3.90.40 \(\int \frac {864+32 x-2 x^2+e^{-3 e^5+3 x} (576+1728 x-216 x^2)+e^{-6 e^5+6 x} (96+576 x-72 x^2)}{2916 x^2+432 e^{-9 e^5+9 x} x^2+36 e^{-12 e^5+12 x} x^2+108 x^3+x^4+e^{-6 e^5+6 x} (1944 x^2+12 x^3)+e^{-3 e^5+3 x} (3888 x^2+72 x^3)} \, dx\)

Optimal. Leaf size=32 \[ \frac {-8+x}{\left (3 \left (3+e^{-3 \left (e^5-x\right )}\right )^2+\frac {x}{2}\right ) x} \]

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Rubi [F]  time = 5.08, 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 {864+32 x-2 x^2+e^{-3 e^5+3 x} \left (576+1728 x-216 x^2\right )+e^{-6 e^5+6 x} \left (96+576 x-72 x^2\right )}{2916 x^2+432 e^{-9 e^5+9 x} x^2+36 e^{-12 e^5+12 x} x^2+108 x^3+x^4+e^{-6 e^5+6 x} \left (1944 x^2+12 x^3\right )+e^{-3 e^5+3 x} \left (3888 x^2+72 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(864 + 32*x - 2*x^2 + E^(-3*E^5 + 3*x)*(576 + 1728*x - 216*x^2) + E^(-6*E^5 + 6*x)*(96 + 576*x - 72*x^2))/
(2916*x^2 + 432*E^(-9*E^5 + 9*x)*x^2 + 36*E^(-12*E^5 + 12*x)*x^2 + 108*x^3 + x^4 + E^(-6*E^5 + 6*x)*(1944*x^2
+ 12*x^3) + E^(-3*E^5 + 3*x)*(3888*x^2 + 72*x^3)),x]

[Out]

550*E^(12*E^5)*Defer[Int][(54*E^(6*E^5) + 6*E^(6*x) + 36*E^(3*E^5 + 3*x) + E^(6*E^5)*x)^(-2), x] + 216*E^(9*E^
5)*Defer[Int][E^(3*x)/(54*E^(6*E^5) + 6*E^(6*x) + 36*E^(3*E^5 + 3*x) + E^(6*E^5)*x)^2, x] - 5168*E^(12*E^5)*De
fer[Int][1/(x*(54*E^(6*E^5) + 6*E^(6*x) + 36*E^(3*E^5 + 3*x) + E^(6*E^5)*x)^2), x] - 1728*E^(9*E^5)*Defer[Int]
[E^(3*x)/(x*(54*E^(6*E^5) + 6*E^(6*x) + 36*E^(3*E^5 + 3*x) + E^(6*E^5)*x)^2), x] + 12*E^(12*E^5)*Defer[Int][x/
(54*E^(6*E^5) + 6*E^(6*x) + 36*E^(3*E^5 + 3*x) + E^(6*E^5)*x)^2, x] - 12*E^(6*E^5)*Defer[Int][(54*E^(6*E^5) +
6*E^(6*x) + 36*E^(3*E^5 + 3*x) + E^(6*E^5)*x)^(-1), x] + 16*E^(6*E^5)*Defer[Int][1/(x^2*(54*E^(6*E^5) + 6*E^(6
*x) + 36*E^(3*E^5 + 3*x) + E^(6*E^5)*x)), x] + 96*E^(6*E^5)*Defer[Int][1/(x*(54*E^(6*E^5) + 6*E^(6*x) + 36*E^(
3*E^5 + 3*x) + E^(6*E^5)*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 e^{6 e^5} \left (-e^{6 e^5} \left (-432-16 x+x^2\right )-36 e^{3 \left (e^5+x\right )} \left (-8-24 x+3 x^2\right )-12 e^{6 x} \left (-4-24 x+3 x^2\right )\right )}{x^2 \left (6 e^{6 x}+36 e^{3 \left (e^5+x\right )}+e^{6 e^5} (54+x)\right )^2} \, dx\\ &=\left (2 e^{6 e^5}\right ) \int \frac {-e^{6 e^5} \left (-432-16 x+x^2\right )-36 e^{3 \left (e^5+x\right )} \left (-8-24 x+3 x^2\right )-12 e^{6 x} \left (-4-24 x+3 x^2\right )}{x^2 \left (6 e^{6 x}+36 e^{3 \left (e^5+x\right )}+e^{6 e^5} (54+x)\right )^2} \, dx\\ &=\left (2 e^{6 e^5}\right ) \int \left (\frac {e^{3 e^5} (-8+x) \left (323 e^{3 e^5}+108 e^{3 x}+6 e^{3 e^5} x\right )}{x \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2}-\frac {2 \left (-4-24 x+3 x^2\right )}{x^2 \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )}\right ) \, dx\\ &=-\left (\left (4 e^{6 e^5}\right ) \int \frac {-4-24 x+3 x^2}{x^2 \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )} \, dx\right )+\left (2 e^{9 e^5}\right ) \int \frac {(-8+x) \left (323 e^{3 e^5}+108 e^{3 x}+6 e^{3 e^5} x\right )}{x \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2} \, dx\\ &=-\left (\left (4 e^{6 e^5}\right ) \int \left (\frac {3}{54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x}-\frac {4}{x^2 \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )}-\frac {24}{x \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )}\right ) \, dx\right )+\left (2 e^{9 e^5}\right ) \int \left (\frac {323 e^{3 e^5}+108 e^{3 x}+6 e^{3 e^5} x}{\left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2}-\frac {8 \left (323 e^{3 e^5}+108 e^{3 x}+6 e^{3 e^5} x\right )}{x \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2}\right ) \, dx\\ &=-\left (\left (12 e^{6 e^5}\right ) \int \frac {1}{54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x} \, dx\right )+\left (16 e^{6 e^5}\right ) \int \frac {1}{x^2 \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )} \, dx+\left (96 e^{6 e^5}\right ) \int \frac {1}{x \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )} \, dx+\left (2 e^{9 e^5}\right ) \int \frac {323 e^{3 e^5}+108 e^{3 x}+6 e^{3 e^5} x}{\left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2} \, dx-\left (16 e^{9 e^5}\right ) \int \frac {323 e^{3 e^5}+108 e^{3 x}+6 e^{3 e^5} x}{x \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2} \, dx\\ &=-\left (\left (12 e^{6 e^5}\right ) \int \frac {1}{54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x} \, dx\right )+\left (16 e^{6 e^5}\right ) \int \frac {1}{x^2 \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )} \, dx+\left (96 e^{6 e^5}\right ) \int \frac {1}{x \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )} \, dx+\left (2 e^{9 e^5}\right ) \int \left (\frac {323 e^{3 e^5}}{\left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2}+\frac {108 e^{3 x}}{\left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2}+\frac {6 e^{3 e^5} x}{\left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2}\right ) \, dx-\left (16 e^{9 e^5}\right ) \int \left (\frac {6 e^{3 e^5}}{\left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2}+\frac {323 e^{3 e^5}}{x \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2}+\frac {108 e^{3 x}}{x \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2}\right ) \, dx\\ &=-\left (\left (12 e^{6 e^5}\right ) \int \frac {1}{54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x} \, dx\right )+\left (16 e^{6 e^5}\right ) \int \frac {1}{x^2 \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )} \, dx+\left (96 e^{6 e^5}\right ) \int \frac {1}{x \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )} \, dx+\left (216 e^{9 e^5}\right ) \int \frac {e^{3 x}}{\left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2} \, dx-\left (1728 e^{9 e^5}\right ) \int \frac {e^{3 x}}{x \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2} \, dx+\left (12 e^{12 e^5}\right ) \int \frac {x}{\left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2} \, dx-\left (96 e^{12 e^5}\right ) \int \frac {1}{\left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2} \, dx+\left (646 e^{12 e^5}\right ) \int \frac {1}{\left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2} \, dx-\left (5168 e^{12 e^5}\right ) \int \frac {1}{x \left (54 e^{6 e^5}+6 e^{6 x}+36 e^{3 e^5+3 x}+e^{6 e^5} x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 49, normalized size = 1.53 \begin {gather*} -\frac {2 e^{6 e^5} (8-x)}{x \left (6 e^{6 x}+36 e^{3 \left (e^5+x\right )}+e^{6 e^5} (54+x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(864 + 32*x - 2*x^2 + E^(-3*E^5 + 3*x)*(576 + 1728*x - 216*x^2) + E^(-6*E^5 + 6*x)*(96 + 576*x - 72*
x^2))/(2916*x^2 + 432*E^(-9*E^5 + 9*x)*x^2 + 36*E^(-12*E^5 + 12*x)*x^2 + 108*x^3 + x^4 + E^(-6*E^5 + 6*x)*(194
4*x^2 + 12*x^3) + E^(-3*E^5 + 3*x)*(3888*x^2 + 72*x^3)),x]

[Out]

(-2*E^(6*E^5)*(8 - x))/(x*(6*E^(6*x) + 36*E^(3*(E^5 + x)) + E^(6*E^5)*(54 + x)))

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fricas [A]  time = 0.52, size = 38, normalized size = 1.19 \begin {gather*} \frac {2 \, {\left (x - 8\right )}}{x^{2} + 6 \, x e^{\left (6 \, x - 6 \, e^{5}\right )} + 36 \, x e^{\left (3 \, x - 3 \, e^{5}\right )} + 54 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*x^2+576*x+96)*exp(-3*exp(5)+3*x)^2+(-216*x^2+1728*x+576)*exp(-3*exp(5)+3*x)-2*x^2+32*x+864)/(3
6*x^2*exp(-3*exp(5)+3*x)^4+432*x^2*exp(-3*exp(5)+3*x)^3+(12*x^3+1944*x^2)*exp(-3*exp(5)+3*x)^2+(72*x^3+3888*x^
2)*exp(-3*exp(5)+3*x)+x^4+108*x^3+2916*x^2),x, algorithm="fricas")

[Out]

2*(x - 8)/(x^2 + 6*x*e^(6*x - 6*e^5) + 36*x*e^(3*x - 3*e^5) + 54*x)

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giac [B]  time = 0.39, size = 368, normalized size = 11.50 \begin {gather*} \frac {4 \, {\left (36 \, x^{5} e^{\left (6 \, e^{5}\right )} + 216 \, x^{4} e^{\left (6 \, x\right )} + 1296 \, x^{4} e^{\left (3 \, x + 3 \, e^{5}\right )} + 3588 \, x^{4} e^{\left (6 \, e^{5}\right )} + 9864 \, x^{3} e^{\left (6 \, x\right )} + 59184 \, x^{3} e^{\left (3 \, x + 3 \, e^{5}\right )} + 73321 \, x^{3} e^{\left (6 \, e^{5}\right )} - 92730 \, x^{2} e^{\left (6 \, x\right )} - 556380 \, x^{2} e^{\left (3 \, x + 3 \, e^{5}\right )} - 834578 \, x^{2} e^{\left (6 \, e^{5}\right )} - 48 \, x e^{\left (6 \, x\right )} - 288 \, x e^{\left (3 \, x + 3 \, e^{5}\right )} - 432 \, x e^{\left (6 \, e^{5}\right )}\right )}}{36 \, x^{6} e^{\left (6 \, e^{5}\right )} + 432 \, x^{5} e^{\left (6 \, x\right )} + 2592 \, x^{5} e^{\left (3 \, x + 3 \, e^{5}\right )} + 5820 \, x^{5} e^{\left (6 \, e^{5}\right )} + 93168 \, x^{4} e^{\left (6 \, x\right )} + 1296 \, x^{4} e^{\left (12 \, x - 6 \, e^{5}\right )} + 15552 \, x^{4} e^{\left (9 \, x - 3 \, e^{5}\right )} + 279072 \, x^{4} e^{\left (3 \, x + 3 \, e^{5}\right )} + 313633 \, x^{4} e^{\left (6 \, e^{5}\right )} + 3755820 \, x^{3} e^{\left (6 \, x\right )} + 69552 \, x^{3} e^{\left (12 \, x - 6 \, e^{5}\right )} + 834624 \, x^{3} e^{\left (9 \, x - 3 \, e^{5}\right )} + 7511688 \, x^{3} e^{\left (3 \, x + 3 \, e^{5}\right )} + 5633820 \, x^{3} e^{\left (6 \, e^{5}\right )} + 1944 \, x^{2} e^{\left (6 \, x\right )} + 36 \, x^{2} e^{\left (12 \, x - 6 \, e^{5}\right )} + 432 \, x^{2} e^{\left (9 \, x - 3 \, e^{5}\right )} + 3888 \, x^{2} e^{\left (3 \, x + 3 \, e^{5}\right )} + 2916 \, x^{2} e^{\left (6 \, e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*x^2+576*x+96)*exp(-3*exp(5)+3*x)^2+(-216*x^2+1728*x+576)*exp(-3*exp(5)+3*x)-2*x^2+32*x+864)/(3
6*x^2*exp(-3*exp(5)+3*x)^4+432*x^2*exp(-3*exp(5)+3*x)^3+(12*x^3+1944*x^2)*exp(-3*exp(5)+3*x)^2+(72*x^3+3888*x^
2)*exp(-3*exp(5)+3*x)+x^4+108*x^3+2916*x^2),x, algorithm="giac")

[Out]

4*(36*x^5*e^(6*e^5) + 216*x^4*e^(6*x) + 1296*x^4*e^(3*x + 3*e^5) + 3588*x^4*e^(6*e^5) + 9864*x^3*e^(6*x) + 591
84*x^3*e^(3*x + 3*e^5) + 73321*x^3*e^(6*e^5) - 92730*x^2*e^(6*x) - 556380*x^2*e^(3*x + 3*e^5) - 834578*x^2*e^(
6*e^5) - 48*x*e^(6*x) - 288*x*e^(3*x + 3*e^5) - 432*x*e^(6*e^5))/(36*x^6*e^(6*e^5) + 432*x^5*e^(6*x) + 2592*x^
5*e^(3*x + 3*e^5) + 5820*x^5*e^(6*e^5) + 93168*x^4*e^(6*x) + 1296*x^4*e^(12*x - 6*e^5) + 15552*x^4*e^(9*x - 3*
e^5) + 279072*x^4*e^(3*x + 3*e^5) + 313633*x^4*e^(6*e^5) + 3755820*x^3*e^(6*x) + 69552*x^3*e^(12*x - 6*e^5) +
834624*x^3*e^(9*x - 3*e^5) + 7511688*x^3*e^(3*x + 3*e^5) + 5633820*x^3*e^(6*e^5) + 1944*x^2*e^(6*x) + 36*x^2*e
^(12*x - 6*e^5) + 432*x^2*e^(9*x - 3*e^5) + 3888*x^2*e^(3*x + 3*e^5) + 2916*x^2*e^(6*e^5))

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maple [A]  time = 1.62, size = 36, normalized size = 1.12

method	result	size
risch	\(\frac {2 x -16}{x \left (6 \,{\mathrm e}^{-6 \,{\mathrm e}^{5}+6 x}+36 \,{\mathrm e}^{-3 \,{\mathrm e}^{5}+3 x}+x +54\right )}\)	\(36\)
norman	\(\frac {2 x -16}{x \left (6 \,{\mathrm e}^{-6 \,{\mathrm e}^{5}+6 x}+36 \,{\mathrm e}^{-3 \,{\mathrm e}^{5}+3 x}+x +54\right )}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-72*x^2+576*x+96)*exp(-3*exp(5)+3*x)^2+(-216*x^2+1728*x+576)*exp(-3*exp(5)+3*x)-2*x^2+32*x+864)/(36*x^2*
exp(-3*exp(5)+3*x)^4+432*x^2*exp(-3*exp(5)+3*x)^3+(12*x^3+1944*x^2)*exp(-3*exp(5)+3*x)^2+(72*x^3+3888*x^2)*exp
(-3*exp(5)+3*x)+x^4+108*x^3+2916*x^2),x,method=_RETURNVERBOSE)

[Out]

2*(-8+x)/x/(6*exp(-6*exp(5)+6*x)+36*exp(-3*exp(5)+3*x)+x+54)

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maxima [B]  time = 0.43, size = 56, normalized size = 1.75 \begin {gather*} \frac {2 \, {\left (x e^{\left (6 \, e^{5}\right )} - 8 \, e^{\left (6 \, e^{5}\right )}\right )}}{x^{2} e^{\left (6 \, e^{5}\right )} + 6 \, x e^{\left (6 \, x\right )} + 36 \, x e^{\left (3 \, x + 3 \, e^{5}\right )} + 54 \, x e^{\left (6 \, e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*x^2+576*x+96)*exp(-3*exp(5)+3*x)^2+(-216*x^2+1728*x+576)*exp(-3*exp(5)+3*x)-2*x^2+32*x+864)/(3
6*x^2*exp(-3*exp(5)+3*x)^4+432*x^2*exp(-3*exp(5)+3*x)^3+(12*x^3+1944*x^2)*exp(-3*exp(5)+3*x)^2+(72*x^3+3888*x^
2)*exp(-3*exp(5)+3*x)+x^4+108*x^3+2916*x^2),x, algorithm="maxima")

[Out]

2*(x*e^(6*e^5) - 8*e^(6*e^5))/(x^2*e^(6*e^5) + 6*x*e^(6*x) + 36*x*e^(3*x + 3*e^5) + 54*x*e^(6*e^5))

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mupad [B]  time = 7.07, size = 63, normalized size = 1.97 \begin {gather*} -\frac {2\,\left (-36\,x^4-1644\,x^3+15455\,x^2+8\,x\right )}{x^2\,\left (36\,x^2+1932\,x+1\right )\,\left (x+36\,{\mathrm {e}}^{3\,x-3\,{\mathrm {e}}^5}+6\,{\mathrm {e}}^{6\,x-6\,{\mathrm {e}}^5}+54\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((32*x + exp(6*x - 6*exp(5))*(576*x - 72*x^2 + 96) + exp(3*x - 3*exp(5))*(1728*x - 216*x^2 + 576) - 2*x^2 +
864)/(exp(6*x - 6*exp(5))*(1944*x^2 + 12*x^3) + exp(3*x - 3*exp(5))*(3888*x^2 + 72*x^3) + 432*x^2*exp(9*x - 9
*exp(5)) + 36*x^2*exp(12*x - 12*exp(5)) + 2916*x^2 + 108*x^3 + x^4),x)

[Out]

-(2*(8*x + 15455*x^2 - 1644*x^3 - 36*x^4))/(x^2*(1932*x + 36*x^2 + 1)*(x + 36*exp(3*x - 3*exp(5)) + 6*exp(6*x
- 6*exp(5)) + 54))

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sympy [A]  time = 0.23, size = 37, normalized size = 1.16 \begin {gather*} \frac {2 x - 16}{x^{2} + 36 x e^{3 x - 3 e^{5}} + 6 x e^{6 x - 6 e^{5}} + 54 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-72*x**2+576*x+96)*exp(-3*exp(5)+3*x)**2+(-216*x**2+1728*x+576)*exp(-3*exp(5)+3*x)-2*x**2+32*x+864
)/(36*x**2*exp(-3*exp(5)+3*x)**4+432*x**2*exp(-3*exp(5)+3*x)**3+(12*x**3+1944*x**2)*exp(-3*exp(5)+3*x)**2+(72*
x**3+3888*x**2)*exp(-3*exp(5)+3*x)+x**4+108*x**3+2916*x**2),x)

[Out]

(2*x - 16)/(x**2 + 36*x*exp(3*x - 3*exp(5)) + 6*x*exp(6*x - 6*exp(5)) + 54*x)

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