3.66.45 \(\int \frac {100-160 x-20 x^2+6 x^3+e^x (150 x+30 x^2-10 x^3)}{(-25 x+30 x^2-x^4+e^x (-25 x^2+5 x^3)) \log ^2(\frac {25 x^4-50 x^5+15 x^6+25 e^{2 x} x^6+10 x^7+x^8+e^x (50 x^5-50 x^6-10 x^7)}{625-250 x+25 x^2})} \, dx\)

Optimal. Leaf size=34 \[ \frac {1}{\log \left (\frac {x^2 \left (-x+\left (1-e^x+\frac {x}{5}\right ) x^2\right )^2}{(-5+x)^2}\right )} \]

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Rubi [F]  time = 4.23, 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 {100-160 x-20 x^2+6 x^3+e^x \left (150 x+30 x^2-10 x^3\right )}{\left (-25 x+30 x^2-x^4+e^x \left (-25 x^2+5 x^3\right )\right ) \log ^2\left (\frac {25 x^4-50 x^5+15 x^6+25 e^{2 x} x^6+10 x^7+x^8+e^x \left (50 x^5-50 x^6-10 x^7\right )}{625-250 x+25 x^2}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(100 - 160*x - 20*x^2 + 6*x^3 + E^x*(150*x + 30*x^2 - 10*x^3))/((-25*x + 30*x^2 - x^4 + E^x*(-25*x^2 + 5*x
^3))*Log[(25*x^4 - 50*x^5 + 15*x^6 + 25*E^(2*x)*x^6 + 10*x^7 + x^8 + E^x*(50*x^5 - 50*x^6 - 10*x^7))/(625 - 25
0*x + 25*x^2)]^2),x]

[Out]

-2*Defer[Int][Log[(x^4*(-5 - 5*(-1 + E^x)*x + x^2)^2)/(25*(-5 + x)^2)]^(-2), x] + 2*Defer[Int][1/((-5 + x)*Log
[(x^4*(-5 - 5*(-1 + E^x)*x + x^2)^2)/(25*(-5 + x)^2)]^2), x] - 6*Defer[Int][1/(x*Log[(x^4*(-5 - 5*(-1 + E^x)*x
+ x^2)^2)/(25*(-5 + x)^2)]^2), x] - 10*Defer[Int][1/((-5 + 5*x - 5*E^x*x + x^2)*Log[(x^4*(-5 - 5*(-1 + E^x)*x
+ x^2)^2)/(25*(-5 + x)^2)]^2), x] - 10*Defer[Int][1/(x*(-5 + 5*x - 5*E^x*x + x^2)*Log[(x^4*(-5 - 5*(-1 + E^x)
*x + x^2)^2)/(25*(-5 + x)^2)]^2), x] + 8*Defer[Int][x/((-5 + 5*x - 5*E^x*x + x^2)*Log[(x^4*(-5 - 5*(-1 + E^x)*
x + x^2)^2)/(25*(-5 + x)^2)]^2), x] + 2*Defer[Int][x^2/((-5 + 5*x - 5*E^x*x + x^2)*Log[(x^4*(-5 - 5*(-1 + E^x)
*x + x^2)^2)/(25*(-5 + x)^2)]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-50+\left (80-75 e^x\right ) x-5 \left (-2+3 e^x\right ) x^2+\left (-3+5 e^x\right ) x^3\right )}{(5-x) x \left (5+5 \left (-1+e^x\right ) x-x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx\\ &=2 \int \frac {-50+\left (80-75 e^x\right ) x-5 \left (-2+3 e^x\right ) x^2+\left (-3+5 e^x\right ) x^3}{(5-x) x \left (5+5 \left (-1+e^x\right ) x-x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx\\ &=2 \int \left (\frac {15+3 x-x^2}{(-5+x) x \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}+\frac {-5-5 x+4 x^2+x^3}{x \left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}\right ) \, dx\\ &=2 \int \frac {15+3 x-x^2}{(-5+x) x \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx+2 \int \frac {-5-5 x+4 x^2+x^3}{x \left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx\\ &=2 \int \left (-\frac {1}{\log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}+\frac {1}{(-5+x) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}-\frac {3}{x \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}\right ) \, dx+2 \int \left (-\frac {5}{\left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}-\frac {5}{x \left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}+\frac {4 x}{\left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}+\frac {x^2}{\left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )}\right ) \, dx\\ &=-\left (2 \int \frac {1}{\log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx\right )+2 \int \frac {1}{(-5+x) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx+2 \int \frac {x^2}{\left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx-6 \int \frac {1}{x \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx+8 \int \frac {x}{\left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx-10 \int \frac {1}{\left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx-10 \int \frac {1}{x \left (-5+5 x-5 e^x x+x^2\right ) \log ^2\left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 30, normalized size = 0.88 \begin {gather*} \frac {1}{\log \left (\frac {x^4 \left (-5-5 \left (-1+e^x\right ) x+x^2\right )^2}{25 (-5+x)^2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(100 - 160*x - 20*x^2 + 6*x^3 + E^x*(150*x + 30*x^2 - 10*x^3))/((-25*x + 30*x^2 - x^4 + E^x*(-25*x^2
+ 5*x^3))*Log[(25*x^4 - 50*x^5 + 15*x^6 + 25*E^(2*x)*x^6 + 10*x^7 + x^8 + E^x*(50*x^5 - 50*x^6 - 10*x^7))/(62
5 - 250*x + 25*x^2)]^2),x]

[Out]

Log[(x^4*(-5 - 5*(-1 + E^x)*x + x^2)^2)/(25*(-5 + x)^2)]^(-1)

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fricas [B]  time = 0.76, size = 66, normalized size = 1.94 \begin {gather*} \frac {1}{\log \left (\frac {x^{8} + 10 \, x^{7} + 25 \, x^{6} e^{\left (2 \, x\right )} + 15 \, x^{6} - 50 \, x^{5} + 25 \, x^{4} - 10 \, {\left (x^{7} + 5 \, x^{6} - 5 \, x^{5}\right )} e^{x}}{25 \, {\left (x^{2} - 10 \, x + 25\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^3+30*x^2+150*x)*exp(x)+6*x^3-20*x^2-160*x+100)/((5*x^3-25*x^2)*exp(x)-x^4+30*x^2-25*x)/log((
25*x^6*exp(x)^2+(-10*x^7-50*x^6+50*x^5)*exp(x)+x^8+10*x^7+15*x^6-50*x^5+25*x^4)/(25*x^2-250*x+625))^2,x, algor
ithm="fricas")

[Out]

1/log(1/25*(x^8 + 10*x^7 + 25*x^6*e^(2*x) + 15*x^6 - 50*x^5 + 25*x^4 - 10*(x^7 + 5*x^6 - 5*x^5)*e^x)/(x^2 - 10
*x + 25))

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^3+30*x^2+150*x)*exp(x)+6*x^3-20*x^2-160*x+100)/((5*x^3-25*x^2)*exp(x)-x^4+30*x^2-25*x)/log((
25*x^6*exp(x)^2+(-10*x^7-50*x^6+50*x^5)*exp(x)+x^8+10*x^7+15*x^6-50*x^5+25*x^4)/(25*x^2-250*x+625))^2,x, algor
ithm="giac")

[Out]

Timed out

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maple [C]  time = 0.34, size = 786, normalized size = 23.12

method	result	size
risch	\(\frac {2 i}{8 i \ln \relax (x )-4 i \ln \relax (5)+\pi \,\mathrm {csgn}\left (\frac {i \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}}{\left (x -5\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{4} \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}}{\left (x -5\right )^{2}}\right ) \mathrm {csgn}\left (i x^{4}\right )+\pi \,\mathrm {csgn}\left (\frac {i}{\left (x -5\right )^{2}}\right ) \mathrm {csgn}\left (i \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}}{\left (x -5\right )^{2}}\right )+\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )+\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )+\pi \mathrm {csgn}\left (i x^{2}\right )^{3}+\pi \mathrm {csgn}\left (i x^{4}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i x^{4} \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}}{\left (x -5\right )^{2}}\right )^{3}+\pi \mathrm {csgn}\left (i x^{3}\right )^{3}+\pi \mathrm {csgn}\left (i \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}\right )^{3}+\pi \mathrm {csgn}\left (\frac {i \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}}{\left (x -5\right )^{2}}\right )^{3}-\pi \mathrm {csgn}\left (i \left (x -5\right )^{2}\right )^{3}-4 i \ln \left (x -5\right )+4 i \ln \left (-5+x^{2}+\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )+\pi \mathrm {csgn}\left (i \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )\right )^{2} \mathrm {csgn}\left (i \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2}-\pi \,\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2}-\pi \mathrm {csgn}\left (i \left (x -5\right )\right )^{2} \mathrm {csgn}\left (i \left (x -5\right )^{2}\right )+2 \pi \,\mathrm {csgn}\left (i \left (x -5\right )\right ) \mathrm {csgn}\left (i \left (x -5\right )^{2}\right )^{2}-\pi \,\mathrm {csgn}\left (\frac {i \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}}{\left (x -5\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i x^{4} \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}}{\left (x -5\right )^{2}}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i x^{4} \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}}{\left (x -5\right )^{2}}\right )^{2} \mathrm {csgn}\left (i x^{4}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{\left (x -5\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}}{\left (x -5\right )^{2}}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}\right ) \mathrm {csgn}\left (\frac {i \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}}{\left (x -5\right )^{2}}\right )^{2}+2 \pi \,\mathrm {csgn}\left (i \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )\right ) \mathrm {csgn}\left (i \left (5-x^{2}-\left (-5 \,{\mathrm e}^{x}+5\right ) x \right )^{2}\right )^{2}-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2}-\pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2}+\pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}}\)	\(786\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*x^3+30*x^2+150*x)*exp(x)+6*x^3-20*x^2-160*x+100)/((5*x^3-25*x^2)*exp(x)-x^4+30*x^2-25*x)/ln((25*x^6*
exp(x)^2+(-10*x^7-50*x^6+50*x^5)*exp(x)+x^8+10*x^7+15*x^6-50*x^5+25*x^4)/(25*x^2-250*x+625))^2,x,method=_RETUR
NVERBOSE)

[Out]

2*I/(Pi*csgn(I/(x-5)^2*(5-x^2-(-5*exp(x)+5)*x)^2)*csgn(I*x^4*(5-x^2-(-5*exp(x)+5)*x)^2/(x-5)^2)*csgn(I*x^4)+Pi
*csgn(I/(x-5)^2)*csgn(I*(5-x^2-(-5*exp(x)+5)*x)^2)*csgn(I/(x-5)^2*(5-x^2-(-5*exp(x)+5)*x)^2)+Pi*csgn(I*(5-x^2-
(-5*exp(x)+5)*x))^2*csgn(I*(5-x^2-(-5*exp(x)+5)*x)^2)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)*csgn(I*x^2)*csgn(I*x^3)+Pi
*csgn(I*x)*csgn(I*x^3)*csgn(I*x^4)-Pi*csgn(I*x)*csgn(I*x^4)^2-Pi*csgn(I*x^3)*csgn(I*x^4)^2+Pi*csgn(I*x^4)^3-Pi
*csgn(I*(x-5))^2*csgn(I*(x-5)^2)+2*Pi*csgn(I*(x-5))*csgn(I*(x-5)^2)^2+Pi*csgn(I*x^4*(5-x^2-(-5*exp(x)+5)*x)^2/
(x-5)^2)^3-Pi*csgn(I/(x-5)^2*(5-x^2-(-5*exp(x)+5)*x)^2)*csgn(I*x^4*(5-x^2-(-5*exp(x)+5)*x)^2/(x-5)^2)^2-Pi*csg
n(I*x^4*(5-x^2-(-5*exp(x)+5)*x)^2/(x-5)^2)^2*csgn(I*x^4)-Pi*csgn(I/(x-5)^2)*csgn(I/(x-5)^2*(5-x^2-(-5*exp(x)+5
)*x)^2)^2-Pi*csgn(I*(5-x^2-(-5*exp(x)+5)*x)^2)*csgn(I/(x-5)^2*(5-x^2-(-5*exp(x)+5)*x)^2)^2+2*Pi*csgn(I*(5-x^2-
(-5*exp(x)+5)*x))*csgn(I*(5-x^2-(-5*exp(x)+5)*x)^2)^2+Pi*csgn(I*x^3)^3-Pi*csgn(I*x)*csgn(I*x^3)^2-Pi*csgn(I*x^
2)*csgn(I*x^3)^2+8*I*ln(x)+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2+Pi*csgn(I*(5-x^2-(-5*exp(x)
+5)*x)^2)^3+Pi*csgn(I/(x-5)^2*(5-x^2-(-5*exp(x)+5)*x)^2)^3-Pi*csgn(I*(x-5)^2)^3-4*I*ln(5)-4*I*ln(x-5)+4*I*ln(-
5+x^2+(-5*exp(x)+5)*x))

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maxima [A]  time = 0.95, size = 33, normalized size = 0.97 \begin {gather*} -\frac {1}{2 \, {\left (\log \relax (5) - \log \left (-x^{2} + 5 \, x e^{x} - 5 \, x + 5\right ) + \log \left (x - 5\right ) - 2 \, \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^3+30*x^2+150*x)*exp(x)+6*x^3-20*x^2-160*x+100)/((5*x^3-25*x^2)*exp(x)-x^4+30*x^2-25*x)/log((
25*x^6*exp(x)^2+(-10*x^7-50*x^6+50*x^5)*exp(x)+x^8+10*x^7+15*x^6-50*x^5+25*x^4)/(25*x^2-250*x+625))^2,x, algor
ithm="maxima")

[Out]

-1/2/(log(5) - log(-x^2 + 5*x*e^x - 5*x + 5) + log(x - 5) - 2*log(x))

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mupad [B]  time = 5.21, size = 69, normalized size = 2.03 \begin {gather*} \frac {1}{\ln \left (\frac {25\,x^6\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^x\,\left (10\,x^7+50\,x^6-50\,x^5\right )+25\,x^4-50\,x^5+15\,x^6+10\,x^7+x^8}{25\,x^2-250\,x+625}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x^3 - 20*x^2 - 160*x + exp(x)*(150*x + 30*x^2 - 10*x^3) + 100)/(log((25*x^6*exp(2*x) - exp(x)*(50*x^6
- 50*x^5 + 10*x^7) + 25*x^4 - 50*x^5 + 15*x^6 + 10*x^7 + x^8)/(25*x^2 - 250*x + 625))^2*(25*x + exp(x)*(25*x^2
- 5*x^3) - 30*x^2 + x^4)),x)

[Out]

1/log((25*x^6*exp(2*x) - exp(x)*(50*x^6 - 50*x^5 + 10*x^7) + 25*x^4 - 50*x^5 + 15*x^6 + 10*x^7 + x^8)/(25*x^2
- 250*x + 625))

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sympy [B]  time = 0.51, size = 65, normalized size = 1.91 \begin {gather*} \frac {1}{\log {\left (\frac {x^{8} + 10 x^{7} + 25 x^{6} e^{2 x} + 15 x^{6} - 50 x^{5} + 25 x^{4} + \left (- 10 x^{7} - 50 x^{6} + 50 x^{5}\right ) e^{x}}{25 x^{2} - 250 x + 625} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x**3+30*x**2+150*x)*exp(x)+6*x**3-20*x**2-160*x+100)/((5*x**3-25*x**2)*exp(x)-x**4+30*x**2-25*
x)/ln((25*x**6*exp(x)**2+(-10*x**7-50*x**6+50*x**5)*exp(x)+x**8+10*x**7+15*x**6-50*x**5+25*x**4)/(25*x**2-250*
x+625))**2,x)

[Out]

1/log((x**8 + 10*x**7 + 25*x**6*exp(2*x) + 15*x**6 - 50*x**5 + 25*x**4 + (-10*x**7 - 50*x**6 + 50*x**5)*exp(x)
)/(25*x**2 - 250*x + 625))

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