3.30.6 \(\int \frac {-1+e^x (-120 x^3+60 x^4)}{3+e^x (1440 x-1440 x^2+360 x^3)+e^{2 x} (172800 x^2-345600 x^3+259200 x^4-86400 x^5+10800 x^6)} \, dx\)

Optimal. Leaf size=24 \[ e^5-\frac {x}{3 \left (1+60 e^x (-2+x)^2 x\right )} \]

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Rubi [F]  time = 1.80, 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 {-1+e^x \left (-120 x^3+60 x^4\right )}{3+e^x \left (1440 x-1440 x^2+360 x^3\right )+e^{2 x} \left (172800 x^2-345600 x^3+259200 x^4-86400 x^5+10800 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + E^x*(-120*x^3 + 60*x^4))/(3 + E^x*(1440*x - 1440*x^2 + 360*x^3) + E^(2*x)*(172800*x^2 - 345600*x^3 +
259200*x^4 - 86400*x^5 + 10800*x^6)),x]

[Out]

-Defer[Int][(1 + 60*E^x*(-2 + x)^2*x)^(-2), x] - (4*Defer[Int][1/((-2 + x)*(1 + 60*E^x*(-2 + x)^2*x)^2), x])/3
- Defer[Int][x/(1 + 60*E^x*(-2 + x)^2*x)^2, x]/3 + (2*Defer[Int][(1 + 60*E^x*(-2 + x)^2*x)^(-1), x])/3 + (4*D
efer[Int][1/((-2 + x)*(1 + 60*E^x*(-2 + x)^2*x)), x])/3 + Defer[Int][x/(1 + 60*E^x*(-2 + x)^2*x), x]/3

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+60 e^x (-2+x) x^3}{3 \left (1+60 e^x (-2+x)^2 x\right )^2} \, dx\\ &=\frac {1}{3} \int \frac {-1+60 e^x (-2+x) x^3}{\left (1+60 e^x (-2+x)^2 x\right )^2} \, dx\\ &=\frac {1}{3} \int \left (-\frac {-2+x+x^2}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2}+\frac {x^2}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {-2+x+x^2}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2} \, dx\right )+\frac {1}{3} \int \frac {x^2}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )} \, dx\\ &=\frac {1}{3} \int \frac {x^2}{(-2+x) \left (1+60 e^x (-2+x)^2 x\right )} \, dx-\frac {1}{3} \int \left (\frac {3}{\left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2}+\frac {4}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2}+\frac {x}{\left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2}\right ) \, dx\\ &=-\left (\frac {1}{3} \int \frac {x}{\left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2} \, dx\right )+\frac {1}{3} \int \left (\frac {2}{1+240 e^x x-240 e^x x^2+60 e^x x^3}+\frac {4}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )}+\frac {x}{1+240 e^x x-240 e^x x^2+60 e^x x^3}\right ) \, dx-\frac {4}{3} \int \frac {1}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2} \, dx-\int \frac {1}{\left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )^2} \, dx\\ &=-\left (\frac {1}{3} \int \frac {x}{\left (1+60 e^x (-2+x)^2 x\right )^2} \, dx\right )+\frac {1}{3} \int \frac {x}{1+240 e^x x-240 e^x x^2+60 e^x x^3} \, dx+\frac {2}{3} \int \frac {1}{1+240 e^x x-240 e^x x^2+60 e^x x^3} \, dx-\frac {4}{3} \int \frac {1}{(-2+x) \left (1+60 e^x (-2+x)^2 x\right )^2} \, dx+\frac {4}{3} \int \frac {1}{(-2+x) \left (1+240 e^x x-240 e^x x^2+60 e^x x^3\right )} \, dx-\int \frac {1}{\left (1+60 e^x (-2+x)^2 x\right )^2} \, dx\\ &=-\left (\frac {1}{3} \int \frac {x}{\left (1+60 e^x (-2+x)^2 x\right )^2} \, dx\right )+\frac {1}{3} \int \frac {x}{1+60 e^x (-2+x)^2 x} \, dx+\frac {2}{3} \int \frac {1}{1+60 e^x (-2+x)^2 x} \, dx-\frac {4}{3} \int \frac {1}{(-2+x) \left (1+60 e^x (-2+x)^2 x\right )^2} \, dx+\frac {4}{3} \int \frac {1}{(-2+x) \left (1+60 e^x (-2+x)^2 x\right )} \, dx-\int \frac {1}{\left (1+60 e^x (-2+x)^2 x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 20, normalized size = 0.83 \begin {gather*} -\frac {x}{3 \left (1+60 e^x (-2+x)^2 x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + E^x*(-120*x^3 + 60*x^4))/(3 + E^x*(1440*x - 1440*x^2 + 360*x^3) + E^(2*x)*(172800*x^2 - 345600
*x^3 + 259200*x^4 - 86400*x^5 + 10800*x^6)),x]

[Out]

-1/3*x/(1 + 60*E^x*(-2 + x)^2*x)

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fricas [A]  time = 0.95, size = 23, normalized size = 0.96 \begin {gather*} -\frac {x}{3 \, {\left (60 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{x} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^4-120*x^3)*exp(x)-1)/((10800*x^6-86400*x^5+259200*x^4-345600*x^3+172800*x^2)*exp(x)^2+(360*x^
3-1440*x^2+1440*x)*exp(x)+3),x, algorithm="fricas")

[Out]

-1/3*x/(60*(x^3 - 4*x^2 + 4*x)*e^x + 1)

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giac [A]  time = 0.28, size = 26, normalized size = 1.08 \begin {gather*} -\frac {x}{3 \, {\left (60 \, x^{3} e^{x} - 240 \, x^{2} e^{x} + 240 \, x e^{x} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^4-120*x^3)*exp(x)-1)/((10800*x^6-86400*x^5+259200*x^4-345600*x^3+172800*x^2)*exp(x)^2+(360*x^
3-1440*x^2+1440*x)*exp(x)+3),x, algorithm="giac")

[Out]

-1/3*x/(60*x^3*e^x - 240*x^2*e^x + 240*x*e^x + 1)

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maple [A]  time = 0.06, size = 27, normalized size = 1.12

method	result	size
norman	\(-\frac {x}{3 \left (60 \,{\mathrm e}^{x} x^{3}-240 \,{\mathrm e}^{x} x^{2}+240 \,{\mathrm e}^{x} x +1\right )}\)	\(27\)
risch	\(-\frac {x}{3 \left (60 \,{\mathrm e}^{x} x^{3}-240 \,{\mathrm e}^{x} x^{2}+240 \,{\mathrm e}^{x} x +1\right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((60*x^4-120*x^3)*exp(x)-1)/((10800*x^6-86400*x^5+259200*x^4-345600*x^3+172800*x^2)*exp(x)^2+(360*x^3-1440
*x^2+1440*x)*exp(x)+3),x,method=_RETURNVERBOSE)

[Out]

-1/3*x/(60*exp(x)*x^3-240*exp(x)*x^2+240*exp(x)*x+1)

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maxima [A]  time = 0.49, size = 23, normalized size = 0.96 \begin {gather*} -\frac {x}{3 \, {\left (60 \, {\left (x^{3} - 4 \, x^{2} + 4 \, x\right )} e^{x} + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x^4-120*x^3)*exp(x)-1)/((10800*x^6-86400*x^5+259200*x^4-345600*x^3+172800*x^2)*exp(x)^2+(360*x^
3-1440*x^2+1440*x)*exp(x)+3),x, algorithm="maxima")

[Out]

-1/3*x/(60*(x^3 - 4*x^2 + 4*x)*e^x + 1)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {{\mathrm {e}}^x\,\left (120\,x^3-60\,x^4\right )+1}{{\mathrm {e}}^{2\,x}\,\left (10800\,x^6-86400\,x^5+259200\,x^4-345600\,x^3+172800\,x^2\right )+{\mathrm {e}}^x\,\left (360\,x^3-1440\,x^2+1440\,x\right )+3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x)*(120*x^3 - 60*x^4) + 1)/(exp(2*x)*(172800*x^2 - 345600*x^3 + 259200*x^4 - 86400*x^5 + 10800*x^6)
+ exp(x)*(1440*x - 1440*x^2 + 360*x^3) + 3),x)

[Out]

int(-(exp(x)*(120*x^3 - 60*x^4) + 1)/(exp(2*x)*(172800*x^2 - 345600*x^3 + 259200*x^4 - 86400*x^5 + 10800*x^6)
+ exp(x)*(1440*x - 1440*x^2 + 360*x^3) + 3), x)

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sympy [A]  time = 0.23, size = 20, normalized size = 0.83 \begin {gather*} - \frac {x}{\left (180 x^{3} - 720 x^{2} + 720 x\right ) e^{x} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((60*x**4-120*x**3)*exp(x)-1)/((10800*x**6-86400*x**5+259200*x**4-345600*x**3+172800*x**2)*exp(x)**2
+(360*x**3-1440*x**2+1440*x)*exp(x)+3),x)

[Out]

-x/((180*x**3 - 720*x**2 + 720*x)*exp(x) + 3)

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