\left (4 \, x^{2} - 20 \, x + 25\right )} e}{13 \, x^{2} - {\left (4 \, x^{2} - 20 \, x + 25\right )} \log \left (\frac {x^{2} - 2 \, x + 1}{x^{2}}\right ) - 60 \, x + 75} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-52*x^4+390*x^3-1250*x^2+2000*x-1250)*exp(1)/((16*x^6-176*x^5+760*x^4-1600*x^3+1625*x^2-625*x)*log(
(x^2-2*x+1)/x^2)^2+(-104*x^6+1104*x^5-4650*x^4+9650*x^3-9750*x^2+3750*x)*log((x^2-2*x+1)/x^2)+169*x^6-1729*x^5
+7110*x^4-14550*x^3+14625*x^2-5625*x),x, algorithm="fricas")

[Out]

-(4*x^2 - 20*x + 25)*e/(13*x^2 - (4*x^2 - 20*x + 25)*log((x^2 - 2*x + 1)/x^2) - 60*x + 75)

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giac [B]  time = 0.60, size = 74, normalized size = 2.31 \begin {gather*} \frac {{\left (4 \, x^{2} - 20 \, x + 25\right )} e}{4 \, x^{2} \log \left (\frac {x^{2} - 2 \, x + 1}{x^{2}}\right ) - 13 \, x^{2} - 20 \, x \log \left (\frac {x^{2} - 2 \, x + 1}{x^{2}}\right ) + 60 \, x + 25 \, \log \left (\frac {x^{2} - 2 \, x + 1}{x^{2}}\right ) - 75} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-52*x^4+390*x^3-1250*x^2+2000*x-1250)*exp(1)/((16*x^6-176*x^5+760*x^4-1600*x^3+1625*x^2-625*x)*log(
(x^2-2*x+1)/x^2)^2+(-104*x^6+1104*x^5-4650*x^4+9650*x^3-9750*x^2+3750*x)*log((x^2-2*x+1)/x^2)+169*x^6-1729*x^5
+7110*x^4-14550*x^3+14625*x^2-5625*x),x, algorithm="giac")

[Out]

(4*x^2 - 20*x + 25)*e/(4*x^2*log((x^2 - 2*x + 1)/x^2) - 13*x^2 - 20*x*log((x^2 - 2*x + 1)/x^2) + 60*x + 25*log
((x^2 - 2*x + 1)/x^2) - 75)

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maple [B]  time = 0.20, size = 72, normalized size = 2.25

method	result	size
risch	\(\frac {{\mathrm e} \left (2 x -5\right )^{2}}{4 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right ) x^{2}-20 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right ) x -13 x^{2}+25 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right )+60 x -75}\)	\(72\)
norman	\(\frac {-20 x \,{\mathrm e}+4 x^{2} {\mathrm e}+25 \,{\mathrm e}}{4 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right ) x^{2}-20 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right ) x -13 x^{2}+25 \ln \left (\frac {x^{2}-2 x +1}{x^{2}}\right )+60 x -75}\)	\(80\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-52*x^4+390*x^3-1250*x^2+2000*x-1250)*exp(1)/((16*x^6-176*x^5+760*x^4-1600*x^3+1625*x^2-625*x)*ln((x^2-2*
x+1)/x^2)^2+(-104*x^6+1104*x^5-4650*x^4+9650*x^3-9750*x^2+3750*x)*ln((x^2-2*x+1)/x^2)+169*x^6-1729*x^5+7110*x^
4-14550*x^3+14625*x^2-5625*x),x,method=_RETURNVERBOSE)

[Out]

exp(1)*(2*x-5)^2/(4*ln((x^2-2*x+1)/x^2)*x^2-20*ln((x^2-2*x+1)/x^2)*x-13*x^2+25*ln((x^2-2*x+1)/x^2)+60*x-75)

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maxima [A]  time = 0.43, size = 56, normalized size = 1.75 \begin {gather*} -\frac {{\left (4 \, x^{2} - 20 \, x + 25\right )} e}{13 \, x^{2} - 2 \, {\left (4 \, x^{2} - 20 \, x + 25\right )} \log \left (x - 1\right ) + 2 \, {\left (4 \, x^{2} - 20 \, x + 25\right )} \log \relax (x) - 60 \, x + 75} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-52*x^4+390*x^3-1250*x^2+2000*x-1250)*exp(1)/((16*x^6-176*x^5+760*x^4-1600*x^3+1625*x^2-625*x)*log(
(x^2-2*x+1)/x^2)^2+(-104*x^6+1104*x^5-4650*x^4+9650*x^3-9750*x^2+3750*x)*log((x^2-2*x+1)/x^2)+169*x^6-1729*x^5
+7110*x^4-14550*x^3+14625*x^2-5625*x),x, algorithm="maxima")

[Out]

-(4*x^2 - 20*x + 25)*e/(13*x^2 - 2*(4*x^2 - 20*x + 25)*log(x - 1) + 2*(4*x^2 - 20*x + 25)*log(x) - 60*x + 75)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {e}\,\left (52\,x^4-390\,x^3+1250\,x^2-2000\,x+1250\right )}{5625\,x-\ln \left (\frac {x^2-2\,x+1}{x^2}\right )\,\left (-104\,x^6+1104\,x^5-4650\,x^4+9650\,x^3-9750\,x^2+3750\,x\right )+{\ln \left (\frac {x^2-2\,x+1}{x^2}\right )}^2\,\left (-16\,x^6+176\,x^5-760\,x^4+1600\,x^3-1625\,x^2+625\,x\right )-14625\,x^2+14550\,x^3-7110\,x^4+1729\,x^5-169\,x^6} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1)*(1250*x^2 - 2000*x - 390*x^3 + 52*x^4 + 1250))/(5625*x - log((x^2 - 2*x + 1)/x^2)*(3750*x - 9750*x
^2 + 9650*x^3 - 4650*x^4 + 1104*x^5 - 104*x^6) + log((x^2 - 2*x + 1)/x^2)^2*(625*x - 1625*x^2 + 1600*x^3 - 760
*x^4 + 176*x^5 - 16*x^6) - 14625*x^2 + 14550*x^3 - 7110*x^4 + 1729*x^5 - 169*x^6),x)

[Out]

int((exp(1)*(1250*x^2 - 2000*x - 390*x^3 + 52*x^4 + 1250))/(5625*x - log((x^2 - 2*x + 1)/x^2)*(3750*x - 9750*x
^2 + 9650*x^3 - 4650*x^4 + 1104*x^5 - 104*x^6) + log((x^2 - 2*x + 1)/x^2)^2*(625*x - 1625*x^2 + 1600*x^3 - 760
*x^4 + 176*x^5 - 16*x^6) - 14625*x^2 + 14550*x^3 - 7110*x^4 + 1729*x^5 - 169*x^6), x)

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sympy [B]  time = 0.37, size = 53, normalized size = 1.66 \begin {gather*} \frac {4 e x^{2} - 20 e x + 25 e}{- 13 x^{2} + 60 x + \left (4 x^{2} - 20 x + 25\right ) \log {\left (\frac {x^{2} - 2 x + 1}{x^{2}} \right )} - 75} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-52*x**4+390*x**3-1250*x**2+2000*x-1250)*exp(1)/((16*x**6-176*x**5+760*x**4-1600*x**3+1625*x**2-625
*x)*ln((x**2-2*x+1)/x**2)**2+(-104*x**6+1104*x**5-4650*x**4+9650*x**3-9750*x**2+3750*x)*ln((x**2-2*x+1)/x**2)+
169*x**6-1729*x**5+7110*x**4-14550*x**3+14625*x**2-5625*x),x)

[Out]

(4*E*x**2 - 20*E*x + 25*E)/(-13*x**2 + 60*x + (4*x**2 - 20*x + 25)*log((x**2 - 2*x + 1)/x**2) - 75)

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