3.42.32 \(\int \frac {e^{\frac {100+80 x^2-60 x^3-4 x^4+e^{2 x} x^4-24 x^5+x^6+6 x^7+x^8+e^x (20 x^2+8 x^4-6 x^5-2 x^6)}{x^4}} (-400-160 x^2+60 x^3+x^4-24 x^5+2 e^{2 x} x^5+2 x^6+18 x^7+4 x^8+e^x (-40 x^2+20 x^3+2 x^5-10 x^6-2 x^7))}{x^4} \, dx\)

Optimal. Leaf size=25 \[ 1+e^{\left (4+e^x+\frac {10}{x^2}+x-x (4+x)\right )^2} x \]

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Rubi [B]  time = 29.82, antiderivative size = 330, normalized size of antiderivative = 13.20, number of steps used = 1, number of rules used = 1, integrand size = 152, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.007, Rules used = {2288} \begin {gather*} -\frac {\left (-2 x^8-9 x^7-x^6-e^{2 x} x^5+12 x^5-30 x^3+80 x^2+e^x \left (x^7+5 x^6-x^5-10 x^3+20 x^2\right )+200\right ) \exp \left (\frac {x^8+6 x^7+x^6-24 x^5+e^{2 x} x^4-4 x^4-60 x^3+80 x^2+2 e^x \left (-x^6-3 x^5+4 x^4+10 x^2\right )+100}{x^4}\right )}{x^4 \left (\frac {4 x^7+21 x^6+3 x^5+e^{2 x} x^4-60 x^4+2 e^{2 x} x^3-8 x^3-90 x^2+e^x \left (-6 x^5-15 x^4+16 x^3+20 x\right )+e^x \left (-x^6-3 x^5+4 x^4+10 x^2\right )+80 x}{x^4}-\frac {2 \left (x^8+6 x^7+x^6-24 x^5+e^{2 x} x^4-4 x^4-60 x^3+80 x^2+2 e^x \left (-x^6-3 x^5+4 x^4+10 x^2\right )+100\right )}{x^5}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^((100 + 80*x^2 - 60*x^3 - 4*x^4 + E^(2*x)*x^4 - 24*x^5 + x^6 + 6*x^7 + x^8 + E^x*(20*x^2 + 8*x^4 - 6*x^
5 - 2*x^6))/x^4)*(-400 - 160*x^2 + 60*x^3 + x^4 - 24*x^5 + 2*E^(2*x)*x^5 + 2*x^6 + 18*x^7 + 4*x^8 + E^x*(-40*x
^2 + 20*x^3 + 2*x^5 - 10*x^6 - 2*x^7)))/x^4,x]

[Out]

-((E^((100 + 80*x^2 - 60*x^3 - 4*x^4 + E^(2*x)*x^4 - 24*x^5 + x^6 + 6*x^7 + x^8 + 2*E^x*(10*x^2 + 4*x^4 - 3*x^
5 - x^6))/x^4)*(200 + 80*x^2 - 30*x^3 + 12*x^5 - E^(2*x)*x^5 - x^6 - 9*x^7 - 2*x^8 + E^x*(20*x^2 - 10*x^3 - x^
5 + 5*x^6 + x^7)))/(x^4*((80*x - 90*x^2 - 8*x^3 + 2*E^(2*x)*x^3 - 60*x^4 + E^(2*x)*x^4 + 3*x^5 + 21*x^6 + 4*x^
7 + E^x*(20*x + 16*x^3 - 15*x^4 - 6*x^5) + E^x*(10*x^2 + 4*x^4 - 3*x^5 - x^6))/x^4 - (2*(100 + 80*x^2 - 60*x^3
- 4*x^4 + E^(2*x)*x^4 - 24*x^5 + x^6 + 6*x^7 + x^8 + 2*E^x*(10*x^2 + 4*x^4 - 3*x^5 - x^6)))/x^5)))

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\frac {\exp \left (\frac {100+80 x^2-60 x^3-4 x^4+e^{2 x} x^4-24 x^5+x^6+6 x^7+x^8+2 e^x \left (10 x^2+4 x^4-3 x^5-x^6\right )}{x^4}\right ) \left (200+80 x^2-30 x^3+12 x^5-e^{2 x} x^5-x^6-9 x^7-2 x^8+e^x \left (20 x^2-10 x^3-x^5+5 x^6+x^7\right )\right )}{x^4 \left (\frac {80 x-90 x^2-8 x^3+2 e^{2 x} x^3-60 x^4+e^{2 x} x^4+3 x^5+21 x^6+4 x^7+e^x \left (20 x+16 x^3-15 x^4-6 x^5\right )+e^x \left (10 x^2+4 x^4-3 x^5-x^6\right )}{x^4}-\frac {2 \left (100+80 x^2-60 x^3-4 x^4+e^{2 x} x^4-24 x^5+x^6+6 x^7+x^8+2 e^x \left (10 x^2+4 x^4-3 x^5-x^6\right )\right )}{x^5}\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 30, normalized size = 1.20 \begin {gather*} e^{\frac {\left (-10-\left (4+e^x\right ) x^2+3 x^3+x^4\right )^2}{x^4}} x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((100 + 80*x^2 - 60*x^3 - 4*x^4 + E^(2*x)*x^4 - 24*x^5 + x^6 + 6*x^7 + x^8 + E^x*(20*x^2 + 8*x^4
- 6*x^5 - 2*x^6))/x^4)*(-400 - 160*x^2 + 60*x^3 + x^4 - 24*x^5 + 2*E^(2*x)*x^5 + 2*x^6 + 18*x^7 + 4*x^8 + E^x*
(-40*x^2 + 20*x^3 + 2*x^5 - 10*x^6 - 2*x^7)))/x^4,x]

[Out]

E^((-10 - (4 + E^x)*x^2 + 3*x^3 + x^4)^2/x^4)*x

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fricas [B]  time = 0.58, size = 71, normalized size = 2.84 \begin {gather*} x e^{\left (\frac {x^{8} + 6 \, x^{7} + x^{6} - 24 \, x^{5} + x^{4} e^{\left (2 \, x\right )} - 4 \, x^{4} - 60 \, x^{3} + 80 \, x^{2} - 2 \, {\left (x^{6} + 3 \, x^{5} - 4 \, x^{4} - 10 \, x^{2}\right )} e^{x} + 100}{x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5*exp(x)^2+(-2*x^7-10*x^6+2*x^5+20*x^3-40*x^2)*exp(x)+4*x^8+18*x^7+2*x^6-24*x^5+x^4+60*x^3-160*
x^2-400)*exp((exp(x)^2*x^4+(-2*x^6-6*x^5+8*x^4+20*x^2)*exp(x)+x^8+6*x^7+x^6-24*x^5-4*x^4-60*x^3+80*x^2+100)/x^
4)/x^4,x, algorithm="fricas")

[Out]

x*e^((x^8 + 6*x^7 + x^6 - 24*x^5 + x^4*e^(2*x) - 4*x^4 - 60*x^3 + 80*x^2 - 2*(x^6 + 3*x^5 - 4*x^4 - 10*x^2)*e^
x + 100)/x^4)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (4 \, x^{8} + 18 \, x^{7} + 2 \, x^{6} + 2 \, x^{5} e^{\left (2 \, x\right )} - 24 \, x^{5} + x^{4} + 60 \, x^{3} - 160 \, x^{2} - 2 \, {\left (x^{7} + 5 \, x^{6} - x^{5} - 10 \, x^{3} + 20 \, x^{2}\right )} e^{x} - 400\right )} e^{\left (\frac {x^{8} + 6 \, x^{7} + x^{6} - 24 \, x^{5} + x^{4} e^{\left (2 \, x\right )} - 4 \, x^{4} - 60 \, x^{3} + 80 \, x^{2} - 2 \, {\left (x^{6} + 3 \, x^{5} - 4 \, x^{4} - 10 \, x^{2}\right )} e^{x} + 100}{x^{4}}\right )}}{x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5*exp(x)^2+(-2*x^7-10*x^6+2*x^5+20*x^3-40*x^2)*exp(x)+4*x^8+18*x^7+2*x^6-24*x^5+x^4+60*x^3-160*
x^2-400)*exp((exp(x)^2*x^4+(-2*x^6-6*x^5+8*x^4+20*x^2)*exp(x)+x^8+6*x^7+x^6-24*x^5-4*x^4-60*x^3+80*x^2+100)/x^
4)/x^4,x, algorithm="giac")

[Out]

integrate((4*x^8 + 18*x^7 + 2*x^6 + 2*x^5*e^(2*x) - 24*x^5 + x^4 + 60*x^3 - 160*x^2 - 2*(x^7 + 5*x^6 - x^5 - 1
0*x^3 + 20*x^2)*e^x - 400)*e^((x^8 + 6*x^7 + x^6 - 24*x^5 + x^4*e^(2*x) - 4*x^4 - 60*x^3 + 80*x^2 - 2*(x^6 + 3
*x^5 - 4*x^4 - 10*x^2)*e^x + 100)/x^4)/x^4, x)

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maple [B]  time = 0.31, size = 83, normalized size = 3.32

method	result	size
risch	\(x \,{\mathrm e}^{-\frac {-x^{8}+2 x^{6} {\mathrm e}^{x}-6 x^{7}+6 x^{5} {\mathrm e}^{x}-x^{6}-8 \,{\mathrm e}^{x} x^{4}-{\mathrm e}^{2 x} x^{4}+24 x^{5}+4 x^{4}-20 \,{\mathrm e}^{x} x^{2}+60 x^{3}-80 x^{2}-100}{x^{4}}}\)	\(83\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x^5*exp(x)^2+(-2*x^7-10*x^6+2*x^5+20*x^3-40*x^2)*exp(x)+4*x^8+18*x^7+2*x^6-24*x^5+x^4+60*x^3-160*x^2-40
0)*exp((exp(x)^2*x^4+(-2*x^6-6*x^5+8*x^4+20*x^2)*exp(x)+x^8+6*x^7+x^6-24*x^5-4*x^4-60*x^3+80*x^2+100)/x^4)/x^4
,x,method=_RETURNVERBOSE)

[Out]

x*exp(-(-x^8+2*x^6*exp(x)-6*x^7+6*x^5*exp(x)-x^6-8*exp(x)*x^4-exp(2*x)*x^4+24*x^5+4*x^4-20*exp(x)*x^2+60*x^3-8
0*x^2-100)/x^4)

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maxima [B]  time = 0.65, size = 61, normalized size = 2.44 \begin {gather*} x e^{\left (x^{4} + 6 \, x^{3} - 2 \, x^{2} e^{x} + x^{2} - 6 \, x e^{x} - 24 \, x - \frac {60}{x} + \frac {20 \, e^{x}}{x^{2}} + \frac {80}{x^{2}} + \frac {100}{x^{4}} + e^{\left (2 \, x\right )} + 8 \, e^{x} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x^5*exp(x)^2+(-2*x^7-10*x^6+2*x^5+20*x^3-40*x^2)*exp(x)+4*x^8+18*x^7+2*x^6-24*x^5+x^4+60*x^3-160*
x^2-400)*exp((exp(x)^2*x^4+(-2*x^6-6*x^5+8*x^4+20*x^2)*exp(x)+x^8+6*x^7+x^6-24*x^5-4*x^4-60*x^3+80*x^2+100)/x^
4)/x^4,x, algorithm="maxima")

[Out]

x*e^(x^4 + 6*x^3 - 2*x^2*e^x + x^2 - 6*x*e^x - 24*x - 60/x + 20*e^x/x^2 + 80/x^2 + 100/x^4 + e^(2*x) + 8*e^x -
4)

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mupad [B]  time = 3.55, size = 72, normalized size = 2.88 \begin {gather*} x\,{\mathrm {e}}^{-6\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{-24\,x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{-4}\,{\mathrm {e}}^{-2\,x^2\,{\mathrm {e}}^x}\,{\mathrm {e}}^{\frac {20\,{\mathrm {e}}^x}{x^2}}\,{\mathrm {e}}^{6\,x^3}\,{\mathrm {e}}^{-\frac {60}{x}}\,{\mathrm {e}}^{\frac {80}{x^2}}\,{\mathrm {e}}^{\frac {100}{x^4}}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,{\mathrm {e}}^{8\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((x^4*exp(2*x) + exp(x)*(20*x^2 + 8*x^4 - 6*x^5 - 2*x^6) + 80*x^2 - 60*x^3 - 4*x^4 - 24*x^5 + x^6 + 6*
x^7 + x^8 + 100)/x^4)*(2*x^5*exp(2*x) - exp(x)*(40*x^2 - 20*x^3 - 2*x^5 + 10*x^6 + 2*x^7) - 160*x^2 + 60*x^3 +
x^4 - 24*x^5 + 2*x^6 + 18*x^7 + 4*x^8 - 400))/x^4,x)

[Out]

x*exp(-6*x*exp(x))*exp(-24*x)*exp(x^2)*exp(x^4)*exp(-4)*exp(-2*x^2*exp(x))*exp((20*exp(x))/x^2)*exp(6*x^3)*exp
(-60/x)*exp(80/x^2)*exp(100/x^4)*exp(exp(2*x))*exp(8*exp(x))

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sympy [B]  time = 4.07, size = 71, normalized size = 2.84 \begin {gather*} x e^{\frac {x^{8} + 6 x^{7} + x^{6} - 24 x^{5} + x^{4} e^{2 x} - 4 x^{4} - 60 x^{3} + 80 x^{2} + \left (- 2 x^{6} - 6 x^{5} + 8 x^{4} + 20 x^{2}\right ) e^{x} + 100}{x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*x**5*exp(x)**2+(-2*x**7-10*x**6+2*x**5+20*x**3-40*x**2)*exp(x)+4*x**8+18*x**7+2*x**6-24*x**5+x**4
+60*x**3-160*x**2-400)*exp((exp(x)**2*x**4+(-2*x**6-6*x**5+8*x**4+20*x**2)*exp(x)+x**8+6*x**7+x**6-24*x**5-4*x
**4-60*x**3+80*x**2+100)/x**4)/x**4,x)

[Out]

x*exp((x**8 + 6*x**7 + x**6 - 24*x**5 + x**4*exp(2*x) - 4*x**4 - 60*x**3 + 80*x**2 + (-2*x**6 - 6*x**5 + 8*x**
4 + 20*x**2)*exp(x) + 100)/x**4)

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