∫ −625x−1250x3+1875x4+ex(625+1250x2−1875x3)+(−2+2ex) log(ex−x)+(1250x2−2500x3+ex(−1250x+2500x2)) log(x)+(−625exx+625x2) log2(x)+log2(ex−x)(−2x2+3x3+ex(2x−3x2)+(2x−4x2+ex(−2+4x)) log(x)+(−ex+x) log2(x))_______________________________________________________________________________________________________________________________________________________________________

3.18.89 \(\int \frac {-625 x-1250 x^3+1875 x^4+e^x (625+1250 x^2-1875 x^3)+(-2+2 e^x) \log (e^x-x)+(1250 x^2-2500 x^3+e^x (-1250 x+2500 x^2)) \log (x)+(-625 e^x x+625 x^2) \log ^2(x)+\log ^2(e^x-x) (-2 x^2+3 x^3+e^x (2 x-3 x^2)+(2 x-4 x^2+e^x (-2+4 x)) \log (x)+(-e^x+x) \log ^2(x))}{625 e^x x-625 x^2+(e^x-x) \log ^2(e^x-x)} \, dx\)

Optimal. Leaf size=31 \[ -x (-x+\log (x))^2+\log \left (-x-\frac {1}{625} \log ^2\left (e^x-x\right )\right ) \]

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Rubi [F] time = 3.50, 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 {-625 x-1250 x^3+1875 x^4+e^x \left (625+1250 x^2-1875 x^3\right )+\left (-2+2 e^x\right ) \log \left (e^x-x\right )+\left (1250 x^2-2500 x^3+e^x \left (-1250 x+2500 x^2\right )\right ) \log (x)+\left (-625 e^x x+625 x^2\right ) \log ^2(x)+\log ^2\left (e^x-x\right ) \left (-2 x^2+3 x^3+e^x \left (2 x-3 x^2\right )+\left (2 x-4 x^2+e^x (-2+4 x)\right ) \log (x)+\left (-e^x+x\right ) \log ^2(x)\right )}{625 e^x x-625 x^2+\left (e^x-x\right ) \log ^2\left (e^x-x\right )} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

Int[(-625*x - 1250*x^3 + 1875*x^4 + E^x*(625 + 1250*x^2 - 1875*x^3) + (-2 + 2*E^x)*Log[E^x - x] + (1250*x^2 -

2500*x^3 + E^x*(-1250*x + 2500*x^2))*Log[x] + (-625*E^x*x + 625*x^2)*Log[x]^2 + Log[E^x - x]^2*(-2*x^2 + 3*x^3

+ E^x*(2*x - 3*x^2) + (2*x - 4*x^2 + E^x*(-2 + 4*x))*Log[x] + (-E^x + x)*Log[x]^2))/(625*E^x*x - 625*x^2 + (E

^x - x)*Log[E^x - x]^2),x]

[Out]

-x^3 + 2*x*Log[x] - 2*(x - x^2)*Log[x] - x*Log[x]^2 + 625*Defer[Int][(625*x + Log[E^x - x]^2)^(-1), x] + 2*Def

er[Int][Log[E^x - x]/(625*x + Log[E^x - x]^2), x] - 2*Defer[Int][Log[E^x - x]/((E^x - x)*(625*x + Log[E^x - x]

^2)), x] + 2*Defer[Int][(x*Log[E^x - x])/((E^x - x)*(625*x + Log[E^x - x]^2)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-1+e^x\right ) \log \left (e^x-x\right )-\left (e^x-x\right ) \log ^2\left (e^x-x\right ) \left (x (-2+3 x)+(2-4 x) \log (x)+\log ^2(x)\right )-625 \left (e^x-x\right ) \left (-1-2 x^2+3 x^3+2 (1-2 x) x \log (x)+x \log ^2(x)\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx\\ &=\int \left (\frac {2 (-1+x) \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )}+\frac {625+1250 x^2-1875 x^3+2 \log \left (e^x-x\right )+2 x \log ^2\left (e^x-x\right )-3 x^2 \log ^2\left (e^x-x\right )-1250 x \log (x)+2500 x^2 \log (x)-2 \log ^2\left (e^x-x\right ) \log (x)+4 x \log ^2\left (e^x-x\right ) \log (x)-625 x \log ^2(x)-\log ^2\left (e^x-x\right ) \log ^2(x)}{625 x+\log ^2\left (e^x-x\right )}\right ) \, dx\\ &=2 \int \frac {(-1+x) \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+\int \frac {625+1250 x^2-1875 x^3+2 \log \left (e^x-x\right )+2 x \log ^2\left (e^x-x\right )-3 x^2 \log ^2\left (e^x-x\right )-1250 x \log (x)+2500 x^2 \log (x)-2 \log ^2\left (e^x-x\right ) \log (x)+4 x \log ^2\left (e^x-x\right ) \log (x)-625 x \log ^2(x)-\log ^2\left (e^x-x\right ) \log ^2(x)}{625 x+\log ^2\left (e^x-x\right )} \, dx\\ &=2 \int \left (-\frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )}+\frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )}\right ) \, dx+\int \frac {2 \log \left (e^x-x\right )-\log ^2\left (e^x-x\right ) \left (x (-2+3 x)+(2-4 x) \log (x)+\log ^2(x)\right )-625 \left (-1-2 x^2+3 x^3+2 (1-2 x) x \log (x)+x \log ^2(x)\right )}{625 x+\log ^2\left (e^x-x\right )} \, dx\\ &=-\left (2 \int \frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx\right )+2 \int \frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+\int \left (\frac {625+1250 x^2-1875 x^3+2 \log \left (e^x-x\right )+2 x \log ^2\left (e^x-x\right )-3 x^2 \log ^2\left (e^x-x\right )}{625 x+\log ^2\left (e^x-x\right )}+2 (-1+2 x) \log (x)-\log ^2(x)\right ) \, dx\\ &=-\left (2 \int \frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx\right )+2 \int \frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+2 \int (-1+2 x) \log (x) \, dx+\int \frac {625+1250 x^2-1875 x^3+2 \log \left (e^x-x\right )+2 x \log ^2\left (e^x-x\right )-3 x^2 \log ^2\left (e^x-x\right )}{625 x+\log ^2\left (e^x-x\right )} \, dx-\int \log ^2(x) \, dx\\ &=-2 \left (x-x^2\right ) \log (x)-x \log ^2(x)-2 \int (-1+x) \, dx-2 \int \frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+2 \int \frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+2 \int \log (x) \, dx+\int \left (-x (-2+3 x)+\frac {625+2 \log \left (e^x-x\right )}{625 x+\log ^2\left (e^x-x\right )}\right ) \, dx\\ &=-x^2+2 x \log (x)-2 \left (x-x^2\right ) \log (x)-x \log ^2(x)-2 \int \frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+2 \int \frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx-\int x (-2+3 x) \, dx+\int \frac {625+2 \log \left (e^x-x\right )}{625 x+\log ^2\left (e^x-x\right )} \, dx\\ &=-x^2+2 x \log (x)-2 \left (x-x^2\right ) \log (x)-x \log ^2(x)-2 \int \frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+2 \int \frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx-\int \left (-2 x+3 x^2\right ) \, dx+\int \left (\frac {625}{625 x+\log ^2\left (e^x-x\right )}+\frac {2 \log \left (e^x-x\right )}{625 x+\log ^2\left (e^x-x\right )}\right ) \, dx\\ &=-x^3+2 x \log (x)-2 \left (x-x^2\right ) \log (x)-x \log ^2(x)+2 \int \frac {\log \left (e^x-x\right )}{625 x+\log ^2\left (e^x-x\right )} \, dx-2 \int \frac {\log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+2 \int \frac {x \log \left (e^x-x\right )}{\left (e^x-x\right ) \left (625 x+\log ^2\left (e^x-x\right )\right )} \, dx+625 \int \frac {1}{625 x+\log ^2\left (e^x-x\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A] time = 0.12, size = 35, normalized size = 1.13 \begin {gather*} -x^3+2 x^2 \log (x)-x \log ^2(x)+\log \left (625 x+\log ^2\left (e^x-x\right )\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-625*x - 1250*x^3 + 1875*x^4 + E^x*(625 + 1250*x^2 - 1875*x^3) + (-2 + 2*E^x)*Log[E^x - x] + (1250*

x^2 - 2500*x^3 + E^x*(-1250*x + 2500*x^2))*Log[x] + (-625*E^x*x + 625*x^2)*Log[x]^2 + Log[E^x - x]^2*(-2*x^2 +

3*x^3 + E^x*(2*x - 3*x^2) + (2*x - 4*x^2 + E^x*(-2 + 4*x))*Log[x] + (-E^x + x)*Log[x]^2))/(625*E^x*x - 625*x^

2 + (E^x - x)*Log[E^x - x]^2),x]

[Out]

-x^3 + 2*x^2*Log[x] - x*Log[x]^2 + Log[625*x + Log[E^x - x]^2]

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fricas [A] time = 0.82, size = 34, normalized size = 1.10 \begin {gather*} -x^{3} + 2 \, x^{2} \log \relax (x) - x \log \relax (x)^{2} + \log \left (\log \left (-x + e^{x}\right )^{2} + 625 \, x\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-exp(x))*log(x)^2+((4*x-2)*exp(x)-4*x^2+2*x)*log(x)+(-3*x^2+2*x)*exp(x)+3*x^3-2*x^2)*log(exp(x)-

x)^2+(2*exp(x)-2)*log(exp(x)-x)+(-625*exp(x)*x+625*x^2)*log(x)^2+((2500*x^2-1250*x)*exp(x)-2500*x^3+1250*x^2)*

log(x)+(-1875*x^3+1250*x^2+625)*exp(x)+1875*x^4-1250*x^3-625*x)/((exp(x)-x)*log(exp(x)-x)^2+625*exp(x)*x-625*x

^2),x, algorithm="fricas")

[Out]

-x^3 + 2*x^2*log(x) - x*log(x)^2 + log(log(-x + e^x)^2 + 625*x)

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giac [A] time = 0.86, size = 34, normalized size = 1.10 \begin {gather*} -x^{3} + 2 \, x^{2} \log \relax (x) - x \log \relax (x)^{2} + \log \left (\log \left (-x + e^{x}\right )^{2} + 625 \, x\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-exp(x))*log(x)^2+((4*x-2)*exp(x)-4*x^2+2*x)*log(x)+(-3*x^2+2*x)*exp(x)+3*x^3-2*x^2)*log(exp(x)-

x)^2+(2*exp(x)-2)*log(exp(x)-x)+(-625*exp(x)*x+625*x^2)*log(x)^2+((2500*x^2-1250*x)*exp(x)-2500*x^3+1250*x^2)*

log(x)+(-1875*x^3+1250*x^2+625)*exp(x)+1875*x^4-1250*x^3-625*x)/((exp(x)-x)*log(exp(x)-x)^2+625*exp(x)*x-625*x

^2),x, algorithm="giac")

[Out]

-x^3 + 2*x^2*log(x) - x*log(x)^2 + log(log(-x + e^x)^2 + 625*x)

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maple [A] time = 0.04, size = 35, normalized size = 1.13


method	result	size

risch	\(-x^{3}+2 x^{2} \ln \relax (x )-x \ln \relax (x )^{2}+\ln \left (\ln \left ({\mathrm e}^{x}-x \right )^{2}+625 x \right )\)	\(35\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((((x-exp(x))*ln(x)^2+((4*x-2)*exp(x)-4*x^2+2*x)*ln(x)+(-3*x^2+2*x)*exp(x)+3*x^3-2*x^2)*ln(exp(x)-x)^2+(2*e

xp(x)-2)*ln(exp(x)-x)+(-625*exp(x)*x+625*x^2)*ln(x)^2+((2500*x^2-1250*x)*exp(x)-2500*x^3+1250*x^2)*ln(x)+(-187

5*x^3+1250*x^2+625)*exp(x)+1875*x^4-1250*x^3-625*x)/((exp(x)-x)*ln(exp(x)-x)^2+625*exp(x)*x-625*x^2),x,method=

_RETURNVERBOSE)

[Out]

-x^3+2*x^2*ln(x)-x*ln(x)^2+ln(ln(exp(x)-x)^2+625*x)

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maxima [A] time = 0.58, size = 34, normalized size = 1.10 \begin {gather*} -x^{3} + 2 \, x^{2} \log \relax (x) - x \log \relax (x)^{2} + \log \left (\log \left (-x + e^{x}\right )^{2} + 625 \, x\right ) \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-exp(x))*log(x)^2+((4*x-2)*exp(x)-4*x^2+2*x)*log(x)+(-3*x^2+2*x)*exp(x)+3*x^3-2*x^2)*log(exp(x)-

x)^2+(2*exp(x)-2)*log(exp(x)-x)+(-625*exp(x)*x+625*x^2)*log(x)^2+((2500*x^2-1250*x)*exp(x)-2500*x^3+1250*x^2)*

log(x)+(-1875*x^3+1250*x^2+625)*exp(x)+1875*x^4-1250*x^3-625*x)/((exp(x)-x)*log(exp(x)-x)^2+625*exp(x)*x-625*x

^2),x, algorithm="maxima")

[Out]

-x^3 + 2*x^2*log(x) - x*log(x)^2 + log(log(-x + e^x)^2 + 625*x)

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mupad [B] time = 1.30, size = 34, normalized size = 1.10 \begin {gather*} \ln \left ({\ln \left ({\mathrm {e}}^x-x\right )}^2+625\,x\right )-x\,{\ln \relax (x)}^2+2\,x^2\,\ln \relax (x)-x^3 \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((625*x - exp(x)*(1250*x^2 - 1875*x^3 + 625) - log(exp(x) - x)*(2*exp(x) - 2) - log(exp(x) - x)^2*(log(x)*(

2*x + exp(x)*(4*x - 2) - 4*x^2) + exp(x)*(2*x - 3*x^2) + log(x)^2*(x - exp(x)) - 2*x^2 + 3*x^3) + log(x)*(exp(

x)*(1250*x - 2500*x^2) - 1250*x^2 + 2500*x^3) + 1250*x^3 - 1875*x^4 + log(x)^2*(625*x*exp(x) - 625*x^2))/(log(

exp(x) - x)^2*(x - exp(x)) - 625*x*exp(x) + 625*x^2),x)

[Out]

log(625*x + log(exp(x) - x)^2) - x*log(x)^2 + 2*x^2*log(x) - x^3

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sympy [A] time = 0.64, size = 31, normalized size = 1.00 \begin {gather*} - x^{3} + 2 x^{2} \log {\relax (x )} - x \log {\relax (x )}^{2} + \log {\left (625 x + \log {\left (- x + e^{x} \right )}^{2} \right )} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x-exp(x))*ln(x)**2+((4*x-2)*exp(x)-4*x**2+2*x)*ln(x)+(-3*x**2+2*x)*exp(x)+3*x**3-2*x**2)*ln(exp(x

)-x)**2+(2*exp(x)-2)*ln(exp(x)-x)+(-625*exp(x)*x+625*x**2)*ln(x)**2+((2500*x**2-1250*x)*exp(x)-2500*x**3+1250*

x**2)*ln(x)+(-1875*x**3+1250*x**2+625)*exp(x)+1875*x**4-1250*x**3-625*x)/((exp(x)-x)*ln(exp(x)-x)**2+625*exp(x

)*x-625*x**2),x)

[Out]

-x**3 + 2*x**2*log(x) - x*log(x)**2 + log(625*x + log(-x + exp(x))**2)

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