3.5.64 \(\int \frac {34 x^2+15 x^3+e^{2 x} (-36+51 x+18 x^2)+e^x (18 x-66 x^2-18 x^3)+(-9 x^2-6 x^3+e^{2 x} (9-18 x-6 x^2)+e^x (24 x^2+6 x^3)) \log (x)}{-43 x^3-9 x^4+e^{2 x} (-27 x-9 x^2)+e^x (54 x^2+18 x^3)+(9 x^3+3 x^4+e^{2 x} (9 x+3 x^2)+e^x (-18 x^2-6 x^3)) \log (x)} \, dx\)

Optimal. Leaf size=30 \[ \log \left (\frac {1}{x+\frac {3 \left (e^x-x\right )^2 (3+x) (3-\log (x))}{16 x}}\right ) \]

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Rubi [F]  time = 180.00, 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*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

Int[(34*x^2 + 15*x^3 + E^(2*x)*(-36 + 51*x + 18*x^2) + E^x*(18*x - 66*x^2 - 18*x^3) + (-9*x^2 - 6*x^3 + E^(2*x
)*(9 - 18*x - 6*x^2) + E^x*(24*x^2 + 6*x^3))*Log[x])/(-43*x^3 - 9*x^4 + E^(2*x)*(-27*x - 9*x^2) + E^x*(54*x^2
+ 18*x^3) + (9*x^3 + 3*x^4 + E^(2*x)*(9*x + 3*x^2) + E^x*(-18*x^2 - 6*x^3))*Log[x]),x]

[Out]

$Aborted

Rubi steps

Aborted

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Mathematica [B]  time = 0.25, size = 97, normalized size = 3.23 \begin {gather*} \log (x)-\log \left (-27 e^{2 x}+54 e^x x-9 e^{2 x} x-43 x^2+18 e^x x^2-9 x^3+9 e^{2 x} \log (x)-18 e^x x \log (x)+3 e^{2 x} x \log (x)+9 x^2 \log (x)-6 e^x x^2 \log (x)+3 x^3 \log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(34*x^2 + 15*x^3 + E^(2*x)*(-36 + 51*x + 18*x^2) + E^x*(18*x - 66*x^2 - 18*x^3) + (-9*x^2 - 6*x^3 +
E^(2*x)*(9 - 18*x - 6*x^2) + E^x*(24*x^2 + 6*x^3))*Log[x])/(-43*x^3 - 9*x^4 + E^(2*x)*(-27*x - 9*x^2) + E^x*(5
4*x^2 + 18*x^3) + (9*x^3 + 3*x^4 + E^(2*x)*(9*x + 3*x^2) + E^x*(-18*x^2 - 6*x^3))*Log[x]),x]

[Out]

Log[x] - Log[-27*E^(2*x) + 54*E^x*x - 9*E^(2*x)*x - 43*x^2 + 18*E^x*x^2 - 9*x^3 + 9*E^(2*x)*Log[x] - 18*E^x*x*
Log[x] + 3*E^(2*x)*x*Log[x] + 9*x^2*Log[x] - 6*E^x*x^2*Log[x] + 3*x^3*Log[x]]

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fricas [B]  time = 0.65, size = 116, normalized size = 3.87 \begin {gather*} -\log \left (x + 3\right ) + \log \relax (x) - 2 \, \log \left (-x + e^{x}\right ) - \log \left (-\frac {9 \, x^{3} + 43 \, x^{2} + 9 \, {\left (x + 3\right )} e^{\left (2 \, x\right )} - 18 \, {\left (x^{2} + 3 \, x\right )} e^{x} - 3 \, {\left (x^{3} + 3 \, x^{2} + {\left (x + 3\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{2} + 3 \, x\right )} e^{x}\right )} \log \relax (x)}{x^{3} + 3 \, x^{2} + {\left (x + 3\right )} e^{\left (2 \, x\right )} - 2 \, {\left (x^{2} + 3 \, x\right )} e^{x}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x^2-18*x+9)*exp(x)^2+(6*x^3+24*x^2)*exp(x)-6*x^3-9*x^2)*log(x)+(18*x^2+51*x-36)*exp(x)^2+(-18*
x^3-66*x^2+18*x)*exp(x)+15*x^3+34*x^2)/(((3*x^2+9*x)*exp(x)^2+(-6*x^3-18*x^2)*exp(x)+3*x^4+9*x^3)*log(x)+(-9*x
^2-27*x)*exp(x)^2+(18*x^3+54*x^2)*exp(x)-9*x^4-43*x^3),x, algorithm="fricas")

[Out]

-log(x + 3) + log(x) - 2*log(-x + e^x) - log(-(9*x^3 + 43*x^2 + 9*(x + 3)*e^(2*x) - 18*(x^2 + 3*x)*e^x - 3*(x^
3 + 3*x^2 + (x + 3)*e^(2*x) - 2*(x^2 + 3*x)*e^x)*log(x))/(x^3 + 3*x^2 + (x + 3)*e^(2*x) - 2*(x^2 + 3*x)*e^x))

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giac [B]  time = 1.28, size = 89, normalized size = 2.97 \begin {gather*} -\log \left (3 \, x^{3} \log \relax (x) - 6 \, x^{2} e^{x} \log \relax (x) - 9 \, x^{3} + 18 \, x^{2} e^{x} + 9 \, x^{2} \log \relax (x) + 3 \, x e^{\left (2 \, x\right )} \log \relax (x) - 18 \, x e^{x} \log \relax (x) - 43 \, x^{2} - 9 \, x e^{\left (2 \, x\right )} + 54 \, x e^{x} + 9 \, e^{\left (2 \, x\right )} \log \relax (x) - 27 \, e^{\left (2 \, x\right )}\right ) + \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x^2-18*x+9)*exp(x)^2+(6*x^3+24*x^2)*exp(x)-6*x^3-9*x^2)*log(x)+(18*x^2+51*x-36)*exp(x)^2+(-18*
x^3-66*x^2+18*x)*exp(x)+15*x^3+34*x^2)/(((3*x^2+9*x)*exp(x)^2+(-6*x^3-18*x^2)*exp(x)+3*x^4+9*x^3)*log(x)+(-9*x
^2-27*x)*exp(x)^2+(18*x^3+54*x^2)*exp(x)-9*x^4-43*x^3),x, algorithm="giac")

[Out]

-log(3*x^3*log(x) - 6*x^2*e^x*log(x) - 9*x^3 + 18*x^2*e^x + 9*x^2*log(x) + 3*x*e^(2*x)*log(x) - 18*x*e^x*log(x
) - 43*x^2 - 9*x*e^(2*x) + 54*x*e^x + 9*e^(2*x)*log(x) - 27*e^(2*x)) + log(x)

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maple [B]  time = 0.08, size = 98, normalized size = 3.27

method	result	size
risch	\(\ln \relax (x )-\ln \left (3+x \right )-2 \ln \left ({\mathrm e}^{x}-x \right )-\ln \left (\ln \relax (x )-\frac {9 x^{3}-18 \,{\mathrm e}^{x} x^{2}+9 x \,{\mathrm e}^{2 x}+43 x^{2}-54 \,{\mathrm e}^{x} x +27 \,{\mathrm e}^{2 x}}{3 \left (x^{3}-2 \,{\mathrm e}^{x} x^{2}+x \,{\mathrm e}^{2 x}+3 x^{2}-6 \,{\mathrm e}^{x} x +3 \,{\mathrm e}^{2 x}\right )}\right )\)	\(98\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-6*x^2-18*x+9)*exp(x)^2+(6*x^3+24*x^2)*exp(x)-6*x^3-9*x^2)*ln(x)+(18*x^2+51*x-36)*exp(x)^2+(-18*x^3-66*
x^2+18*x)*exp(x)+15*x^3+34*x^2)/(((3*x^2+9*x)*exp(x)^2+(-6*x^3-18*x^2)*exp(x)+3*x^4+9*x^3)*ln(x)+(-9*x^2-27*x)
*exp(x)^2+(18*x^3+54*x^2)*exp(x)-9*x^4-43*x^3),x,method=_RETURNVERBOSE)

[Out]

ln(x)-ln(3+x)-2*ln(exp(x)-x)-ln(ln(x)-1/3*(9*x^3-18*exp(x)*x^2+9*x*exp(2*x)+43*x^2-54*exp(x)*x+27*exp(2*x))/(x
^3-2*exp(x)*x^2+x*exp(2*x)+3*x^2-6*exp(x)*x+3*exp(2*x)))

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maxima [B]  time = 0.48, size = 99, normalized size = 3.30 \begin {gather*} -\log \left (x + 3\right ) + \log \relax (x) - \log \left (-\frac {9 \, x^{3} + 43 \, x^{2} - 3 \, {\left ({\left (x + 3\right )} \log \relax (x) - 3 \, x - 9\right )} e^{\left (2 \, x\right )} - 6 \, {\left (3 \, x^{2} - {\left (x^{2} + 3 \, x\right )} \log \relax (x) + 9 \, x\right )} e^{x} - 3 \, {\left (x^{3} + 3 \, x^{2}\right )} \log \relax (x)}{3 \, {\left ({\left (x + 3\right )} \log \relax (x) - 3 \, x - 9\right )}}\right ) - \log \left (\log \relax (x) - 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x^2-18*x+9)*exp(x)^2+(6*x^3+24*x^2)*exp(x)-6*x^3-9*x^2)*log(x)+(18*x^2+51*x-36)*exp(x)^2+(-18*
x^3-66*x^2+18*x)*exp(x)+15*x^3+34*x^2)/(((3*x^2+9*x)*exp(x)^2+(-6*x^3-18*x^2)*exp(x)+3*x^4+9*x^3)*log(x)+(-9*x
^2-27*x)*exp(x)^2+(18*x^3+54*x^2)*exp(x)-9*x^4-43*x^3),x, algorithm="maxima")

[Out]

-log(x + 3) + log(x) - log(-1/3*(9*x^3 + 43*x^2 - 3*((x + 3)*log(x) - 3*x - 9)*e^(2*x) - 6*(3*x^2 - (x^2 + 3*x
)*log(x) + 9*x)*e^x - 3*(x^3 + 3*x^2)*log(x))/((x + 3)*log(x) - 3*x - 9)) - log(log(x) - 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{2\,x}\,\left (18\,x^2+51\,x-36\right )-\ln \relax (x)\,\left ({\mathrm {e}}^{2\,x}\,\left (6\,x^2+18\,x-9\right )-{\mathrm {e}}^x\,\left (6\,x^3+24\,x^2\right )+9\,x^2+6\,x^3\right )+34\,x^2+15\,x^3-{\mathrm {e}}^x\,\left (18\,x^3+66\,x^2-18\,x\right )}{{\mathrm {e}}^{2\,x}\,\left (9\,x^2+27\,x\right )-{\mathrm {e}}^x\,\left (18\,x^3+54\,x^2\right )-\ln \relax (x)\,\left ({\mathrm {e}}^{2\,x}\,\left (3\,x^2+9\,x\right )-{\mathrm {e}}^x\,\left (6\,x^3+18\,x^2\right )+9\,x^3+3\,x^4\right )+43\,x^3+9\,x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x)*(51*x + 18*x^2 - 36) - log(x)*(exp(2*x)*(18*x + 6*x^2 - 9) - exp(x)*(24*x^2 + 6*x^3) + 9*x^2 +
6*x^3) + 34*x^2 + 15*x^3 - exp(x)*(66*x^2 - 18*x + 18*x^3))/(exp(2*x)*(27*x + 9*x^2) - exp(x)*(54*x^2 + 18*x^3
) - log(x)*(exp(2*x)*(9*x + 3*x^2) - exp(x)*(18*x^2 + 6*x^3) + 9*x^3 + 3*x^4) + 43*x^3 + 9*x^4),x)

[Out]

int(-(exp(2*x)*(51*x + 18*x^2 - 36) - log(x)*(exp(2*x)*(18*x + 6*x^2 - 9) - exp(x)*(24*x^2 + 6*x^3) + 9*x^2 +
6*x^3) + 34*x^2 + 15*x^3 - exp(x)*(66*x^2 - 18*x + 18*x^3))/(exp(2*x)*(27*x + 9*x^2) - exp(x)*(54*x^2 + 18*x^3
) - log(x)*(exp(2*x)*(9*x + 3*x^2) - exp(x)*(18*x^2 + 6*x^3) + 9*x^3 + 3*x^4) + 43*x^3 + 9*x^4), x)

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sympy [B]  time = 5.92, size = 71, normalized size = 2.37 \begin {gather*} \log {\relax (x )} - \log {\left (x + 3 \right )} - \log {\left (\log {\relax (x )} - 3 \right )} - \log {\left (- 2 x e^{x} + e^{2 x} + \frac {3 x^{3} \log {\relax (x )} - 9 x^{3} + 9 x^{2} \log {\relax (x )} - 43 x^{2}}{3 x \log {\relax (x )} - 9 x + 9 \log {\relax (x )} - 27} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-6*x**2-18*x+9)*exp(x)**2+(6*x**3+24*x**2)*exp(x)-6*x**3-9*x**2)*ln(x)+(18*x**2+51*x-36)*exp(x)**
2+(-18*x**3-66*x**2+18*x)*exp(x)+15*x**3+34*x**2)/(((3*x**2+9*x)*exp(x)**2+(-6*x**3-18*x**2)*exp(x)+3*x**4+9*x
**3)*ln(x)+(-9*x**2-27*x)*exp(x)**2+(18*x**3+54*x**2)*exp(x)-9*x**4-43*x**3),x)

[Out]

log(x) - log(x + 3) - log(log(x) - 3) - log(-2*x*exp(x) + exp(2*x) + (3*x**3*log(x) - 9*x**3 + 9*x**2*log(x) -
43*x**2)/(3*x*log(x) - 9*x + 9*log(x) - 27))

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