∫ −10e5− 10x__ e5−2x +2e10x−8e5x2+8x3+e− 5x__ e5−2x (2e10−18e5x+8x2) ___________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 144x2−48x4+4x6+e− 20x__ e5−2x (e10−4e5x+4x2)+e− 15x__ e5−2x (4e10x−16e5x2+16x3)+e10(36−12x2+x4)+e5(−144x+48x3−4x5)+e− 10x__ e5−2x (−48x2+24x4+e10(−12+6x2)+e5(48x−24x3))+e− 5x__ e5−2x (−96x3+16x5+e10(−24x+4x3)+e5(96x2−16x4)) dx

3.104.11 $\int \frac {-10 e^{5-\frac {10 x}{e^5-2 x}}+2 e^{10} x-8 e^5 x^2+8 x^3+e^{-\frac {5 x}{e^5-2 x}} (2 e^{10}-18 e^5 x+8 x^2)}{144 x^2-48 x^4+4 x^6+e^{-\frac {20 x}{e^5-2 x}} (e^{10}-4 e^5 x+4 x^2)+e^{-\frac {15 x}{e^5-2 x}} (4 e^{10} x-16 e^5 x^2+16 x^3)+e^{10} (36-12 x^2+x^4)+e^5 (-144 x+48 x^3-4 x^5)+e^{-\frac {10 x}{e^5-2 x}} (-48 x^2+24 x^4+e^{10} (-12+6 x^2)+e^5 (48 x-24 x^3))+e^{-\frac {5 x}{e^5-2 x}} (-96 x^3+16 x^5+e^{10} (-24 x+4 x^3)+e^5 (96 x^2-16 x^4))} \, dx$

Optimal. Leaf size=28 \[ 1+\frac {1}{6-\left (e^{\frac {5 x}{-e^5+2 x}}+x\right )^2} \]

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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[(-10*E^(5 - (10*x)/(E^5 - 2*x)) + 2*E^10*x - 8*E^5*x^2 + 8*x^3 + (2*E^10 - 18*E^5*x + 8*x^2)/E^((5*x)/(E^5

- 2*x)))/(144*x^2 - 48*x^4 + 4*x^6 + (E^10 - 4*E^5*x + 4*x^2)/E^((20*x)/(E^5 - 2*x)) + (4*E^10*x - 16*E^5*x^2

+ 16*x^3)/E^((15*x)/(E^5 - 2*x)) + E^10*(36 - 12*x^2 + x^4) + E^5*(-144*x + 48*x^3 - 4*x^5) + (-48*x^2 + 24*x

^4 + E^10*(-12 + 6*x^2) + E^5*(48*x - 24*x^3))/E^((10*x)/(E^5 - 2*x)) + (-96*x^3 + 16*x^5 + E^10*(-24*x + 4*x^

3) + E^5*(96*x^2 - 16*x^4))/E^((5*x)/(E^5 - 2*x))),x]

[Out]

$Aborted

Rubi steps

Aborted

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Mathematica [A] time = 0.31, size = 40, normalized size = 1.43 \begin {gather*} -\frac {1}{-6+e^{-\frac {10 x}{e^5-2 x}}+2 e^{-\frac {5 x}{e^5-2 x}} x+x^2} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

Integrate[(-10*E^(5 - (10*x)/(E^5 - 2*x)) + 2*E^10*x - 8*E^5*x^2 + 8*x^3 + (2*E^10 - 18*E^5*x + 8*x^2)/E^((5*x

)/(E^5 - 2*x)))/(144*x^2 - 48*x^4 + 4*x^6 + (E^10 - 4*E^5*x + 4*x^2)/E^((20*x)/(E^5 - 2*x)) + (4*E^10*x - 16*E

^5*x^2 + 16*x^3)/E^((15*x)/(E^5 - 2*x)) + E^10*(36 - 12*x^2 + x^4) + E^5*(-144*x + 48*x^3 - 4*x^5) + (-48*x^2

+ 24*x^4 + E^10*(-12 + 6*x^2) + E^5*(48*x - 24*x^3))/E^((10*x)/(E^5 - 2*x)) + (-96*x^3 + 16*x^5 + E^10*(-24*x

+ 4*x^3) + E^5*(96*x^2 - 16*x^4))/E^((5*x)/(E^5 - 2*x))),x]

[Out]

-(-6 + E^((-10*x)/(E^5 - 2*x)) + (2*x)/E^((5*x)/(E^5 - 2*x)) + x^2)^(-1)

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fricas [A] time = 1.11, size = 40, normalized size = 1.43 \begin {gather*} -\frac {1}{x^{2} + 2 \, x e^{\left (\frac {5 \, x}{2 \, x - e^{5}}\right )} + e^{\left (\frac {10 \, x}{2 \, x - e^{5}}\right )} - 6} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*exp(5)*exp(-5*x/(exp(5)-2*x))^2+(2*exp(5)^2-18*x*exp(5)+8*x^2)*exp(-5*x/(exp(5)-2*x))+2*x*exp(5

)^2-8*x^2*exp(5)+8*x^3)/((exp(5)^2-4*x*exp(5)+4*x^2)*exp(-5*x/(exp(5)-2*x))^4+(4*x*exp(5)^2-16*x^2*exp(5)+16*x

^3)*exp(-5*x/(exp(5)-2*x))^3+((6*x^2-12)*exp(5)^2+(-24*x^3+48*x)*exp(5)+24*x^4-48*x^2)*exp(-5*x/(exp(5)-2*x))^

2+((4*x^3-24*x)*exp(5)^2+(-16*x^4+96*x^2)*exp(5)+16*x^5-96*x^3)*exp(-5*x/(exp(5)-2*x))+(x^4-12*x^2+36)*exp(5)^

2+(-4*x^5+48*x^3-144*x)*exp(5)+4*x^6-48*x^4+144*x^2),x, algorithm="fricas")

[Out]

-1/(x^2 + 2*x*e^(5*x/(2*x - e^5)) + e^(10*x/(2*x - e^5)) - 6)

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giac [B] time = 32.17, size = 219, normalized size = 7.82 \begin {gather*} \frac {2 \, {\left (\frac {4 \, x e^{5}}{2 \, x - e^{5}} - \frac {4 \, x^{2} e^{5}}{{\left (2 \, x - e^{5}\right )}^{2}} - e^{5}\right )} e^{\left (-5\right )}}{\frac {x^{2} e^{10}}{{\left (2 \, x - e^{5}\right )}^{2}} - \frac {4 \, x e^{\left (\frac {10 \, x}{2 \, x - e^{5}}\right )}}{2 \, x - e^{5}} + \frac {4 \, x^{2} e^{\left (\frac {10 \, x}{2 \, x - e^{5}}\right )}}{{\left (2 \, x - e^{5}\right )}^{2}} - \frac {2 \, x e^{\left (\frac {5 \, x}{2 \, x - e^{5}} + 5\right )}}{2 \, x - e^{5}} + \frac {4 \, x^{2} e^{\left (\frac {5 \, x}{2 \, x - e^{5}} + 5\right )}}{{\left (2 \, x - e^{5}\right )}^{2}} + \frac {24 \, x}{2 \, x - e^{5}} - \frac {24 \, x^{2}}{{\left (2 \, x - e^{5}\right )}^{2}} + e^{\left (\frac {10 \, x}{2 \, x - e^{5}}\right )} - 6} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*exp(5)*exp(-5*x/(exp(5)-2*x))^2+(2*exp(5)^2-18*x*exp(5)+8*x^2)*exp(-5*x/(exp(5)-2*x))+2*x*exp(5

)^2-8*x^2*exp(5)+8*x^3)/((exp(5)^2-4*x*exp(5)+4*x^2)*exp(-5*x/(exp(5)-2*x))^4+(4*x*exp(5)^2-16*x^2*exp(5)+16*x

^3)*exp(-5*x/(exp(5)-2*x))^3+((6*x^2-12)*exp(5)^2+(-24*x^3+48*x)*exp(5)+24*x^4-48*x^2)*exp(-5*x/(exp(5)-2*x))^

2+((4*x^3-24*x)*exp(5)^2+(-16*x^4+96*x^2)*exp(5)+16*x^5-96*x^3)*exp(-5*x/(exp(5)-2*x))+(x^4-12*x^2+36)*exp(5)^

2+(-4*x^5+48*x^3-144*x)*exp(5)+4*x^6-48*x^4+144*x^2),x, algorithm="giac")

[Out]

2*(4*x*e^5/(2*x - e^5) - 4*x^2*e^5/(2*x - e^5)^2 - e^5)*e^(-5)/(x^2*e^10/(2*x - e^5)^2 - 4*x*e^(10*x/(2*x - e^

5))/(2*x - e^5) + 4*x^2*e^(10*x/(2*x - e^5))/(2*x - e^5)^2 - 2*x*e^(5*x/(2*x - e^5) + 5)/(2*x - e^5) + 4*x^2*e

^(5*x/(2*x - e^5) + 5)/(2*x - e^5)^2 + 24*x/(2*x - e^5) - 24*x^2/(2*x - e^5)^2 + e^(10*x/(2*x - e^5)) - 6)

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maple [A] time = 0.99, size = 37, normalized size = 1.32


method	result	size

risch	$-\frac {1}{{\mathrm e}^{-\frac {10 x}{{\mathrm e}^{5}-2 x}}+2 \,{\mathrm e}^{-\frac {5 x}{{\mathrm e}^{5}-2 x}} x +x^{2}-6}$	$37$

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((-10*exp(5)*exp(-5*x/(exp(5)-2*x))^2+(2*exp(5)^2-18*x*exp(5)+8*x^2)*exp(-5*x/(exp(5)-2*x))+2*x*exp(5)^2-8*

x^2*exp(5)+8*x^3)/((exp(5)^2-4*x*exp(5)+4*x^2)*exp(-5*x/(exp(5)-2*x))^4+(4*x*exp(5)^2-16*x^2*exp(5)+16*x^3)*ex

p(-5*x/(exp(5)-2*x))^3+((6*x^2-12)*exp(5)^2+(-24*x^3+48*x)*exp(5)+24*x^4-48*x^2)*exp(-5*x/(exp(5)-2*x))^2+((4*

x^3-24*x)*exp(5)^2+(-16*x^4+96*x^2)*exp(5)+16*x^5-96*x^3)*exp(-5*x/(exp(5)-2*x))+(x^4-12*x^2+36)*exp(5)^2+(-4*

x^5+48*x^3-144*x)*exp(5)+4*x^6-48*x^4+144*x^2),x,method=_RETURNVERBOSE)

[Out]

-1/(exp(-10*x/(exp(5)-2*x))+2*exp(-5*x/(exp(5)-2*x))*x+x^2-6)

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maxima [A] time = 0.47, size = 46, normalized size = 1.64 \begin {gather*} -\frac {1}{x^{2} + 2 \, x e^{\left (\frac {5 \, e^{5}}{2 \, {\left (2 \, x - e^{5}\right )}} + \frac {5}{2}\right )} + e^{\left (\frac {5 \, e^{5}}{2 \, x - e^{5}} + 5\right )} - 6} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*exp(5)*exp(-5*x/(exp(5)-2*x))^2+(2*exp(5)^2-18*x*exp(5)+8*x^2)*exp(-5*x/(exp(5)-2*x))+2*x*exp(5

)^2-8*x^2*exp(5)+8*x^3)/((exp(5)^2-4*x*exp(5)+4*x^2)*exp(-5*x/(exp(5)-2*x))^4+(4*x*exp(5)^2-16*x^2*exp(5)+16*x

^3)*exp(-5*x/(exp(5)-2*x))^3+((6*x^2-12)*exp(5)^2+(-24*x^3+48*x)*exp(5)+24*x^4-48*x^2)*exp(-5*x/(exp(5)-2*x))^

2+((4*x^3-24*x)*exp(5)^2+(-16*x^4+96*x^2)*exp(5)+16*x^5-96*x^3)*exp(-5*x/(exp(5)-2*x))+(x^4-12*x^2+36)*exp(5)^

2+(-4*x^5+48*x^3-144*x)*exp(5)+4*x^6-48*x^4+144*x^2),x, algorithm="maxima")

[Out]

-1/(x^2 + 2*x*e^(5/2*e^5/(2*x - e^5) + 5/2) + e^(5*e^5/(2*x - e^5) + 5) - 6)

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mupad [B] time = 9.27, size = 40, normalized size = 1.43 \begin {gather*} -\frac {1}{{\mathrm {e}}^{\frac {10\,x}{2\,x-{\mathrm {e}}^5}}+2\,x\,{\mathrm {e}}^{\frac {5\,x}{2\,x-{\mathrm {e}}^5}}+x^2-6} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*x*exp(10) - 10*exp(5)*exp((10*x)/(2*x - exp(5))) + exp((5*x)/(2*x - exp(5)))*(2*exp(10) - 18*x*exp(5) +

8*x^2) - 8*x^2*exp(5) + 8*x^3)/(exp((20*x)/(2*x - exp(5)))*(exp(10) - 4*x*exp(5) + 4*x^2) + exp(10)*(x^4 - 12

*x^2 + 36) + exp((10*x)/(2*x - exp(5)))*(exp(5)*(48*x - 24*x^3) + exp(10)*(6*x^2 - 12) - 48*x^2 + 24*x^4) + ex

p((15*x)/(2*x - exp(5)))*(4*x*exp(10) - 16*x^2*exp(5) + 16*x^3) - exp(5)*(144*x - 48*x^3 + 4*x^5) - exp((5*x)/

(2*x - exp(5)))*(exp(10)*(24*x - 4*x^3) - exp(5)*(96*x^2 - 16*x^4) + 96*x^3 - 16*x^5) + 144*x^2 - 48*x^4 + 4*x

^6),x)

[Out]

-1/(exp((10*x)/(2*x - exp(5))) + 2*x*exp((5*x)/(2*x - exp(5))) + x^2 - 6)

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sympy [A] time = 0.42, size = 36, normalized size = 1.29 \begin {gather*} - \frac {1}{x^{2} + 2 x e^{- \frac {5 x}{- 2 x + e^{5}}} - 6 + e^{- \frac {10 x}{- 2 x + e^{5}}}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((-10*exp(5)*exp(-5*x/(exp(5)-2*x))**2+(2*exp(5)**2-18*x*exp(5)+8*x**2)*exp(-5*x/(exp(5)-2*x))+2*x*ex

p(5)**2-8*x**2*exp(5)+8*x**3)/((exp(5)**2-4*x*exp(5)+4*x**2)*exp(-5*x/(exp(5)-2*x))**4+(4*x*exp(5)**2-16*x**2*

exp(5)+16*x**3)*exp(-5*x/(exp(5)-2*x))**3+((6*x**2-12)*exp(5)**2+(-24*x**3+48*x)*exp(5)+24*x**4-48*x**2)*exp(-

5*x/(exp(5)-2*x))**2+((4*x**3-24*x)*exp(5)**2+(-16*x**4+96*x**2)*exp(5)+16*x**5-96*x**3)*exp(-5*x/(exp(5)-2*x)

)+(x**4-12*x**2+36)*exp(5)**2+(-4*x**5+48*x**3-144*x)*exp(5)+4*x**6-48*x**4+144*x**2),x)

[Out]

-1/(x**2 + 2*x*exp(-5*x/(-2*x + exp(5))) - 6 + exp(-10*x/(-2*x + exp(5))))

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