3.4.53 \(\int \frac {-120+120 x+e^{e^{2 x}} (300-300 x+e^{2 x} (300-600 x+300 x^2))}{4-20 e^{e^{2 x}}+25 e^{2 e^{2 x}}} \, dx\)

Optimal. Leaf size=24 \[ \frac {6 (1-x)^2}{\frac {2}{5}-e^{e^{2 x}}} \]

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Rubi [F]  time = 1.62, 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 {-120+120 x+e^{e^{2 x}} \left (300-300 x+e^{2 x} \left (300-600 x+300 x^2\right )\right )}{4-20 e^{e^{2 x}}+25 e^{2 e^{2 x}}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-120 + 120*x + E^E^(2*x)*(300 - 300*x + E^(2*x)*(300 - 600*x + 300*x^2)))/(4 - 20*E^E^(2*x) + 25*E^(2*E^(
2*x))),x]

[Out]

30/(2 - 5*E^E^(2*x)) + 30/(E^(2*x)*(2 - 5*E^E^(2*x))) + 120*Defer[Int][x/(-2 + 5*E^E^(2*x))^2, x] - 300*Defer[
Int][(E^E^(2*x)*x)/(-2 + 5*E^E^(2*x))^2, x] - 600*Defer[Int][(E^(E^(2*x) + 2*x)*x)/(-2 + 5*E^E^(2*x))^2, x] +
300*Defer[Int][(E^(E^(2*x) + 2*x)*x^2)/(-2 + 5*E^E^(2*x))^2, x] - 30*Defer[Subst][Defer[Int][1/((-2 + 5*E^x)*x
^2), x], x, E^(2*x)] - 60*Defer[Subst][Defer[Int][1/((-2 + 5*E^x)^2*x), x], x, E^(2*x)]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-120+120 x+e^{e^{2 x}} \left (300-300 x+e^{2 x} \left (300-600 x+300 x^2\right )\right )}{\left (2-5 e^{e^{2 x}}\right )^2} \, dx\\ &=\int \left (-\frac {120}{\left (-2+5 e^{e^{2 x}}\right )^2}+\frac {300 e^{e^{2 x}}}{\left (-2+5 e^{e^{2 x}}\right )^2}+\frac {300 e^{e^{2 x}+2 x} (-1+x)^2}{\left (-2+5 e^{e^{2 x}}\right )^2}+\frac {120 x}{\left (-2+5 e^{e^{2 x}}\right )^2}-\frac {300 e^{e^{2 x}} x}{\left (-2+5 e^{e^{2 x}}\right )^2}\right ) \, dx\\ &=-\left (120 \int \frac {1}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx\right )+120 \int \frac {x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx+300 \int \frac {e^{e^{2 x}}}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx+300 \int \frac {e^{e^{2 x}+2 x} (-1+x)^2}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx-300 \int \frac {e^{e^{2 x}} x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx\\ &=-\left (60 \operatorname {Subst}\left (\int \frac {1}{\left (-2+5 e^x\right )^2 x} \, dx,x,e^{2 x}\right )\right )+120 \int \frac {x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx+150 \operatorname {Subst}\left (\int \frac {e^x}{\left (-2+5 e^x\right )^2 x} \, dx,x,e^{2 x}\right )-300 \int \frac {e^{e^{2 x}} x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx+300 \int \left (\frac {e^{e^{2 x}+2 x}}{\left (-2+5 e^{e^{2 x}}\right )^2}-\frac {2 e^{e^{2 x}+2 x} x}{\left (-2+5 e^{e^{2 x}}\right )^2}+\frac {e^{e^{2 x}+2 x} x^2}{\left (-2+5 e^{e^{2 x}}\right )^2}\right ) \, dx\\ &=\frac {30 e^{-2 x}}{2-5 e^{e^{2 x}}}-30 \operatorname {Subst}\left (\int \frac {1}{\left (-2+5 e^x\right ) x^2} \, dx,x,e^{2 x}\right )-60 \operatorname {Subst}\left (\int \frac {1}{\left (-2+5 e^x\right )^2 x} \, dx,x,e^{2 x}\right )+120 \int \frac {x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx+300 \int \frac {e^{e^{2 x}+2 x}}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx-300 \int \frac {e^{e^{2 x}} x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx+300 \int \frac {e^{e^{2 x}+2 x} x^2}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx-600 \int \frac {e^{e^{2 x}+2 x} x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx\\ &=\frac {30 e^{-2 x}}{2-5 e^{e^{2 x}}}-30 \operatorname {Subst}\left (\int \frac {1}{\left (-2+5 e^x\right ) x^2} \, dx,x,e^{2 x}\right )-60 \operatorname {Subst}\left (\int \frac {1}{\left (-2+5 e^x\right )^2 x} \, dx,x,e^{2 x}\right )+120 \int \frac {x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx+150 \operatorname {Subst}\left (\int \frac {e^x}{\left (-2+5 e^x\right )^2} \, dx,x,e^{2 x}\right )-300 \int \frac {e^{e^{2 x}} x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx+300 \int \frac {e^{e^{2 x}+2 x} x^2}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx-600 \int \frac {e^{e^{2 x}+2 x} x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx\\ &=\frac {30 e^{-2 x}}{2-5 e^{e^{2 x}}}-30 \operatorname {Subst}\left (\int \frac {1}{\left (-2+5 e^x\right ) x^2} \, dx,x,e^{2 x}\right )-60 \operatorname {Subst}\left (\int \frac {1}{\left (-2+5 e^x\right )^2 x} \, dx,x,e^{2 x}\right )+120 \int \frac {x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx+150 \operatorname {Subst}\left (\int \frac {1}{(-2+5 x)^2} \, dx,x,e^{e^{2 x}}\right )-300 \int \frac {e^{e^{2 x}} x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx+300 \int \frac {e^{e^{2 x}+2 x} x^2}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx-600 \int \frac {e^{e^{2 x}+2 x} x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx\\ &=\frac {30}{2-5 e^{e^{2 x}}}+\frac {30 e^{-2 x}}{2-5 e^{e^{2 x}}}-30 \operatorname {Subst}\left (\int \frac {1}{\left (-2+5 e^x\right ) x^2} \, dx,x,e^{2 x}\right )-60 \operatorname {Subst}\left (\int \frac {1}{\left (-2+5 e^x\right )^2 x} \, dx,x,e^{2 x}\right )+120 \int \frac {x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx-300 \int \frac {e^{e^{2 x}} x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx+300 \int \frac {e^{e^{2 x}+2 x} x^2}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx-600 \int \frac {e^{e^{2 x}+2 x} x}{\left (-2+5 e^{e^{2 x}}\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 20, normalized size = 0.83 \begin {gather*} -\frac {30 (-1+x)^2}{-2+5 e^{e^{2 x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-120 + 120*x + E^E^(2*x)*(300 - 300*x + E^(2*x)*(300 - 600*x + 300*x^2)))/(4 - 20*E^E^(2*x) + 25*E^
(2*E^(2*x))),x]

[Out]

(-30*(-1 + x)^2)/(-2 + 5*E^E^(2*x))

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fricas [A]  time = 0.67, size = 21, normalized size = 0.88 \begin {gather*} -\frac {30 \, {\left (x^{2} - 2 \, x + 1\right )}}{5 \, e^{\left (e^{\left (2 \, x\right )}\right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((300*x^2-600*x+300)*exp(x)^2-300*x+300)*exp(exp(x)^2)+120*x-120)/(25*exp(exp(x)^2)^2-20*exp(exp(x)
^2)+4),x, algorithm="fricas")

[Out]

-30*(x^2 - 2*x + 1)/(5*e^(e^(2*x)) - 2)

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giac [A]  time = 0.35, size = 21, normalized size = 0.88 \begin {gather*} -\frac {30 \, {\left (x^{2} - 2 \, x + 1\right )}}{5 \, e^{\left (e^{\left (2 \, x\right )}\right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((300*x^2-600*x+300)*exp(x)^2-300*x+300)*exp(exp(x)^2)+120*x-120)/(25*exp(exp(x)^2)^2-20*exp(exp(x)
^2)+4),x, algorithm="giac")

[Out]

-30*(x^2 - 2*x + 1)/(5*e^(e^(2*x)) - 2)

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maple [A]  time = 0.08, size = 22, normalized size = 0.92

method	result	size
risch	\(-\frac {30 \left (x^{2}-2 x +1\right )}{5 \,{\mathrm e}^{{\mathrm e}^{2 x}}-2}\)	\(22\)
norman	\(\frac {-75 \,{\mathrm e}^{{\mathrm e}^{2 x}}+60 x -30 x^{2}}{5 \,{\mathrm e}^{{\mathrm e}^{2 x}}-2}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((300*x^2-600*x+300)*exp(x)^2-300*x+300)*exp(exp(x)^2)+120*x-120)/(25*exp(exp(x)^2)^2-20*exp(exp(x)^2)+4)
,x,method=_RETURNVERBOSE)

[Out]

-30*(x^2-2*x+1)/(5*exp(exp(2*x))-2)

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maxima [A]  time = 0.44, size = 21, normalized size = 0.88 \begin {gather*} -\frac {30 \, {\left (x^{2} - 2 \, x + 1\right )}}{5 \, e^{\left (e^{\left (2 \, x\right )}\right )} - 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((300*x^2-600*x+300)*exp(x)^2-300*x+300)*exp(exp(x)^2)+120*x-120)/(25*exp(exp(x)^2)^2-20*exp(exp(x)
^2)+4),x, algorithm="maxima")

[Out]

-30*(x^2 - 2*x + 1)/(5*e^(e^(2*x)) - 2)

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mupad [B]  time = 0.43, size = 23, normalized size = 0.96 \begin {gather*} -\frac {30\,x^2-60\,x+30}{5\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}-2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((120*x + exp(exp(2*x))*(exp(2*x)*(300*x^2 - 600*x + 300) - 300*x + 300) - 120)/(25*exp(2*exp(2*x)) - 20*ex
p(exp(2*x)) + 4),x)

[Out]

-(30*x^2 - 60*x + 30)/(5*exp(exp(2*x)) - 2)

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sympy [A]  time = 0.14, size = 19, normalized size = 0.79 \begin {gather*} \frac {- 6 x^{2} + 12 x - 6}{e^{e^{2 x}} - \frac {2}{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((300*x**2-600*x+300)*exp(x)**2-300*x+300)*exp(exp(x)**2)+120*x-120)/(25*exp(exp(x)**2)**2-20*exp(e
xp(x)**2)+4),x)

[Out]

(-6*x**2 + 12*x - 6)/(exp(exp(2*x)) - 2/5)

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