3.46.36 \(\int \frac {e^{-2+\frac {2}{e^2 \log (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3})}} (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5)}{(-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6) \log ^2(\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3})} \, dx\)

Optimal. Leaf size=32 \[ e^{\frac {2}{e^2 \log \left (\frac {75}{x^2}-x+\frac {x^2}{-x+x^2}\right )}} \]

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Rubi [F]  time = 4.73, 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 {\exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right ) \left (-300+600 x-300 x^2-4 x^3+4 x^4-2 x^5\right )}{\left (-75 x+150 x^2-75 x^3+2 x^4-3 x^5+x^6\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-2 + 2/(E^2*Log[(-75 + 75*x + 2*x^3 - x^4)/(-x^2 + x^3)]))*(-300 + 600*x - 300*x^2 - 4*x^3 + 4*x^4 - 2
*x^5))/((-75*x + 150*x^2 - 75*x^3 + 2*x^4 - 3*x^5 + x^6)*Log[(-75 + 75*x + 2*x^3 - x^4)/(-x^2 + x^3)]^2),x]

[Out]

2*Defer[Int][E^(-2 + 2/(E^2*Log[(-75 + 75*x + 2*x^3 - x^4)/(-x^2 + x^3)]))/((-1 + x)*Log[-((75 - 75*x - 2*x^3
+ x^4)/((-1 + x)*x^2))]^2), x] + 4*Defer[Int][E^(-2 + 2/(E^2*Log[(-75 + 75*x + 2*x^3 - x^4)/(-x^2 + x^3)]))/(x
*Log[-((75 - 75*x - 2*x^3 + x^4)/((-1 + x)*x^2))]^2), x] + 150*Defer[Int][E^(-2 + 2/(E^2*Log[(-75 + 75*x + 2*x
^3 - x^4)/(-x^2 + x^3)]))/((75 - 75*x - 2*x^3 + x^4)*Log[-((75 - 75*x - 2*x^3 + x^4)/((-1 + x)*x^2))]^2), x] +
12*Defer[Int][(E^(-2 + 2/(E^2*Log[(-75 + 75*x + 2*x^3 - x^4)/(-x^2 + x^3)]))*x^2)/((75 - 75*x - 2*x^3 + x^4)*
Log[-((75 - 75*x - 2*x^3 + x^4)/((-1 + x)*x^2))]^2), x] - 8*Defer[Int][(E^(-2 + 2/(E^2*Log[(-75 + 75*x + 2*x^3
- x^4)/(-x^2 + x^3)]))*x^3)/((75 - 75*x - 2*x^3 + x^4)*Log[-((75 - 75*x - 2*x^3 + x^4)/((-1 + x)*x^2))]^2), x
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right ) \left (150-300 x+150 x^2+2 x^3-2 x^4+x^5\right )}{x \left (75-150 x+75 x^2-2 x^3+3 x^4-x^5\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{(-1+x) x^2}\right )} \, dx\\ &=2 \int \frac {\exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right ) \left (150-300 x+150 x^2+2 x^3-2 x^4+x^5\right )}{x \left (75-150 x+75 x^2-2 x^3+3 x^4-x^5\right ) \log ^2\left (\frac {-75+75 x+2 x^3-x^4}{(-1+x) x^2}\right )} \, dx\\ &=2 \int \left (\frac {\exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right )}{(-1+x) \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )}+\frac {2 \exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right )}{x \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )}+\frac {\exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right ) \left (75+6 x^2-4 x^3\right )}{\left (75-75 x-2 x^3+x^4\right ) \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )}\right ) \, dx\\ &=2 \int \frac {\exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right )}{(-1+x) \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )} \, dx+2 \int \frac {\exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right ) \left (75+6 x^2-4 x^3\right )}{\left (75-75 x-2 x^3+x^4\right ) \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )} \, dx+4 \int \frac {\exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right )}{x \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )} \, dx\\ &=2 \int \left (\frac {75 \exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right )}{\left (75-75 x-2 x^3+x^4\right ) \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )}+\frac {6 \exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right ) x^2}{\left (75-75 x-2 x^3+x^4\right ) \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )}-\frac {4 \exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right ) x^3}{\left (75-75 x-2 x^3+x^4\right ) \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )}\right ) \, dx+2 \int \frac {\exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right )}{(-1+x) \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )} \, dx+4 \int \frac {\exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right )}{x \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )} \, dx\\ &=2 \int \frac {\exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right )}{(-1+x) \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )} \, dx+4 \int \frac {\exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right )}{x \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )} \, dx-8 \int \frac {\exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right ) x^3}{\left (75-75 x-2 x^3+x^4\right ) \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )} \, dx+12 \int \frac {\exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right ) x^2}{\left (75-75 x-2 x^3+x^4\right ) \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )} \, dx+150 \int \frac {\exp \left (-2+\frac {2}{e^2 \log \left (\frac {-75+75 x+2 x^3-x^4}{-x^2+x^3}\right )}\right )}{\left (75-75 x-2 x^3+x^4\right ) \log ^2\left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 33, normalized size = 1.03 \begin {gather*} e^{\frac {2}{e^2 \log \left (-\frac {75-75 x-2 x^3+x^4}{(-1+x) x^2}\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-2 + 2/(E^2*Log[(-75 + 75*x + 2*x^3 - x^4)/(-x^2 + x^3)]))*(-300 + 600*x - 300*x^2 - 4*x^3 + 4*x
^4 - 2*x^5))/((-75*x + 150*x^2 - 75*x^3 + 2*x^4 - 3*x^5 + x^6)*Log[(-75 + 75*x + 2*x^3 - x^4)/(-x^2 + x^3)]^2)
,x]

[Out]

E^(2/(E^2*Log[-((75 - 75*x - 2*x^3 + x^4)/((-1 + x)*x^2))]))

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fricas [B]  time = 0.58, size = 68, normalized size = 2.12 \begin {gather*} e^{\left (-\frac {2 \, {\left (e^{2} \log \left (-\frac {x^{4} - 2 \, x^{3} - 75 \, x + 75}{x^{3} - x^{2}}\right ) - 1\right )} e^{\left (-2\right )}}{\log \left (-\frac {x^{4} - 2 \, x^{3} - 75 \, x + 75}{x^{3} - x^{2}}\right )} + 2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5+4*x^4-4*x^3-300*x^2+600*x-300)*exp(2/exp(2)/log((-x^4+2*x^3+75*x-75)/(x^3-x^2)))/(x^6-3*x^5+
2*x^4-75*x^3+150*x^2-75*x)/exp(2)/log((-x^4+2*x^3+75*x-75)/(x^3-x^2))^2,x, algorithm="fricas")

[Out]

e^(-2*(e^2*log(-(x^4 - 2*x^3 - 75*x + 75)/(x^3 - x^2)) - 1)*e^(-2)/log(-(x^4 - 2*x^3 - 75*x + 75)/(x^3 - x^2))
+ 2)

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giac [B]  time = 0.39, size = 68, normalized size = 2.12 \begin {gather*} e^{\left (\frac {2 \, e^{\left (-2\right )}}{\log \left (-\frac {x^{4}}{x^{3} - x^{2}} + \frac {2 \, x^{3}}{x^{3} - x^{2}} + \frac {75 \, x}{x^{3} - x^{2}} - \frac {75}{x^{3} - x^{2}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5+4*x^4-4*x^3-300*x^2+600*x-300)*exp(2/exp(2)/log((-x^4+2*x^3+75*x-75)/(x^3-x^2)))/(x^6-3*x^5+
2*x^4-75*x^3+150*x^2-75*x)/exp(2)/log((-x^4+2*x^3+75*x-75)/(x^3-x^2))^2,x, algorithm="giac")

[Out]

e^(2*e^(-2)/log(-x^4/(x^3 - x^2) + 2*x^3/(x^3 - x^2) + 75*x/(x^3 - x^2) - 75/(x^3 - x^2)))

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maple [A]  time = 0.04, size = 36, normalized size = 1.12

method	result	size
risch	\({\mathrm e}^{\frac {2 \,{\mathrm e}^{-2}}{\ln \left (\frac {-x^{4}+2 x^{3}+75 x -75}{x^{3}-x^{2}}\right )}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^5+4*x^4-4*x^3-300*x^2+600*x-300)*exp(2/exp(2)/ln((-x^4+2*x^3+75*x-75)/(x^3-x^2)))/(x^6-3*x^5+2*x^4-7
5*x^3+150*x^2-75*x)/exp(2)/ln((-x^4+2*x^3+75*x-75)/(x^3-x^2))^2,x,method=_RETURNVERBOSE)

[Out]

exp(2*exp(-2)/ln((-x^4+2*x^3+75*x-75)/(x^3-x^2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -2 \, \int \frac {{\left (x^{5} - 2 \, x^{4} + 2 \, x^{3} + 150 \, x^{2} - 300 \, x + 150\right )} e^{\left (\frac {2 \, e^{\left (-2\right )}}{\log \left (-\frac {x^{4} - 2 \, x^{3} - 75 \, x + 75}{x^{3} - x^{2}}\right )} - 2\right )}}{{\left (x^{6} - 3 \, x^{5} + 2 \, x^{4} - 75 \, x^{3} + 150 \, x^{2} - 75 \, x\right )} \log \left (-\frac {x^{4} - 2 \, x^{3} - 75 \, x + 75}{x^{3} - x^{2}}\right )^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5+4*x^4-4*x^3-300*x^2+600*x-300)*exp(2/exp(2)/log((-x^4+2*x^3+75*x-75)/(x^3-x^2)))/(x^6-3*x^5+
2*x^4-75*x^3+150*x^2-75*x)/exp(2)/log((-x^4+2*x^3+75*x-75)/(x^3-x^2))^2,x, algorithm="maxima")

[Out]

-2*integrate((x^5 - 2*x^4 + 2*x^3 + 150*x^2 - 300*x + 150)*e^(2*e^(-2)/log(-(x^4 - 2*x^3 - 75*x + 75)/(x^3 - x
^2)) - 2)/((x^6 - 3*x^5 + 2*x^4 - 75*x^3 + 150*x^2 - 75*x)*log(-(x^4 - 2*x^3 - 75*x + 75)/(x^3 - x^2))^2), x)

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mupad [B]  time = 4.12, size = 36, normalized size = 1.12 \begin {gather*} {\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{-2}}{\ln \left (-\frac {-x^4+2\,x^3+75\,x-75}{x^2-x^3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-2)*exp((2*exp(-2))/log(-(75*x + 2*x^3 - x^4 - 75)/(x^2 - x^3)))*(300*x^2 - 600*x + 4*x^3 - 4*x^4 + 2
*x^5 + 300))/(log(-(75*x + 2*x^3 - x^4 - 75)/(x^2 - x^3))^2*(75*x - 150*x^2 + 75*x^3 - 2*x^4 + 3*x^5 - x^6)),x
)

[Out]

exp((2*exp(-2))/log(-(75*x + 2*x^3 - x^4 - 75)/(x^2 - x^3)))

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sympy [A]  time = 0.80, size = 27, normalized size = 0.84 \begin {gather*} e^{\frac {2}{e^{2} \log {\left (\frac {- x^{4} + 2 x^{3} + 75 x - 75}{x^{3} - x^{2}} \right )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**5+4*x**4-4*x**3-300*x**2+600*x-300)*exp(2/exp(2)/ln((-x**4+2*x**3+75*x-75)/(x**3-x**2)))/(x**
6-3*x**5+2*x**4-75*x**3+150*x**2-75*x)/exp(2)/ln((-x**4+2*x**3+75*x-75)/(x**3-x**2))**2,x)

[Out]

exp(2*exp(-2)/log((-x**4 + 2*x**3 + 75*x - 75)/(x**3 - x**2)))

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