3.12.63 \(\int \frac {e^{\frac {4096 x^2-x^3+8192 x \log (4)+4096 \log ^2(4)+(-8192 x-8192 \log (4)) \log (x)+4096 \log ^2(x)}{x^2}} (-24576 x-3 x^3+(-24576-24576 x) \log (4)-24576 \log ^2(4)+(24576+24576 x+49152 \log (4)) \log (x)-24576 \log ^2(x))}{x^3} \, dx\) [1163]

Optimal. Leaf size=23 \[ 3 e^{-x+\frac {4096 (x+\log (4)-\log (x))^2}{x^2}} \]

[Out]

3*exp(4096/x^2*(2*ln(2)+x-ln(x))^2-x)

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Rubi [F]
time = 6.89, 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 {\exp \left (\frac {4096 x^2-x^3+8192 x \log (4)+4096 \log ^2(4)+(-8192 x-8192 \log (4)) \log (x)+4096 \log ^2(x)}{x^2}\right ) \left (-24576 x-3 x^3+(-24576-24576 x) \log (4)-24576 \log ^2(4)+(24576+24576 x+49152 \log (4)) \log (x)-24576 \log ^2(x)\right )}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((4096*x^2 - x^3 + 8192*x*Log[4] + 4096*Log[4]^2 + (-8192*x - 8192*Log[4])*Log[x] + 4096*Log[x]^2)/x^2)
*(-24576*x - 3*x^3 + (-24576 - 24576*x)*Log[4] - 24576*Log[4]^2 + (24576 + 24576*x + 49152*Log[4])*Log[x] - 24
576*Log[x]^2))/x^3,x]

[Out]

-3*Defer[Int][E^(4096 - x + (8192*Log[4])/x + (4096*Log[4]^2)/x^2 + ((-8192*x - 8192*Log[4])*Log[x])/x^2 + (40
96*Log[x]^2)/x^2), x] - 24576*Log[4]*(1 + Log[4])*Defer[Int][E^(4096 - x + (8192*Log[4])/x + (4096*Log[4]^2)/x
^2 + ((-8192*x - 8192*Log[4])*Log[x])/x^2 + (4096*Log[x]^2)/x^2)/x^3, x] - 24576*(1 + Log[4])*Defer[Int][E^(40
96 - x + (8192*Log[4])/x + (4096*Log[4]^2)/x^2 + ((-8192*x - 8192*Log[4])*Log[x])/x^2 + (4096*Log[x]^2)/x^2)/x
^2, x] + 24576*(1 + Log[16])*Defer[Int][(E^(4096 - x + (8192*Log[4])/x + (4096*Log[4]^2)/x^2 + ((-8192*x - 819
2*Log[4])*Log[x])/x^2 + (4096*Log[x]^2)/x^2)*Log[x])/x^3, x] + 24576*Defer[Int][(E^(4096 - x + (8192*Log[4])/x
+ (4096*Log[4]^2)/x^2 + ((-8192*x - 8192*Log[4])*Log[x])/x^2 + (4096*Log[x]^2)/x^2)*Log[x])/x^2, x] - 24576*D
efer[Int][(E^(4096 - x + (8192*Log[4])/x + (4096*Log[4]^2)/x^2 + ((-8192*x - 8192*Log[4])*Log[x])/x^2 + (4096*
Log[x]^2)/x^2)*Log[x]^2)/x^3, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) \left (-24576 x-3 x^3+(-24576-24576 x) \log (4)-24576 \log ^2(4)+(24576+24576 x+49152 \log (4)) \log (x)-24576 \log ^2(x)\right )}{x^3} \, dx\\ &=\int \left (\frac {3 \exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) \left (-x^3-8192 x (1+\log (4))-8192 \log (4) (1+\log (4))\right )}{x^3}+\frac {24576 \exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) (1+x+\log (16)) \log (x)}{x^3}-\frac {24576 \exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) \log ^2(x)}{x^3}\right ) \, dx\\ &=3 \int \frac {\exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) \left (-x^3-8192 x (1+\log (4))-8192 \log (4) (1+\log (4))\right )}{x^3} \, dx+24576 \int \frac {\exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) (1+x+\log (16)) \log (x)}{x^3} \, dx-24576 \int \frac {\exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) \log ^2(x)}{x^3} \, dx\\ &=3 \int \left (-\exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right )-\frac {8192 \exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) (1+\log (4))}{x^2}-\frac {8192 \exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) \log (4) (1+\log (4))}{x^3}\right ) \, dx-24576 \int \frac {\exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) \log ^2(x)}{x^3} \, dx+24576 \int \left (\frac {\exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) \log (x)}{x^2}+\frac {\exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) (1+\log (16)) \log (x)}{x^3}\right ) \, dx\\ &=-\left (3 \int \exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) \, dx\right )+24576 \int \frac {\exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) \log (x)}{x^2} \, dx-24576 \int \frac {\exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) \log ^2(x)}{x^3} \, dx-(24576 (1+\log (4))) \int \frac {\exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right )}{x^2} \, dx-(24576 \log (4) (1+\log (4))) \int \frac {\exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right )}{x^3} \, dx+(24576 (1+\log (16))) \int \frac {\exp \left (4096-x+\frac {8192 \log (4)}{x}+\frac {4096 \log ^2(4)}{x^2}+\frac {(-8192 x-8192 \log (4)) \log (x)}{x^2}+\frac {4096 \log ^2(x)}{x^2}\right ) \log (x)}{x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]
time = 0.60, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{\frac {4096 x^2-x^3+8192 x \log (4)+4096 \log ^2(4)+(-8192 x-8192 \log (4)) \log (x)+4096 \log ^2(x)}{x^2}} \left (-24576 x-3 x^3+(-24576-24576 x) \log (4)-24576 \log ^2(4)+(24576+24576 x+49152 \log (4)) \log (x)-24576 \log ^2(x)\right )}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^((4096*x^2 - x^3 + 8192*x*Log[4] + 4096*Log[4]^2 + (-8192*x - 8192*Log[4])*Log[x] + 4096*Log[x]^2
)/x^2)*(-24576*x - 3*x^3 + (-24576 - 24576*x)*Log[4] - 24576*Log[4]^2 + (24576 + 24576*x + 49152*Log[4])*Log[x
] - 24576*Log[x]^2))/x^3,x]

[Out]

Integrate[(E^((4096*x^2 - x^3 + 8192*x*Log[4] + 4096*Log[4]^2 + (-8192*x - 8192*Log[4])*Log[x] + 4096*Log[x]^2
)/x^2)*(-24576*x - 3*x^3 + (-24576 - 24576*x)*Log[4] - 24576*Log[4]^2 + (24576 + 24576*x + 49152*Log[4])*Log[x
] - 24576*Log[x]^2))/x^3, x]

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Maple [A]
time = 0.06, size = 47, normalized size = 2.04

method	result	size
risch	\(3 \,{\mathrm e}^{\frac {-x^{3}+4096 \ln \left (x \right )^{2}-16384 \ln \left (2\right ) \ln \left (x \right )-8192 x \ln \left (x \right )+16384 \ln \left (2\right )^{2}+16384 x \ln \left (2\right )+4096 x^{2}}{x^{2}}}\)	\(47\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-24576*ln(x)^2+(98304*ln(2)+24576*x+24576)*ln(x)-98304*ln(2)^2+2*(-24576*x-24576)*ln(2)-3*x^3-24576*x)*ex
p((4096*ln(x)^2+(-16384*ln(2)-8192*x)*ln(x)+16384*ln(2)^2+16384*x*ln(2)-x^3+4096*x^2)/x^2)/x^3,x,method=_RETUR
NVERBOSE)

[Out]

3*exp((-x^3+4096*ln(x)^2-16384*ln(2)*ln(x)-8192*x*ln(x)+16384*ln(2)^2+16384*x*ln(2)+4096*x^2)/x^2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
time = 2.15, size = 49, normalized size = 2.13 \begin {gather*} 3 \, e^{\left (-x + \frac {16384 \, \log \left (2\right )}{x} + \frac {16384 \, \log \left (2\right )^{2}}{x^{2}} - \frac {8192 \, \log \left (x\right )}{x} - \frac {16384 \, \log \left (2\right ) \log \left (x\right )}{x^{2}} + \frac {4096 \, \log \left (x\right )^{2}}{x^{2}} + 4096\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24576*log(x)^2+(98304*log(2)+24576*x+24576)*log(x)-98304*log(2)^2+2*(-24576*x-24576)*log(2)-3*x^3-
24576*x)*exp((4096*log(x)^2+(-16384*log(2)-8192*x)*log(x)+16384*log(2)^2+16384*x*log(2)-x^3+4096*x^2)/x^2)/x^3
,x, algorithm="maxima")

[Out]

3*e^(-x + 16384*log(2)/x + 16384*log(2)^2/x^2 - 8192*log(x)/x - 16384*log(2)*log(x)/x^2 + 4096*log(x)^2/x^2 +
4096)

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Fricas [A]
time = 0.36, size = 44, normalized size = 1.91 \begin {gather*} 3 \, e^{\left (-\frac {x^{3} - 4096 \, x^{2} - 16384 \, x \log \left (2\right ) - 16384 \, \log \left (2\right )^{2} + 8192 \, {\left (x + 2 \, \log \left (2\right )\right )} \log \left (x\right ) - 4096 \, \log \left (x\right )^{2}}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24576*log(x)^2+(98304*log(2)+24576*x+24576)*log(x)-98304*log(2)^2+2*(-24576*x-24576)*log(2)-3*x^3-
24576*x)*exp((4096*log(x)^2+(-16384*log(2)-8192*x)*log(x)+16384*log(2)^2+16384*x*log(2)-x^3+4096*x^2)/x^2)/x^3
,x, algorithm="fricas")

[Out]

3*e^(-(x^3 - 4096*x^2 - 16384*x*log(2) - 16384*log(2)^2 + 8192*(x + 2*log(2))*log(x) - 4096*log(x)^2)/x^2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (20) = 40\).
time = 0.22, size = 48, normalized size = 2.09 \begin {gather*} 3 e^{\frac {- x^{3} + 4096 x^{2} + 16384 x \log {\left (2 \right )} + \left (- 8192 x - 16384 \log {\left (2 \right )}\right ) \log {\left (x \right )} + 4096 \log {\left (x \right )}^{2} + 16384 \log {\left (2 \right )}^{2}}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24576*ln(x)**2+(98304*ln(2)+24576*x+24576)*ln(x)-98304*ln(2)**2+2*(-24576*x-24576)*ln(2)-3*x**3-24
576*x)*exp((4096*ln(x)**2+(-16384*ln(2)-8192*x)*ln(x)+16384*ln(2)**2+16384*x*ln(2)-x**3+4096*x**2)/x**2)/x**3,
x)

[Out]

3*exp((-x**3 + 4096*x**2 + 16384*x*log(2) + (-8192*x - 16384*log(2))*log(x) + 4096*log(x)**2 + 16384*log(2)**2
)/x**2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (24) = 48\).
time = 0.41, size = 49, normalized size = 2.13 \begin {gather*} 3 \, e^{\left (-x + \frac {16384 \, \log \left (2\right )}{x} + \frac {16384 \, \log \left (2\right )^{2}}{x^{2}} - \frac {8192 \, \log \left (x\right )}{x} - \frac {16384 \, \log \left (2\right ) \log \left (x\right )}{x^{2}} + \frac {4096 \, \log \left (x\right )^{2}}{x^{2}} + 4096\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-24576*log(x)^2+(98304*log(2)+24576*x+24576)*log(x)-98304*log(2)^2+2*(-24576*x-24576)*log(2)-3*x^3-
24576*x)*exp((4096*log(x)^2+(-16384*log(2)-8192*x)*log(x)+16384*log(2)^2+16384*x*log(2)-x^3+4096*x^2)/x^2)/x^3
,x, algorithm="giac")

[Out]

3*e^(-x + 16384*log(2)/x + 16384*log(2)^2/x^2 - 8192*log(x)/x - 16384*log(2)*log(x)/x^2 + 4096*log(x)^2/x^2 +
4096)

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Mupad [B]
time = 1.09, size = 52, normalized size = 2.26 \begin {gather*} \frac {3\,2^{16384/x}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{4096}\,{\mathrm {e}}^{\frac {16384\,{\ln \left (2\right )}^2}{x^2}}\,{\mathrm {e}}^{\frac {4096\,{\ln \left (x\right )}^2}{x^2}}}{x^{\frac {16384\,\ln \left (2\right )}{x^2}+\frac {8192}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((16384*x*log(2) + 4096*log(x)^2 - log(x)*(8192*x + 16384*log(2)) + 16384*log(2)^2 + 4096*x^2 - x^3)/
x^2)*(24576*x + 2*log(2)*(24576*x + 24576) + 24576*log(x)^2 - log(x)*(24576*x + 98304*log(2) + 24576) + 98304*
log(2)^2 + 3*x^3))/x^3,x)

[Out]

(3*2^(16384/x)*exp(-x)*exp(4096)*exp((16384*log(2)^2)/x^2)*exp((4096*log(x)^2)/x^2))/x^((16384*log(2))/x^2 + 8
192/x)

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