3.60.55 \(\int \frac {e^{-\frac {60+20 x^4-e^x x^4}{2 x^4}} (240+4 x^4+4 e^{\frac {60+20 x^4-e^x x^4}{2 x^4}} x^4+e^x x^5+e^{\frac {60+20 x^4-e^x x^4}{4 x^4}} (-240-8 x^4-e^x x^5))}{2 x^3} \, dx\)

Optimal. Leaf size=23 \[ \left (x-e^{-5+\frac {e^x}{4}-\frac {15}{x^4}} x\right )^2 \]

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Rubi [F]  time = 4.37, 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^{-\frac {60+20 x^4-e^x x^4}{2 x^4}} \left (240+4 x^4+4 e^{\frac {60+20 x^4-e^x x^4}{2 x^4}} x^4+e^x x^5+e^{\frac {60+20 x^4-e^x x^4}{4 x^4}} \left (-240-8 x^4-e^x x^5\right )\right )}{2 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(240 + 4*x^4 + 4*E^((60 + 20*x^4 - E^x*x^4)/(2*x^4))*x^4 + E^x*x^5 + E^((60 + 20*x^4 - E^x*x^4)/(4*x^4))*(
-240 - 8*x^4 - E^x*x^5))/(2*E^((60 + 20*x^4 - E^x*x^4)/(2*x^4))*x^3),x]

[Out]

x^2 + 120*Defer[Int][E^(-10 + E^x/2 - 30/x^4)/x^3, x] - 120*Defer[Int][E^(-5 + E^x/4 - 15/x^4)/x^3, x] + 2*Def
er[Int][E^(-10 + E^x/2 - 30/x^4)*x, x] - 4*Defer[Int][E^(-5 + E^x/4 - 15/x^4)*x, x] + Defer[Int][E^(-10 + E^x/
2 - 30/x^4 + x)*x^2, x]/2 - Defer[Int][E^(-5 - 15/x^4 + (E^x + 4*x)/4)*x^2, x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {e^{-\frac {60+20 x^4-e^x x^4}{2 x^4}} \left (240+4 x^4+4 e^{\frac {60+20 x^4-e^x x^4}{2 x^4}} x^4+e^x x^5+e^{\frac {60+20 x^4-e^x x^4}{4 x^4}} \left (-240-8 x^4-e^x x^5\right )\right )}{x^3} \, dx\\ &=\frac {1}{2} \int \frac {e^{-10-\frac {30}{x^4}} \left (e^{\frac {e^x}{4}}-e^{5+\frac {15}{x^4}}\right ) \left (-4 e^{5+\frac {15}{x^4}} x^4+e^{\frac {e^x}{4}+x} x^5+4 e^{\frac {e^x}{4}} \left (60+x^4\right )\right )}{x^3} \, dx\\ &=\frac {1}{2} \int \left (e^{-10-\frac {30}{x^4}+\frac {1}{4} \left (e^x+4 x\right )} \left (e^{\frac {e^x}{4}}-e^{5+\frac {15}{x^4}}\right ) x^2+\frac {4 e^{-10-\frac {30}{x^4}} \left (60 e^{\frac {e^x}{2}}-60 e^{5+\frac {e^x}{4}+\frac {15}{x^4}}+e^{\frac {e^x}{2}} x^4-2 e^{5+\frac {e^x}{4}+\frac {15}{x^4}} x^4+e^{10+\frac {30}{x^4}} x^4\right )}{x^3}\right ) \, dx\\ &=\frac {1}{2} \int e^{-10-\frac {30}{x^4}+\frac {1}{4} \left (e^x+4 x\right )} \left (e^{\frac {e^x}{4}}-e^{5+\frac {15}{x^4}}\right ) x^2 \, dx+2 \int \frac {e^{-10-\frac {30}{x^4}} \left (60 e^{\frac {e^x}{2}}-60 e^{5+\frac {e^x}{4}+\frac {15}{x^4}}+e^{\frac {e^x}{2}} x^4-2 e^{5+\frac {e^x}{4}+\frac {15}{x^4}} x^4+e^{10+\frac {30}{x^4}} x^4\right )}{x^3} \, dx\\ &=\frac {1}{2} \int \left (e^{-10+\frac {e^x}{4}-\frac {30}{x^4}+\frac {1}{4} \left (e^x+4 x\right )} x^2-e^{-5-\frac {15}{x^4}+\frac {1}{4} \left (e^x+4 x\right )} x^2\right ) \, dx+2 \int \frac {x^4-2 e^{-5+\frac {e^x}{4}-\frac {15}{x^4}} \left (30+x^4\right )+e^{-10+\frac {e^x}{2}-\frac {30}{x^4}} \left (60+x^4\right )}{x^3} \, dx\\ &=\frac {1}{2} \int e^{-10+\frac {e^x}{4}-\frac {30}{x^4}+\frac {1}{4} \left (e^x+4 x\right )} x^2 \, dx-\frac {1}{2} \int e^{-5-\frac {15}{x^4}+\frac {1}{4} \left (e^x+4 x\right )} x^2 \, dx+2 \int \left (x-\frac {2 e^{-5+\frac {e^x}{4}-\frac {15}{x^4}} \left (30+x^4\right )}{x^3}+\frac {e^{-10+\frac {e^x}{2}-\frac {30}{x^4}} \left (60+x^4\right )}{x^3}\right ) \, dx\\ &=x^2+\frac {1}{2} \int e^{-10+\frac {e^x}{2}-\frac {30}{x^4}+x} x^2 \, dx-\frac {1}{2} \int e^{-5-\frac {15}{x^4}+\frac {1}{4} \left (e^x+4 x\right )} x^2 \, dx+2 \int \frac {e^{-10+\frac {e^x}{2}-\frac {30}{x^4}} \left (60+x^4\right )}{x^3} \, dx-4 \int \frac {e^{-5+\frac {e^x}{4}-\frac {15}{x^4}} \left (30+x^4\right )}{x^3} \, dx\\ &=x^2+\frac {1}{2} \int e^{-10+\frac {e^x}{2}-\frac {30}{x^4}+x} x^2 \, dx-\frac {1}{2} \int e^{-5-\frac {15}{x^4}+\frac {1}{4} \left (e^x+4 x\right )} x^2 \, dx+2 \int \left (\frac {60 e^{-10+\frac {e^x}{2}-\frac {30}{x^4}}}{x^3}+e^{-10+\frac {e^x}{2}-\frac {30}{x^4}} x\right ) \, dx-4 \int \left (\frac {30 e^{-5+\frac {e^x}{4}-\frac {15}{x^4}}}{x^3}+e^{-5+\frac {e^x}{4}-\frac {15}{x^4}} x\right ) \, dx\\ &=x^2+\frac {1}{2} \int e^{-10+\frac {e^x}{2}-\frac {30}{x^4}+x} x^2 \, dx-\frac {1}{2} \int e^{-5-\frac {15}{x^4}+\frac {1}{4} \left (e^x+4 x\right )} x^2 \, dx+2 \int e^{-10+\frac {e^x}{2}-\frac {30}{x^4}} x \, dx-4 \int e^{-5+\frac {e^x}{4}-\frac {15}{x^4}} x \, dx+120 \int \frac {e^{-10+\frac {e^x}{2}-\frac {30}{x^4}}}{x^3} \, dx-120 \int \frac {e^{-5+\frac {e^x}{4}-\frac {15}{x^4}}}{x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.07, size = 36, normalized size = 1.57 \begin {gather*} e^{-10-\frac {30}{x^4}} \left (e^{\frac {e^x}{4}}-e^{5+\frac {15}{x^4}}\right )^2 x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(240 + 4*x^4 + 4*E^((60 + 20*x^4 - E^x*x^4)/(2*x^4))*x^4 + E^x*x^5 + E^((60 + 20*x^4 - E^x*x^4)/(4*x
^4))*(-240 - 8*x^4 - E^x*x^5))/(2*E^((60 + 20*x^4 - E^x*x^4)/(2*x^4))*x^3),x]

[Out]

E^(-10 - 30/x^4)*(E^(E^x/4) - E^(5 + 15/x^4))^2*x^2

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fricas [B]  time = 0.68, size = 75, normalized size = 3.26 \begin {gather*} -{\left (2 \, x^{2} e^{\left (-\frac {x^{4} e^{x} - 20 \, x^{4} - 60}{4 \, x^{4}}\right )} - x^{2} e^{\left (-\frac {x^{4} e^{x} - 20 \, x^{4} - 60}{2 \, x^{4}}\right )} - x^{2}\right )} e^{\left (\frac {x^{4} e^{x} - 20 \, x^{4} - 60}{2 \, x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(4*x^4*exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2+(-x^5*exp(x)-8*x^4-240)*exp(1/4*(-exp(x)*x^4+20*x^
4+60)/x^4)+x^5*exp(x)+4*x^4+240)/x^3/exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2,x, algorithm="fricas")

[Out]

-(2*x^2*e^(-1/4*(x^4*e^x - 20*x^4 - 60)/x^4) - x^2*e^(-1/2*(x^4*e^x - 20*x^4 - 60)/x^4) - x^2)*e^(1/2*(x^4*e^x
- 20*x^4 - 60)/x^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{5} e^{x} + 4 \, x^{4} e^{\left (-\frac {x^{4} e^{x} - 20 \, x^{4} - 60}{2 \, x^{4}}\right )} + 4 \, x^{4} - {\left (x^{5} e^{x} + 8 \, x^{4} + 240\right )} e^{\left (-\frac {x^{4} e^{x} - 20 \, x^{4} - 60}{4 \, x^{4}}\right )} + 240\right )} e^{\left (\frac {x^{4} e^{x} - 20 \, x^{4} - 60}{2 \, x^{4}}\right )}}{2 \, x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(4*x^4*exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2+(-x^5*exp(x)-8*x^4-240)*exp(1/4*(-exp(x)*x^4+20*x^
4+60)/x^4)+x^5*exp(x)+4*x^4+240)/x^3/exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2,x, algorithm="giac")

[Out]

integrate(1/2*(x^5*e^x + 4*x^4*e^(-1/2*(x^4*e^x - 20*x^4 - 60)/x^4) + 4*x^4 - (x^5*e^x + 8*x^4 + 240)*e^(-1/4*
(x^4*e^x - 20*x^4 - 60)/x^4) + 240)*e^(1/2*(x^4*e^x - 20*x^4 - 60)/x^4)/x^3, x)

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maple [B]  time = 0.07, size = 52, normalized size = 2.26

method	result	size
risch	\(x^{2}-2 x^{2} {\mathrm e}^{\frac {{\mathrm e}^{x} x^{4}-20 x^{4}-60}{4 x^{4}}}+x^{2} {\mathrm e}^{\frac {{\mathrm e}^{x} x^{4}-20 x^{4}-60}{2 x^{4}}}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(4*x^4*exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2+(-x^5*exp(x)-8*x^4-240)*exp(1/4*(-exp(x)*x^4+20*x^4+60)/
x^4)+x^5*exp(x)+4*x^4+240)/x^3/exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2,x,method=_RETURNVERBOSE)

[Out]

x^2-2*x^2*exp(1/4*(exp(x)*x^4-20*x^4-60)/x^4)+x^2*exp(1/2*(exp(x)*x^4-20*x^4-60)/x^4)

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maxima [A]  time = 0.43, size = 40, normalized size = 1.74 \begin {gather*} x^{2} + {\left (x^{2} e^{\left (\frac {1}{2} \, e^{x}\right )} - 2 \, x^{2} e^{\left (\frac {15}{x^{4}} + \frac {1}{4} \, e^{x} + 5\right )}\right )} e^{\left (-\frac {30}{x^{4}} - 10\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(4*x^4*exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2+(-x^5*exp(x)-8*x^4-240)*exp(1/4*(-exp(x)*x^4+20*x^
4+60)/x^4)+x^5*exp(x)+4*x^4+240)/x^3/exp(1/4*(-exp(x)*x^4+20*x^4+60)/x^4)^2,x, algorithm="maxima")

[Out]

x^2 + (x^2*e^(1/2*e^x) - 2*x^2*e^(15/x^4 + 1/4*e^x + 5))*e^(-30/x^4 - 10)

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mupad [B]  time = 4.24, size = 32, normalized size = 1.39 \begin {gather*} x^2\,{\mathrm {e}}^{\frac {{\mathrm {e}}^x}{2}-\frac {30}{x^4}-10}\,{\left ({\mathrm {e}}^{\frac {15}{x^4}-\frac {{\mathrm {e}}^x}{4}+5}-1\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(2*(5*x^4 - (x^4*exp(x))/4 + 15))/x^4)*((x^5*exp(x))/2 - (exp((5*x^4 - (x^4*exp(x))/4 + 15)/x^4)*(x^
5*exp(x) + 8*x^4 + 240))/2 + 2*x^4*exp((2*(5*x^4 - (x^4*exp(x))/4 + 15))/x^4) + 2*x^4 + 120))/x^3,x)

[Out]

x^2*exp(exp(x)/2 - 30/x^4 - 10)*(exp(15/x^4 - exp(x)/4 + 5) - 1)^2

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sympy [B]  time = 0.26, size = 51, normalized size = 2.22 \begin {gather*} x^{2} - 2 x^{2} e^{- \frac {- \frac {x^{4} e^{x}}{4} + 5 x^{4} + 15}{x^{4}}} + x^{2} e^{- \frac {2 \left (- \frac {x^{4} e^{x}}{4} + 5 x^{4} + 15\right )}{x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(4*x**4*exp(1/4*(-exp(x)*x**4+20*x**4+60)/x**4)**2+(-x**5*exp(x)-8*x**4-240)*exp(1/4*(-exp(x)*x*
*4+20*x**4+60)/x**4)+x**5*exp(x)+4*x**4+240)/x**3/exp(1/4*(-exp(x)*x**4+20*x**4+60)/x**4)**2,x)

[Out]

x**2 - 2*x**2*exp(-(-x**4*exp(x)/4 + 5*x**4 + 15)/x**4) + x**2*exp(-2*(-x**4*exp(x)/4 + 5*x**4 + 15)/x**4)

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