3.76.1 \(\int \frac {(-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x (560-80 x-14 x^2+2 x^3)) \log (x)+(-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x (280-80 x+7 x^2-7 x^3+x^4)) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x (-20480+1024 x^2)} \, dx\)

Optimal. Leaf size=34 \[ \frac {(7-x) x \log ^2(x)}{256 \left (-2+\frac {x^2}{2 \left (5-e^x\right )}\right )} \]

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Rubi [F]  time = 5.93, 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 {\left (-1400+200 x+70 x^2-10 x^3+e^{2 x} (-56+8 x)+e^x \left (560-80 x-14 x^2+2 x^3\right )\right ) \log (x)+\left (-700+200 x-35 x^2+e^{2 x} (-28+8 x)+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log ^2(x)}{51200+2048 e^{2 x}-5120 x^2+128 x^4+e^x \left (-20480+1024 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1400 + 200*x + 70*x^2 - 10*x^3 + E^(2*x)*(-56 + 8*x) + E^x*(560 - 80*x - 14*x^2 + 2*x^3))*Log[x] + (-70
0 + 200*x - 35*x^2 + E^(2*x)*(-28 + 8*x) + E^x*(280 - 80*x + 7*x^2 - 7*x^3 + x^4))*Log[x]^2)/(51200 + 2048*E^(
2*x) - 5120*x^2 + 128*x^4 + E^x*(-20480 + 1024*x^2)),x]

[Out]

(7*x*Log[x])/256 - (x^2*Log[x])/512 - ((14*x - x^2)*Log[x])/512 - (7*x*Log[x]^2)/512 + (x^2*Log[x]^2)/512 + (7
*Log[x]*Defer[Int][x^2/(-20 + 4*E^x + x^2), x])/256 - (Log[x]*Defer[Int][x^3/(-20 + 4*E^x + x^2), x])/256 - (3
5*Defer[Int][(x^3*Log[x]^2)/(-20 + 4*E^x + x^2)^2, x])/128 + (3*Defer[Int][(x^4*Log[x]^2)/(-20 + 4*E^x + x^2)^
2, x])/256 + (9*Defer[Int][(x^5*Log[x]^2)/(-20 + 4*E^x + x^2)^2, x])/512 - Defer[Int][(x^6*Log[x]^2)/(-20 + 4*
E^x + x^2)^2, x]/512 + (21*Defer[Int][(x^2*Log[x]^2)/(-20 + 4*E^x + x^2), x])/512 - (11*Defer[Int][(x^3*Log[x]
^2)/(-20 + 4*E^x + x^2), x])/512 + Defer[Int][(x^4*Log[x]^2)/(-20 + 4*E^x + x^2), x]/512 - (7*Defer[Int][Defer
[Int][x^2/(-20 + 4*E^x + x^2), x]/x, x])/256 + Defer[Int][Defer[Int][x^3/(-20 + 4*E^x + x^2), x]/x, x]/256

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\log (x) \left (2 \left (-5+e^x\right ) (-7+x) \left (-20+4 e^x+x^2\right )+\left (4 e^{2 x} (-7+2 x)-5 \left (140-40 x+7 x^2\right )+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log (x)\right )}{128 \left (20-4 e^x-x^2\right )^2} \, dx\\ &=\frac {1}{128} \int \frac {\log (x) \left (2 \left (-5+e^x\right ) (-7+x) \left (-20+4 e^x+x^2\right )+\left (4 e^{2 x} (-7+2 x)-5 \left (140-40 x+7 x^2\right )+e^x \left (280-80 x+7 x^2-7 x^3+x^4\right )\right ) \log (x)\right )}{\left (20-4 e^x-x^2\right )^2} \, dx\\ &=\frac {1}{128} \int \left (-\frac {x^3 \left (140-6 x-9 x^2+x^3\right ) \log ^2(x)}{4 \left (-20+4 e^x+x^2\right )^2}+\frac {1}{4} \log (x) (-14+2 x-7 \log (x)+2 x \log (x))+\frac {x^2 \log (x) \left (14-2 x+21 \log (x)-11 x \log (x)+x^2 \log (x)\right )}{4 \left (-20+4 e^x+x^2\right )}\right ) \, dx\\ &=-\left (\frac {1}{512} \int \frac {x^3 \left (140-6 x-9 x^2+x^3\right ) \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx\right )+\frac {1}{512} \int \log (x) (-14+2 x-7 \log (x)+2 x \log (x)) \, dx+\frac {1}{512} \int \frac {x^2 \log (x) \left (14-2 x+21 \log (x)-11 x \log (x)+x^2 \log (x)\right )}{-20+4 e^x+x^2} \, dx\\ &=\frac {1}{512} \int \left (2 (-7+x) \log (x)+(-7+2 x) \log ^2(x)\right ) \, dx-\frac {1}{512} \int \left (\frac {140 x^3 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2}-\frac {6 x^4 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2}-\frac {9 x^5 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2}+\frac {x^6 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2}\right ) \, dx+\frac {1}{512} \int \left (\frac {14 x^2 \log (x)}{-20+4 e^x+x^2}-\frac {2 x^3 \log (x)}{-20+4 e^x+x^2}+\frac {21 x^2 \log ^2(x)}{-20+4 e^x+x^2}-\frac {11 x^3 \log ^2(x)}{-20+4 e^x+x^2}+\frac {x^4 \log ^2(x)}{-20+4 e^x+x^2}\right ) \, dx\\ &=\frac {1}{512} \int (-7+2 x) \log ^2(x) \, dx-\frac {1}{512} \int \frac {x^6 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {1}{512} \int \frac {x^4 \log ^2(x)}{-20+4 e^x+x^2} \, dx+\frac {1}{256} \int (-7+x) \log (x) \, dx-\frac {1}{256} \int \frac {x^3 \log (x)}{-20+4 e^x+x^2} \, dx+\frac {3}{256} \int \frac {x^4 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {9}{512} \int \frac {x^5 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {11}{512} \int \frac {x^3 \log ^2(x)}{-20+4 e^x+x^2} \, dx+\frac {7}{256} \int \frac {x^2 \log (x)}{-20+4 e^x+x^2} \, dx+\frac {21}{512} \int \frac {x^2 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {35}{128} \int \frac {x^3 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx\\ &=-\frac {1}{512} \left (14 x-x^2\right ) \log (x)-\frac {1}{512} \int \frac {x^6 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {1}{512} \int \frac {x^4 \log ^2(x)}{-20+4 e^x+x^2} \, dx+\frac {1}{512} \int \left (-7 \log ^2(x)+2 x \log ^2(x)\right ) \, dx-\frac {1}{256} \int \left (-7+\frac {x}{2}\right ) \, dx+\frac {1}{256} \int \frac {\int \frac {x^3}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {3}{256} \int \frac {x^4 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {9}{512} \int \frac {x^5 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {11}{512} \int \frac {x^3 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {7}{256} \int \frac {\int \frac {x^2}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {21}{512} \int \frac {x^2 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {35}{128} \int \frac {x^3 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {1}{256} \log (x) \int \frac {x^3}{-20+4 e^x+x^2} \, dx+\frac {1}{256} (7 \log (x)) \int \frac {x^2}{-20+4 e^x+x^2} \, dx\\ &=\frac {7 x}{256}-\frac {x^2}{1024}-\frac {1}{512} \left (14 x-x^2\right ) \log (x)-\frac {1}{512} \int \frac {x^6 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {1}{512} \int \frac {x^4 \log ^2(x)}{-20+4 e^x+x^2} \, dx+\frac {1}{256} \int x \log ^2(x) \, dx+\frac {1}{256} \int \frac {\int \frac {x^3}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {3}{256} \int \frac {x^4 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {7}{512} \int \log ^2(x) \, dx+\frac {9}{512} \int \frac {x^5 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {11}{512} \int \frac {x^3 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {7}{256} \int \frac {\int \frac {x^2}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {21}{512} \int \frac {x^2 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {35}{128} \int \frac {x^3 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {1}{256} \log (x) \int \frac {x^3}{-20+4 e^x+x^2} \, dx+\frac {1}{256} (7 \log (x)) \int \frac {x^2}{-20+4 e^x+x^2} \, dx\\ &=\frac {7 x}{256}-\frac {x^2}{1024}-\frac {1}{512} \left (14 x-x^2\right ) \log (x)-\frac {7}{512} x \log ^2(x)+\frac {1}{512} x^2 \log ^2(x)-\frac {1}{512} \int \frac {x^6 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {1}{512} \int \frac {x^4 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {1}{256} \int x \log (x) \, dx+\frac {1}{256} \int \frac {\int \frac {x^3}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {3}{256} \int \frac {x^4 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {9}{512} \int \frac {x^5 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {11}{512} \int \frac {x^3 \log ^2(x)}{-20+4 e^x+x^2} \, dx+\frac {7}{256} \int \log (x) \, dx-\frac {7}{256} \int \frac {\int \frac {x^2}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {21}{512} \int \frac {x^2 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {35}{128} \int \frac {x^3 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {1}{256} \log (x) \int \frac {x^3}{-20+4 e^x+x^2} \, dx+\frac {1}{256} (7 \log (x)) \int \frac {x^2}{-20+4 e^x+x^2} \, dx\\ &=\frac {7}{256} x \log (x)-\frac {1}{512} x^2 \log (x)-\frac {1}{512} \left (14 x-x^2\right ) \log (x)-\frac {7}{512} x \log ^2(x)+\frac {1}{512} x^2 \log ^2(x)-\frac {1}{512} \int \frac {x^6 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {1}{512} \int \frac {x^4 \log ^2(x)}{-20+4 e^x+x^2} \, dx+\frac {1}{256} \int \frac {\int \frac {x^3}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {3}{256} \int \frac {x^4 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx+\frac {9}{512} \int \frac {x^5 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {11}{512} \int \frac {x^3 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {7}{256} \int \frac {\int \frac {x^2}{-20+4 e^x+x^2} \, dx}{x} \, dx+\frac {21}{512} \int \frac {x^2 \log ^2(x)}{-20+4 e^x+x^2} \, dx-\frac {35}{128} \int \frac {x^3 \log ^2(x)}{\left (-20+4 e^x+x^2\right )^2} \, dx-\frac {1}{256} \log (x) \int \frac {x^3}{-20+4 e^x+x^2} \, dx+\frac {1}{256} (7 \log (x)) \int \frac {x^2}{-20+4 e^x+x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 29, normalized size = 0.85 \begin {gather*} \frac {\left (-5+e^x\right ) (-7+x) x \log ^2(x)}{128 \left (-20+4 e^x+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-1400 + 200*x + 70*x^2 - 10*x^3 + E^(2*x)*(-56 + 8*x) + E^x*(560 - 80*x - 14*x^2 + 2*x^3))*Log[x]
+ (-700 + 200*x - 35*x^2 + E^(2*x)*(-28 + 8*x) + E^x*(280 - 80*x + 7*x^2 - 7*x^3 + x^4))*Log[x]^2)/(51200 + 20
48*E^(2*x) - 5120*x^2 + 128*x^4 + E^x*(-20480 + 1024*x^2)),x]

[Out]

((-5 + E^x)*(-7 + x)*x*Log[x]^2)/(128*(-20 + 4*E^x + x^2))

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fricas [A]  time = 0.66, size = 37, normalized size = 1.09 \begin {gather*} -\frac {{\left (5 \, x^{2} - {\left (x^{2} - 7 \, x\right )} e^{x} - 35 \, x\right )} \log \relax (x)^{2}}{128 \, {\left (x^{2} + 4 \, e^{x} - 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x-28)*exp(x)^2+(x^4-7*x^3+7*x^2-80*x+280)*exp(x)-35*x^2+200*x-700)*log(x)^2+((8*x-56)*exp(x)^2+
(2*x^3-14*x^2-80*x+560)*exp(x)-10*x^3+70*x^2+200*x-1400)*log(x))/(2048*exp(x)^2+(1024*x^2-20480)*exp(x)+128*x^
4-5120*x^2+51200),x, algorithm="fricas")

[Out]

-1/128*(5*x^2 - (x^2 - 7*x)*e^x - 35*x)*log(x)^2/(x^2 + 4*e^x - 20)

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giac [B]  time = 0.25, size = 49, normalized size = 1.44 \begin {gather*} \frac {x^{2} e^{x} \log \relax (x)^{2} - 5 \, x^{2} \log \relax (x)^{2} - 7 \, x e^{x} \log \relax (x)^{2} + 35 \, x \log \relax (x)^{2}}{128 \, {\left (x^{2} + 4 \, e^{x} - 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x-28)*exp(x)^2+(x^4-7*x^3+7*x^2-80*x+280)*exp(x)-35*x^2+200*x-700)*log(x)^2+((8*x-56)*exp(x)^2+
(2*x^3-14*x^2-80*x+560)*exp(x)-10*x^3+70*x^2+200*x-1400)*log(x))/(2048*exp(x)^2+(1024*x^2-20480)*exp(x)+128*x^
4-5120*x^2+51200),x, algorithm="giac")

[Out]

1/128*(x^2*e^x*log(x)^2 - 5*x^2*log(x)^2 - 7*x*e^x*log(x)^2 + 35*x*log(x)^2)/(x^2 + 4*e^x - 20)

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maple [A]  time = 0.04, size = 32, normalized size = 0.94

method	result	size
risch	\(\frac {x \left ({\mathrm e}^{x} x -5 x -7 \,{\mathrm e}^{x}+35\right ) \ln \relax (x )^{2}}{128 x^{2}+512 \,{\mathrm e}^{x}-2560}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((8*x-28)*exp(x)^2+(x^4-7*x^3+7*x^2-80*x+280)*exp(x)-35*x^2+200*x-700)*ln(x)^2+((8*x-56)*exp(x)^2+(2*x^3-
14*x^2-80*x+560)*exp(x)-10*x^3+70*x^2+200*x-1400)*ln(x))/(2048*exp(x)^2+(1024*x^2-20480)*exp(x)+128*x^4-5120*x
^2+51200),x,method=_RETURNVERBOSE)

[Out]

1/128*x*(exp(x)*x-5*x-7*exp(x)+35)/(x^2+4*exp(x)-20)*ln(x)^2

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maxima [A]  time = 0.42, size = 41, normalized size = 1.21 \begin {gather*} \frac {{\left (x^{2} - 7 \, x\right )} e^{x} \log \relax (x)^{2} - 5 \, {\left (x^{2} - 7 \, x\right )} \log \relax (x)^{2}}{128 \, {\left (x^{2} + 4 \, e^{x} - 20\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x-28)*exp(x)^2+(x^4-7*x^3+7*x^2-80*x+280)*exp(x)-35*x^2+200*x-700)*log(x)^2+((8*x-56)*exp(x)^2+
(2*x^3-14*x^2-80*x+560)*exp(x)-10*x^3+70*x^2+200*x-1400)*log(x))/(2048*exp(x)^2+(1024*x^2-20480)*exp(x)+128*x^
4-5120*x^2+51200),x, algorithm="maxima")

[Out]

1/128*((x^2 - 7*x)*e^x*log(x)^2 - 5*(x^2 - 7*x)*log(x)^2)/(x^2 + 4*e^x - 20)

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mupad [B]  time = 5.10, size = 26, normalized size = 0.76 \begin {gather*} \frac {x\,{\ln \relax (x)}^2\,\left ({\mathrm {e}}^x-5\right )\,\left (x-7\right )}{128\,\left (4\,{\mathrm {e}}^x+x^2-20\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(200*x + exp(2*x)*(8*x - 56) + 70*x^2 - 10*x^3 - exp(x)*(80*x + 14*x^2 - 2*x^3 - 560) - 1400) + lo
g(x)^2*(200*x + exp(x)*(7*x^2 - 80*x - 7*x^3 + x^4 + 280) + exp(2*x)*(8*x - 28) - 35*x^2 - 700))/(2048*exp(2*x
) + exp(x)*(1024*x^2 - 20480) - 5120*x^2 + 128*x^4 + 51200),x)

[Out]

(x*log(x)^2*(exp(x) - 5)*(x - 7))/(128*(4*exp(x) + x^2 - 20))

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sympy [B]  time = 0.39, size = 44, normalized size = 1.29 \begin {gather*} \left (\frac {x^{2}}{512} - \frac {7 x}{512}\right ) \log {\relax (x )}^{2} + \frac {- x^{4} \log {\relax (x )}^{2} + 7 x^{3} \log {\relax (x )}^{2}}{512 x^{2} + 2048 e^{x} - 10240} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((8*x-28)*exp(x)**2+(x**4-7*x**3+7*x**2-80*x+280)*exp(x)-35*x**2+200*x-700)*ln(x)**2+((8*x-56)*exp(
x)**2+(2*x**3-14*x**2-80*x+560)*exp(x)-10*x**3+70*x**2+200*x-1400)*ln(x))/(2048*exp(x)**2+(1024*x**2-20480)*ex
p(x)+128*x**4-5120*x**2+51200),x)

[Out]

(x**2/512 - 7*x/512)*log(x)**2 + (-x**4*log(x)**2 + 7*x**3*log(x)**2)/(512*x**2 + 2048*exp(x) - 10240)

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