3.2.83 \(\int \frac {e^{\frac {16-8 x+7 x^3+x^6+(8 x+2 x^4) \log (\frac {16 x}{3})+x^2 \log ^2(\frac {16 x}{3})}{x^2}} (-32+16 x+7 x^3+2 x^4+4 x^6+(-8 x+2 x^2+4 x^4) \log (\frac {16 x}{3}))}{x^3} \, dx\)

Optimal. Leaf size=28 \[ e^{-\frac {8}{x}-x+\left (\frac {4}{x}+x^2+\log \left (\frac {16 x}{3}\right )\right )^2} \]

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Rubi [F]  time = 11.71, 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 {16-8 x+7 x^3+x^6+\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )+x^2 \log ^2\left (\frac {16 x}{3}\right )}{x^2}\right ) \left (-32+16 x+7 x^3+2 x^4+4 x^6+\left (-8 x+2 x^2+4 x^4\right ) \log \left (\frac {16 x}{3}\right )\right )}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((16 - 8*x + 7*x^3 + x^6 + (8*x + 2*x^4)*Log[(16*x)/3] + x^2*Log[(16*x)/3]^2)/x^2)*(-32 + 16*x + 7*x^3
+ 2*x^4 + 4*x^6 + (-8*x + 2*x^2 + 4*x^4)*Log[(16*x)/3]))/x^3,x]

[Out]

7*Defer[Int][E^(16/x^2 - 8/x + 7*x + x^4 + ((8*x + 2*x^4)*Log[(16*x)/3])/x^2 + Log[(16*x)/3]^2), x] - 32*Defer
[Int][E^(16/x^2 - 8/x + 7*x + x^4 + ((8*x + 2*x^4)*Log[(16*x)/3])/x^2 + Log[(16*x)/3]^2)/x^3, x] + 16*Defer[In
t][E^(16/x^2 - 8/x + 7*x + x^4 + ((8*x + 2*x^4)*Log[(16*x)/3])/x^2 + Log[(16*x)/3]^2)/x^2, x] + 2*Defer[Int][E
^(16/x^2 - 8/x + 7*x + x^4 + ((8*x + 2*x^4)*Log[(16*x)/3])/x^2 + Log[(16*x)/3]^2)*x, x] + 4*Defer[Int][E^(16/x
^2 - 8/x + 7*x + x^4 + ((8*x + 2*x^4)*Log[(16*x)/3])/x^2 + Log[(16*x)/3]^2)*x^3, x] - 8*Defer[Int][(E^(16/x^2
- 8/x + 7*x + x^4 + ((8*x + 2*x^4)*Log[(16*x)/3])/x^2 + Log[(16*x)/3]^2)*Log[(16*x)/3])/x^2, x] + 2*Defer[Int]
[(E^(16/x^2 - 8/x + 7*x + x^4 + ((8*x + 2*x^4)*Log[(16*x)/3])/x^2 + Log[(16*x)/3]^2)*Log[(16*x)/3])/x, x] + 4*
Defer[Int][E^(16/x^2 - 8/x + 7*x + x^4 + ((8*x + 2*x^4)*Log[(16*x)/3])/x^2 + Log[(16*x)/3]^2)*x*Log[(16*x)/3],
x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) \left (-32+16 x+7 x^3+2 x^4+4 x^6+\left (-8 x+2 x^2+4 x^4\right ) \log \left (\frac {16 x}{3}\right )\right )}{x^3} \, dx\\ &=\int \left (\frac {\exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) \left (-32+16 x+7 x^3+2 x^4+4 x^6\right )}{x^3}+\frac {2 \exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) \left (-4+x+2 x^3\right ) \log \left (\frac {16 x}{3}\right )}{x^2}\right ) \, dx\\ &=2 \int \frac {\exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) \left (-4+x+2 x^3\right ) \log \left (\frac {16 x}{3}\right )}{x^2} \, dx+\int \frac {\exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) \left (-32+16 x+7 x^3+2 x^4+4 x^6\right )}{x^3} \, dx\\ &=2 \int \left (-\frac {4 \exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\frac {\exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) \log \left (\frac {16 x}{3}\right )}{x}+2 \exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) x \log \left (\frac {16 x}{3}\right )\right ) \, dx+\int \left (7 \exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right )-\frac {32 \exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right )}{x^3}+\frac {16 \exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right )}{x^2}+2 \exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) x+4 \exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) x^3\right ) \, dx\\ &=2 \int \exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) x \, dx+2 \int \frac {\exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) \log \left (\frac {16 x}{3}\right )}{x} \, dx+4 \int \exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) x^3 \, dx+4 \int \exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) x \log \left (\frac {16 x}{3}\right ) \, dx+7 \int \exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) \, dx-8 \int \frac {\exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right ) \log \left (\frac {16 x}{3}\right )}{x^2} \, dx+16 \int \frac {\exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right )}{x^2} \, dx-32 \int \frac {\exp \left (\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {\left (8 x+2 x^4\right ) \log \left (\frac {16 x}{3}\right )}{x^2}+\log ^2\left (\frac {16 x}{3}\right )\right )}{x^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 43, normalized size = 1.54 \begin {gather*} e^{\frac {16}{x^2}-\frac {8}{x}+7 x+x^4+\frac {2 \left (4+x^3\right ) \log \left (\frac {16 x}{3}\right )}{x}+\log ^2\left (\frac {16 x}{3}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((16 - 8*x + 7*x^3 + x^6 + (8*x + 2*x^4)*Log[(16*x)/3] + x^2*Log[(16*x)/3]^2)/x^2)*(-32 + 16*x +
7*x^3 + 2*x^4 + 4*x^6 + (-8*x + 2*x^2 + 4*x^4)*Log[(16*x)/3]))/x^3,x]

[Out]

E^(16/x^2 - 8/x + 7*x + x^4 + (2*(4 + x^3)*Log[(16*x)/3])/x + Log[(16*x)/3]^2)

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fricas [A]  time = 0.81, size = 41, normalized size = 1.46 \begin {gather*} e^{\left (\frac {x^{6} + x^{2} \log \left (\frac {16}{3} \, x\right )^{2} + 7 \, x^{3} + 2 \, {\left (x^{4} + 4 \, x\right )} \log \left (\frac {16}{3} \, x\right ) - 8 \, x + 16}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+2*x^2-8*x)*log(16/3*x)+4*x^6+2*x^4+7*x^3+16*x-32)*exp((x^2*log(16/3*x)^2+(2*x^4+8*x)*log(16/
3*x)+x^6+7*x^3-8*x+16)/x^2)/x^3,x, algorithm="fricas")

[Out]

e^((x^6 + x^2*log(16/3*x)^2 + 7*x^3 + 2*(x^4 + 4*x)*log(16/3*x) - 8*x + 16)/x^2)

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giac [A]  time = 1.15, size = 42, normalized size = 1.50 \begin {gather*} e^{\left (x^{4} + 2 \, x^{2} \log \left (\frac {16}{3} \, x\right ) + \log \left (\frac {16}{3} \, x\right )^{2} + 7 \, x + \frac {8 \, \log \left (\frac {16}{3} \, x\right )}{x} - \frac {8}{x} + \frac {16}{x^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+2*x^2-8*x)*log(16/3*x)+4*x^6+2*x^4+7*x^3+16*x-32)*exp((x^2*log(16/3*x)^2+(2*x^4+8*x)*log(16/
3*x)+x^6+7*x^3-8*x+16)/x^2)/x^3,x, algorithm="giac")

[Out]

e^(x^4 + 2*x^2*log(16/3*x) + log(16/3*x)^2 + 7*x + 8*log(16/3*x)/x - 8/x + 16/x^2)

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maple [A]  time = 0.04, size = 48, normalized size = 1.71

method	result	size
risch	\(\left (\frac {16 x}{3}\right )^{2 x^{2}} \left (\frac {16 x}{3}\right )^{\frac {8}{x}} {\mathrm e}^{\frac {x^{6}+x^{2} \ln \left (\frac {16 x}{3}\right )^{2}+7 x^{3}-8 x +16}{x^{2}}}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^4+2*x^2-8*x)*ln(16/3*x)+4*x^6+2*x^4+7*x^3+16*x-32)*exp((x^2*ln(16/3*x)^2+(2*x^4+8*x)*ln(16/3*x)+x^6+
7*x^3-8*x+16)/x^2)/x^3,x,method=_RETURNVERBOSE)

[Out]

(16/3*x)^(2*x^2)*(16/3*x)^(8/x)*exp((x^6+x^2*ln(16/3*x)^2+7*x^3-8*x+16)/x^2)

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maxima [B]  time = 1.20, size = 95, normalized size = 3.39 \begin {gather*} \frac {e^{\left (x^{4} - 2 \, x^{2} \log \relax (3) + 8 \, x^{2} \log \relax (2) + 2 \, x^{2} \log \relax (x) + \log \relax (3)^{2} + 16 \, \log \relax (2)^{2} - 2 \, \log \relax (3) \log \relax (x) + 8 \, \log \relax (2) \log \relax (x) + \log \relax (x)^{2} + 7 \, x - \frac {8 \, \log \relax (3)}{x} + \frac {32 \, \log \relax (2)}{x} + \frac {8 \, \log \relax (x)}{x} - \frac {8}{x} + \frac {16}{x^{2}}\right )}}{2^{8 \, \log \relax (3)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^4+2*x^2-8*x)*log(16/3*x)+4*x^6+2*x^4+7*x^3+16*x-32)*exp((x^2*log(16/3*x)^2+(2*x^4+8*x)*log(16/
3*x)+x^6+7*x^3-8*x+16)/x^2)/x^3,x, algorithm="maxima")

[Out]

e^(x^4 - 2*x^2*log(3) + 8*x^2*log(2) + 2*x^2*log(x) + log(3)^2 + 16*log(2)^2 - 2*log(3)*log(x) + 8*log(2)*log(
x) + log(x)^2 + 7*x - 8*log(3)/x + 32*log(2)/x + 8*log(x)/x - 8/x + 16/x^2)/2^(8*log(3))

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mupad [B]  time = 0.38, size = 106, normalized size = 3.79 \begin {gather*} \frac {2^{8\,x^2}\,2^{32/x}\,x^{2\,x^2}\,x^{8/x}\,x^{8\,\ln \relax (2)}\,{\mathrm {e}}^{{\ln \relax (3)}^2}\,{\mathrm {e}}^{7\,x}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{16\,{\ln \relax (2)}^2}\,{\mathrm {e}}^{-\frac {8}{x}}\,{\mathrm {e}}^{\frac {16}{x^2}}\,{\mathrm {e}}^{{\ln \relax (x)}^2}}{2^{8\,\ln \relax (3)}\,3^{2\,x^2}\,3^{8/x}\,x^{2\,\ln \relax (3)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((log((16*x)/3)*(8*x + 2*x^4) - 8*x + 7*x^3 + x^6 + x^2*log((16*x)/3)^2 + 16)/x^2)*(16*x + log((16*x)/
3)*(2*x^2 - 8*x + 4*x^4) + 7*x^3 + 2*x^4 + 4*x^6 - 32))/x^3,x)

[Out]

(2^(8*x^2)*2^(32/x)*x^(2*x^2)*x^(8/x)*x^(8*log(2))*exp(log(3)^2)*exp(7*x)*exp(x^4)*exp(16*log(2)^2)*exp(-8/x)*
exp(16/x^2)*exp(log(x)^2))/(2^(8*log(3))*3^(2*x^2)*3^(8/x)*x^(2*log(3)))

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sympy [B]  time = 0.64, size = 44, normalized size = 1.57 \begin {gather*} e^{\frac {x^{6} + 7 x^{3} + x^{2} \log {\left (\frac {16 x}{3} \right )}^{2} - 8 x + \left (2 x^{4} + 8 x\right ) \log {\left (\frac {16 x}{3} \right )} + 16}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**4+2*x**2-8*x)*ln(16/3*x)+4*x**6+2*x**4+7*x**3+16*x-32)*exp((x**2*ln(16/3*x)**2+(2*x**4+8*x)*l
n(16/3*x)+x**6+7*x**3-8*x+16)/x**2)/x**3,x)

[Out]

exp((x**6 + 7*x**3 + x**2*log(16*x/3)**2 - 8*x + (2*x**4 + 8*x)*log(16*x/3) + 16)/x**2)

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