3.12.75 \(\int \frac {e^{2 e^{x/3}} (720-90 x)+720 x^2-90 x^3+e^{e^{x/3}} (-3-1440 x+e^{x/3} x+180 x^2)}{3 e^{2 e^{x/3}}-6 e^{e^{x/3}} x+3 x^2} \, dx\)

Optimal. Leaf size=26 \[ 15 (16-x) x+\frac {x}{-e^{e^{x/3}}+x} \]

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Rubi [F]  time = 2.10, 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 {e^{2 e^{x/3}} (720-90 x)+720 x^2-90 x^3+e^{e^{x/3}} \left (-3-1440 x+e^{x/3} x+180 x^2\right )}{3 e^{2 e^{x/3}}-6 e^{e^{x/3}} x+3 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*E^(x/3))*(720 - 90*x) + 720*x^2 - 90*x^3 + E^E^(x/3)*(-3 - 1440*x + E^(x/3)*x + 180*x^2))/(3*E^(2*E^
(x/3)) - 6*E^E^(x/3)*x + 3*x^2),x]

[Out]

-480*Defer[Int][(E^E^(x/3)*x)/(E^E^(x/3) - x)^2, x] - 30*Defer[Int][(E^(2*E^(x/3))*x)/(E^E^(x/3) - x)^2, x] +
Defer[Int][(E^((3*E^(x/3) + x)/3)*x)/(E^E^(x/3) - x)^2, x]/3 + 240*Defer[Int][x^2/(E^E^(x/3) - x)^2, x] + 60*D
efer[Int][(E^E^(x/3)*x^2)/(E^E^(x/3) - x)^2, x] - 30*Defer[Int][x^3/(E^E^(x/3) - x)^2, x] - 3*Defer[Subst][Def
er[Int][E^E^x/(E^E^x - 3*x)^2, x], x, x/3] + 720*Defer[Subst][Defer[Int][E^(2*E^x)/(E^E^x - 3*x)^2, x], x, x/3
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 e^{x/3}} (720-90 x)+720 x^2-90 x^3+e^{e^{x/3}} \left (-3-1440 x+e^{x/3} x+180 x^2\right )}{3 \left (e^{e^{x/3}}-x\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {e^{2 e^{x/3}} (720-90 x)+720 x^2-90 x^3+e^{e^{x/3}} \left (-3-1440 x+e^{x/3} x+180 x^2\right )}{\left (e^{e^{x/3}}-x\right )^2} \, dx\\ &=\frac {1}{3} \int \left (-\frac {3 e^{e^{x/3}}}{\left (e^{e^{x/3}}-x\right )^2}-\frac {90 e^{2 e^{x/3}} (-8+x)}{\left (e^{e^{x/3}}-x\right )^2}-\frac {1440 e^{e^{x/3}} x}{\left (e^{e^{x/3}}-x\right )^2}+\frac {e^{\frac {1}{3} \left (3 e^{x/3}+x\right )} x}{\left (e^{e^{x/3}}-x\right )^2}+\frac {720 x^2}{\left (e^{e^{x/3}}-x\right )^2}+\frac {180 e^{e^{x/3}} x^2}{\left (e^{e^{x/3}}-x\right )^2}-\frac {90 x^3}{\left (e^{e^{x/3}}-x\right )^2}\right ) \, dx\\ &=\frac {1}{3} \int \frac {e^{\frac {1}{3} \left (3 e^{x/3}+x\right )} x}{\left (e^{e^{x/3}}-x\right )^2} \, dx-30 \int \frac {e^{2 e^{x/3}} (-8+x)}{\left (e^{e^{x/3}}-x\right )^2} \, dx-30 \int \frac {x^3}{\left (e^{e^{x/3}}-x\right )^2} \, dx+60 \int \frac {e^{e^{x/3}} x^2}{\left (e^{e^{x/3}}-x\right )^2} \, dx+240 \int \frac {x^2}{\left (e^{e^{x/3}}-x\right )^2} \, dx-480 \int \frac {e^{e^{x/3}} x}{\left (e^{e^{x/3}}-x\right )^2} \, dx-\int \frac {e^{e^{x/3}}}{\left (e^{e^{x/3}}-x\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {e^{\frac {1}{3} \left (3 e^{x/3}+x\right )} x}{\left (e^{e^{x/3}}-x\right )^2} \, dx-3 \operatorname {Subst}\left (\int \frac {e^{e^x}}{\left (e^{e^x}-3 x\right )^2} \, dx,x,\frac {x}{3}\right )-30 \int \frac {x^3}{\left (e^{e^{x/3}}-x\right )^2} \, dx-30 \int \left (-\frac {8 e^{2 e^{x/3}}}{\left (e^{e^{x/3}}-x\right )^2}+\frac {e^{2 e^{x/3}} x}{\left (e^{e^{x/3}}-x\right )^2}\right ) \, dx+60 \int \frac {e^{e^{x/3}} x^2}{\left (e^{e^{x/3}}-x\right )^2} \, dx+240 \int \frac {x^2}{\left (e^{e^{x/3}}-x\right )^2} \, dx-480 \int \frac {e^{e^{x/3}} x}{\left (e^{e^{x/3}}-x\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {e^{\frac {1}{3} \left (3 e^{x/3}+x\right )} x}{\left (e^{e^{x/3}}-x\right )^2} \, dx-3 \operatorname {Subst}\left (\int \frac {e^{e^x}}{\left (e^{e^x}-3 x\right )^2} \, dx,x,\frac {x}{3}\right )-30 \int \frac {e^{2 e^{x/3}} x}{\left (e^{e^{x/3}}-x\right )^2} \, dx-30 \int \frac {x^3}{\left (e^{e^{x/3}}-x\right )^2} \, dx+60 \int \frac {e^{e^{x/3}} x^2}{\left (e^{e^{x/3}}-x\right )^2} \, dx+240 \int \frac {e^{2 e^{x/3}}}{\left (e^{e^{x/3}}-x\right )^2} \, dx+240 \int \frac {x^2}{\left (e^{e^{x/3}}-x\right )^2} \, dx-480 \int \frac {e^{e^{x/3}} x}{\left (e^{e^{x/3}}-x\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {e^{\frac {1}{3} \left (3 e^{x/3}+x\right )} x}{\left (e^{e^{x/3}}-x\right )^2} \, dx-3 \operatorname {Subst}\left (\int \frac {e^{e^x}}{\left (e^{e^x}-3 x\right )^2} \, dx,x,\frac {x}{3}\right )-30 \int \frac {e^{2 e^{x/3}} x}{\left (e^{e^{x/3}}-x\right )^2} \, dx-30 \int \frac {x^3}{\left (e^{e^{x/3}}-x\right )^2} \, dx+60 \int \frac {e^{e^{x/3}} x^2}{\left (e^{e^{x/3}}-x\right )^2} \, dx+240 \int \frac {x^2}{\left (e^{e^{x/3}}-x\right )^2} \, dx-480 \int \frac {e^{e^{x/3}} x}{\left (e^{e^{x/3}}-x\right )^2} \, dx+720 \operatorname {Subst}\left (\int \frac {e^{2 e^x}}{\left (e^{e^x}-3 x\right )^2} \, dx,x,\frac {x}{3}\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 31, normalized size = 1.19 \begin {gather*} \frac {1}{3} \left (720 x-\frac {3 x}{e^{e^{x/3}}-x}-45 x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*E^(x/3))*(720 - 90*x) + 720*x^2 - 90*x^3 + E^E^(x/3)*(-3 - 1440*x + E^(x/3)*x + 180*x^2))/(3*E
^(2*E^(x/3)) - 6*E^E^(x/3)*x + 3*x^2),x]

[Out]

(720*x - (3*x)/(E^E^(x/3) - x) - 45*x^2)/3

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fricas [B]  time = 0.65, size = 41, normalized size = 1.58 \begin {gather*} -\frac {15 \, x^{3} - 240 \, x^{2} - 15 \, {\left (x^{2} - 16 \, x\right )} e^{\left (e^{\left (\frac {1}{3} \, x\right )}\right )} - x}{x - e^{\left (e^{\left (\frac {1}{3} \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-90*x+720)*exp(exp(1/3*x))^2+(x*exp(1/3*x)+180*x^2-1440*x-3)*exp(exp(1/3*x))-90*x^3+720*x^2)/(3*ex
p(exp(1/3*x))^2-6*x*exp(exp(1/3*x))+3*x^2),x, algorithm="fricas")

[Out]

-(15*x^3 - 240*x^2 - 15*(x^2 - 16*x)*e^(e^(1/3*x)) - x)/(x - e^(e^(1/3*x)))

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giac [B]  time = 0.96, size = 45, normalized size = 1.73 \begin {gather*} -\frac {15 \, x^{3} - 15 \, x^{2} e^{\left (e^{\left (\frac {1}{3} \, x\right )}\right )} - 240 \, x^{2} + 240 \, x e^{\left (e^{\left (\frac {1}{3} \, x\right )}\right )} - x}{x - e^{\left (e^{\left (\frac {1}{3} \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-90*x+720)*exp(exp(1/3*x))^2+(x*exp(1/3*x)+180*x^2-1440*x-3)*exp(exp(1/3*x))-90*x^3+720*x^2)/(3*ex
p(exp(1/3*x))^2-6*x*exp(exp(1/3*x))+3*x^2),x, algorithm="giac")

[Out]

-(15*x^3 - 15*x^2*e^(e^(1/3*x)) - 240*x^2 + 240*x*e^(e^(1/3*x)) - x)/(x - e^(e^(1/3*x)))

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maple [A]  time = 0.06, size = 23, normalized size = 0.88

method	result	size
risch	\(-15 x^{2}+240 x +\frac {x}{-{\mathrm e}^{{\mathrm e}^{\frac {x}{3}}}+x}\)	\(23\)
norman	\(\frac {{\mathrm e}^{{\mathrm e}^{\frac {x}{3}}}+240 x^{2}-15 x^{3}-240 x \,{\mathrm e}^{{\mathrm e}^{\frac {x}{3}}}+15 \,{\mathrm e}^{{\mathrm e}^{\frac {x}{3}}} x^{2}}{-{\mathrm e}^{{\mathrm e}^{\frac {x}{3}}}+x}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-90*x+720)*exp(exp(1/3*x))^2+(x*exp(1/3*x)+180*x^2-1440*x-3)*exp(exp(1/3*x))-90*x^3+720*x^2)/(3*exp(exp(
1/3*x))^2-6*x*exp(exp(1/3*x))+3*x^2),x,method=_RETURNVERBOSE)

[Out]

-15*x^2+240*x+x/(-exp(exp(1/3*x))+x)

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maxima [B]  time = 0.54, size = 41, normalized size = 1.58 \begin {gather*} -\frac {15 \, x^{3} - 240 \, x^{2} - 15 \, {\left (x^{2} - 16 \, x\right )} e^{\left (e^{\left (\frac {1}{3} \, x\right )}\right )} - x}{x - e^{\left (e^{\left (\frac {1}{3} \, x\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-90*x+720)*exp(exp(1/3*x))^2+(x*exp(1/3*x)+180*x^2-1440*x-3)*exp(exp(1/3*x))-90*x^3+720*x^2)/(3*ex
p(exp(1/3*x))^2-6*x*exp(exp(1/3*x))+3*x^2),x, algorithm="maxima")

[Out]

-(15*x^3 - 240*x^2 - 15*(x^2 - 16*x)*e^(e^(1/3*x)) - x)/(x - e^(e^(1/3*x)))

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mupad [B]  time = 0.84, size = 26, normalized size = 1.00 \begin {gather*} 240\,x+\frac {{\mathrm {e}}^{{\left ({\mathrm {e}}^x\right )}^{1/3}}}{x-{\mathrm {e}}^{{\left ({\mathrm {e}}^x\right )}^{1/3}}}-15\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(exp(x/3))*(1440*x - x*exp(x/3) - 180*x^2 + 3) + exp(2*exp(x/3))*(90*x - 720) - 720*x^2 + 90*x^3)/(3*
exp(2*exp(x/3)) - 6*x*exp(exp(x/3)) + 3*x^2),x)

[Out]

240*x + exp(exp(x)^(1/3))/(x - exp(exp(x)^(1/3))) - 15*x^2

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sympy [A]  time = 0.14, size = 17, normalized size = 0.65 \begin {gather*} - 15 x^{2} + 240 x - \frac {x}{- x + e^{e^{\frac {x}{3}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-90*x+720)*exp(exp(1/3*x))**2+(x*exp(1/3*x)+180*x**2-1440*x-3)*exp(exp(1/3*x))-90*x**3+720*x**2)/(
3*exp(exp(1/3*x))**2-6*x*exp(exp(1/3*x))+3*x**2),x)

[Out]

-15*x**2 + 240*x - x/(-x + exp(exp(x/3)))

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