3.89.33 \(\int \frac {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+(-50 x-18 x^3+36 x^4-18 x^5) \log (x)+(18 x^2-18 x^3) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} (-x^4+3 x^5-3 x^6+x^7)+e^{2+2 x} (3 x^3-6 x^4+3 x^5) \log (x)+e^{2+2 x} (-3 x^2+3 x^3) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx\)

Optimal. Leaf size=28 \[ 5+e^{-2-2 x} \left (3+\frac {25}{(x-(2-x) x+\log (x))^2}\right ) \]

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Rubi [F]  time = 6.54, 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 {-50+50 x-50 x^2-50 x^3+6 x^4-18 x^5+18 x^6-6 x^7+\left (-50 x-18 x^3+36 x^4-18 x^5\right ) \log (x)+\left (18 x^2-18 x^3\right ) \log ^2(x)-6 x \log ^3(x)}{e^{2+2 x} \left (-x^4+3 x^5-3 x^6+x^7\right )+e^{2+2 x} \left (3 x^3-6 x^4+3 x^5\right ) \log (x)+e^{2+2 x} \left (-3 x^2+3 x^3\right ) \log ^2(x)+e^{2+2 x} x \log ^3(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-50 + 50*x - 50*x^2 - 50*x^3 + 6*x^4 - 18*x^5 + 18*x^6 - 6*x^7 + (-50*x - 18*x^3 + 36*x^4 - 18*x^5)*Log[x
] + (18*x^2 - 18*x^3)*Log[x]^2 - 6*x*Log[x]^3)/(E^(2 + 2*x)*(-x^4 + 3*x^5 - 3*x^6 + x^7) + E^(2 + 2*x)*(3*x^3
- 6*x^4 + 3*x^5)*Log[x] + E^(2 + 2*x)*(-3*x^2 + 3*x^3)*Log[x]^2 + E^(2 + 2*x)*x*Log[x]^3),x]

[Out]

3*E^(-2 - 2*x) + 50*Defer[Int][E^(-2 - 2*x)/(-x + x^2 + Log[x])^3, x] - 50*Defer[Int][E^(-2 - 2*x)/(x*(-x + x^
2 + Log[x])^3), x] - 100*Defer[Int][(E^(-2 - 2*x)*x)/(-x + x^2 + Log[x])^3, x] - 50*Defer[Int][E^(-2 - 2*x)/(-
x + x^2 + Log[x])^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{-2-2 x} \left (-25+25 x-25 x^2-25 x^3+3 x^4-9 x^5+9 x^6-3 x^7-x \left (25+9 x^2-18 x^3+9 x^4\right ) \log (x)-9 (-1+x) x^2 \log ^2(x)-3 x \log ^3(x)\right )}{x ((-1+x) x+\log (x))^3} \, dx\\ &=2 \int \frac {e^{-2-2 x} \left (-25+25 x-25 x^2-25 x^3+3 x^4-9 x^5+9 x^6-3 x^7-x \left (25+9 x^2-18 x^3+9 x^4\right ) \log (x)-9 (-1+x) x^2 \log ^2(x)-3 x \log ^3(x)\right )}{x ((-1+x) x+\log (x))^3} \, dx\\ &=2 \int \left (-3 e^{-2-2 x}-\frac {25 e^{-2-2 x} \left (1-x+2 x^2\right )}{x \left (-x+x^2+\log (x)\right )^3}-\frac {25 e^{-2-2 x}}{\left (-x+x^2+\log (x)\right )^2}\right ) \, dx\\ &=-\left (6 \int e^{-2-2 x} \, dx\right )-50 \int \frac {e^{-2-2 x} \left (1-x+2 x^2\right )}{x \left (-x+x^2+\log (x)\right )^3} \, dx-50 \int \frac {e^{-2-2 x}}{\left (-x+x^2+\log (x)\right )^2} \, dx\\ &=3 e^{-2-2 x}-50 \int \frac {e^{-2-2 x}}{\left (-x+x^2+\log (x)\right )^2} \, dx-50 \int \left (-\frac {e^{-2-2 x}}{\left (-x+x^2+\log (x)\right )^3}+\frac {e^{-2-2 x}}{x \left (-x+x^2+\log (x)\right )^3}+\frac {2 e^{-2-2 x} x}{\left (-x+x^2+\log (x)\right )^3}\right ) \, dx\\ &=3 e^{-2-2 x}+50 \int \frac {e^{-2-2 x}}{\left (-x+x^2+\log (x)\right )^3} \, dx-50 \int \frac {e^{-2-2 x}}{x \left (-x+x^2+\log (x)\right )^3} \, dx-50 \int \frac {e^{-2-2 x}}{\left (-x+x^2+\log (x)\right )^2} \, dx-100 \int \frac {e^{-2-2 x} x}{\left (-x+x^2+\log (x)\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 23, normalized size = 0.82 \begin {gather*} -e^{-2 (1+x)} \left (-3-\frac {25}{((-1+x) x+\log (x))^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-50 + 50*x - 50*x^2 - 50*x^3 + 6*x^4 - 18*x^5 + 18*x^6 - 6*x^7 + (-50*x - 18*x^3 + 36*x^4 - 18*x^5)
*Log[x] + (18*x^2 - 18*x^3)*Log[x]^2 - 6*x*Log[x]^3)/(E^(2 + 2*x)*(-x^4 + 3*x^5 - 3*x^6 + x^7) + E^(2 + 2*x)*(
3*x^3 - 6*x^4 + 3*x^5)*Log[x] + E^(2 + 2*x)*(-3*x^2 + 3*x^3)*Log[x]^2 + E^(2 + 2*x)*x*Log[x]^3),x]

[Out]

-((-3 - 25/((-1 + x)*x + Log[x])^2)/E^(2*(1 + x)))

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fricas [B]  time = 0.49, size = 85, normalized size = 3.04 \begin {gather*} \frac {3 \, x^{4} - 6 \, x^{3} + 3 \, x^{2} + 6 \, {\left (x^{2} - x\right )} \log \relax (x) + 3 \, \log \relax (x)^{2} + 25}{2 \, {\left (x^{2} - x\right )} e^{\left (2 \, x + 2\right )} \log \relax (x) + e^{\left (2 \, x + 2\right )} \log \relax (x)^{2} + {\left (x^{4} - 2 \, x^{3} + x^{2}\right )} e^{\left (2 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x*log(x)^3+(-18*x^3+18*x^2)*log(x)^2+(-18*x^5+36*x^4-18*x^3-50*x)*log(x)-6*x^7+18*x^6-18*x^5+6*x
^4-50*x^3-50*x^2+50*x-50)/(x*exp(x+1)^2*log(x)^3+(3*x^3-3*x^2)*exp(x+1)^2*log(x)^2+(3*x^5-6*x^4+3*x^3)*exp(x+1
)^2*log(x)+(x^7-3*x^6+3*x^5-x^4)*exp(x+1)^2),x, algorithm="fricas")

[Out]

(3*x^4 - 6*x^3 + 3*x^2 + 6*(x^2 - x)*log(x) + 3*log(x)^2 + 25)/(2*(x^2 - x)*e^(2*x + 2)*log(x) + e^(2*x + 2)*l
og(x)^2 + (x^4 - 2*x^3 + x^2)*e^(2*x + 2))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {2 \, {\left (3 \, x^{7} - 9 \, x^{6} + 9 \, x^{5} - 3 \, x^{4} + 3 \, x \log \relax (x)^{3} + 25 \, x^{3} + 9 \, {\left (x^{3} - x^{2}\right )} \log \relax (x)^{2} + 25 \, x^{2} + {\left (9 \, x^{5} - 18 \, x^{4} + 9 \, x^{3} + 25 \, x\right )} \log \relax (x) - 25 \, x + 25\right )}}{x e^{\left (2 \, x + 2\right )} \log \relax (x)^{3} + 3 \, {\left (x^{3} - x^{2}\right )} e^{\left (2 \, x + 2\right )} \log \relax (x)^{2} + 3 \, {\left (x^{5} - 2 \, x^{4} + x^{3}\right )} e^{\left (2 \, x + 2\right )} \log \relax (x) + {\left (x^{7} - 3 \, x^{6} + 3 \, x^{5} - x^{4}\right )} e^{\left (2 \, x + 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x*log(x)^3+(-18*x^3+18*x^2)*log(x)^2+(-18*x^5+36*x^4-18*x^3-50*x)*log(x)-6*x^7+18*x^6-18*x^5+6*x
^4-50*x^3-50*x^2+50*x-50)/(x*exp(x+1)^2*log(x)^3+(3*x^3-3*x^2)*exp(x+1)^2*log(x)^2+(3*x^5-6*x^4+3*x^3)*exp(x+1
)^2*log(x)+(x^7-3*x^6+3*x^5-x^4)*exp(x+1)^2),x, algorithm="giac")

[Out]

integrate(-2*(3*x^7 - 9*x^6 + 9*x^5 - 3*x^4 + 3*x*log(x)^3 + 25*x^3 + 9*(x^3 - x^2)*log(x)^2 + 25*x^2 + (9*x^5
- 18*x^4 + 9*x^3 + 25*x)*log(x) - 25*x + 25)/(x*e^(2*x + 2)*log(x)^3 + 3*(x^3 - x^2)*e^(2*x + 2)*log(x)^2 + 3
*(x^5 - 2*x^4 + x^3)*e^(2*x + 2)*log(x) + (x^7 - 3*x^6 + 3*x^5 - x^4)*e^(2*x + 2)), x)

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maple [A]  time = 0.04, size = 29, normalized size = 1.04

method	result	size
risch	\(3 \,{\mathrm e}^{-2 x -2}+\frac {25 \,{\mathrm e}^{-2 x -2}}{\left (x^{2}+\ln \relax (x )-x \right )^{2}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-6*x*ln(x)^3+(-18*x^3+18*x^2)*ln(x)^2+(-18*x^5+36*x^4-18*x^3-50*x)*ln(x)-6*x^7+18*x^6-18*x^5+6*x^4-50*x^3
-50*x^2+50*x-50)/(x*exp(x+1)^2*ln(x)^3+(3*x^3-3*x^2)*exp(x+1)^2*ln(x)^2+(3*x^5-6*x^4+3*x^3)*exp(x+1)^2*ln(x)+(
x^7-3*x^6+3*x^5-x^4)*exp(x+1)^2),x,method=_RETURNVERBOSE)

[Out]

3*exp(-2*x-2)+25/(x^2+ln(x)-x)^2*exp(-2*x-2)

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maxima [B]  time = 0.46, size = 84, normalized size = 3.00 \begin {gather*} \frac {{\left (3 \, x^{4} - 6 \, x^{3} + 3 \, x^{2} + 6 \, {\left (x^{2} - x\right )} \log \relax (x) + 3 \, \log \relax (x)^{2} + 25\right )} e^{\left (-2 \, x\right )}}{x^{4} e^{2} - 2 \, x^{3} e^{2} + x^{2} e^{2} + e^{2} \log \relax (x)^{2} + 2 \, {\left (x^{2} e^{2} - x e^{2}\right )} \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x*log(x)^3+(-18*x^3+18*x^2)*log(x)^2+(-18*x^5+36*x^4-18*x^3-50*x)*log(x)-6*x^7+18*x^6-18*x^5+6*x
^4-50*x^3-50*x^2+50*x-50)/(x*exp(x+1)^2*log(x)^3+(3*x^3-3*x^2)*exp(x+1)^2*log(x)^2+(3*x^5-6*x^4+3*x^3)*exp(x+1
)^2*log(x)+(x^7-3*x^6+3*x^5-x^4)*exp(x+1)^2),x, algorithm="maxima")

[Out]

(3*x^4 - 6*x^3 + 3*x^2 + 6*(x^2 - x)*log(x) + 3*log(x)^2 + 25)*e^(-2*x)/(x^4*e^2 - 2*x^3*e^2 + x^2*e^2 + e^2*l
og(x)^2 + 2*(x^2*e^2 - x*e^2)*log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {6\,x\,{\ln \relax (x)}^3-50\,x+\ln \relax (x)\,\left (18\,x^5-36\,x^4+18\,x^3+50\,x\right )-{\ln \relax (x)}^2\,\left (18\,x^2-18\,x^3\right )+50\,x^2+50\,x^3-6\,x^4+18\,x^5-18\,x^6+6\,x^7+50}{-x\,{\mathrm {e}}^{2\,x+2}\,{\ln \relax (x)}^3+{\mathrm {e}}^{2\,x+2}\,\left (3\,x^2-3\,x^3\right )\,{\ln \relax (x)}^2-{\mathrm {e}}^{2\,x+2}\,\left (3\,x^5-6\,x^4+3\,x^3\right )\,\ln \relax (x)+{\mathrm {e}}^{2\,x+2}\,\left (-x^7+3\,x^6-3\,x^5+x^4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x*log(x)^3 - 50*x + log(x)*(50*x + 18*x^3 - 36*x^4 + 18*x^5) - log(x)^2*(18*x^2 - 18*x^3) + 50*x^2 + 50
*x^3 - 6*x^4 + 18*x^5 - 18*x^6 + 6*x^7 + 50)/(exp(2*x + 2)*(x^4 - 3*x^5 + 3*x^6 - x^7) + exp(2*x + 2)*log(x)^2
*(3*x^2 - 3*x^3) - x*exp(2*x + 2)*log(x)^3 - exp(2*x + 2)*log(x)*(3*x^3 - 6*x^4 + 3*x^5)),x)

[Out]

int((6*x*log(x)^3 - 50*x + log(x)*(50*x + 18*x^3 - 36*x^4 + 18*x^5) - log(x)^2*(18*x^2 - 18*x^3) + 50*x^2 + 50
*x^3 - 6*x^4 + 18*x^5 - 18*x^6 + 6*x^7 + 50)/(exp(2*x + 2)*(x^4 - 3*x^5 + 3*x^6 - x^7) + exp(2*x + 2)*log(x)^2
*(3*x^2 - 3*x^3) - x*exp(2*x + 2)*log(x)^3 - exp(2*x + 2)*log(x)*(3*x^3 - 6*x^4 + 3*x^5)), x)

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sympy [B]  time = 0.46, size = 78, normalized size = 2.79 \begin {gather*} \frac {\left (3 x^{4} - 6 x^{3} + 6 x^{2} \log {\relax (x )} + 3 x^{2} - 6 x \log {\relax (x )} + 3 \log {\relax (x )}^{2} + 25\right ) e^{- 2 x - 2}}{x^{4} - 2 x^{3} + 2 x^{2} \log {\relax (x )} + x^{2} - 2 x \log {\relax (x )} + \log {\relax (x )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-6*x*ln(x)**3+(-18*x**3+18*x**2)*ln(x)**2+(-18*x**5+36*x**4-18*x**3-50*x)*ln(x)-6*x**7+18*x**6-18*x
**5+6*x**4-50*x**3-50*x**2+50*x-50)/(x*exp(x+1)**2*ln(x)**3+(3*x**3-3*x**2)*exp(x+1)**2*ln(x)**2+(3*x**5-6*x**
4+3*x**3)*exp(x+1)**2*ln(x)+(x**7-3*x**6+3*x**5-x**4)*exp(x+1)**2),x)

[Out]

(3*x**4 - 6*x**3 + 6*x**2*log(x) + 3*x**2 - 6*x*log(x) + 3*log(x)**2 + 25)*exp(-2*x - 2)/(x**4 - 2*x**3 + 2*x*
*2*log(x) + x**2 - 2*x*log(x) + log(x)**2)

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