3.23.63 \(\int \frac {(-2 e x-2 e^{1+\frac {4}{x}} x) \log (x)+(e x+e^{1+\frac {4}{x}} x) \log ^2(x)+(-4 e x^2+e^{1+\frac {4}{x}} (16 x-4 x^2)+(e^{1+\frac {4}{x}} (-4+x)+e x) \log ^2(x)) \log (\frac {4 x-\log ^2(x)}{x})}{(-8 x^2+2 x \log ^2(x)) \log ^2(\frac {4 x-\log ^2(x)}{x})} \, dx\) [2263]

Optimal. Leaf size=30 \[ \frac {e \left (x+e^{4/x} x\right )}{2 \log \left (4-\frac {\log ^2(x)}{x}\right )} \]

[Out]

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

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Rubi [F]
time = 2.64, 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 {\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)+\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)+\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{\left (-8 x^2+2 x \log ^2(x)\right ) \log ^2\left (\frac {4 x-\log ^2(x)}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-2*E*x - 2*E^(1 + 4/x)*x)*Log[x] + (E*x + E^(1 + 4/x)*x)*Log[x]^2 + (-4*E*x^2 + E^(1 + 4/x)*(16*x - 4*x^
2) + (E^(1 + 4/x)*(-4 + x) + E*x)*Log[x]^2)*Log[(4*x - Log[x]^2)/x])/((-8*x^2 + 2*x*Log[x]^2)*Log[(4*x - Log[x
]^2)/x]^2),x]

[Out]

(E^(1 + 4/x)*x*(4*x*Log[4 - Log[x]^2/x] - Log[x]^2*Log[4 - Log[x]^2/x]))/(2*(4*x - Log[x]^2)*Log[4 - Log[x]^2/
x]^2) - E*Defer[Int][Log[x]/((-4*x + Log[x]^2)*Log[4 - Log[x]^2/x]^2), x] + (E*Defer[Int][Log[x]^2/((-4*x + Lo
g[x]^2)*Log[4 - Log[x]^2/x]^2), x])/2 + (E*Defer[Int][Log[4 - Log[x]^2/x]^(-1), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)+\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)+\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{x \left (-8 x+2 \log ^2(x)\right ) \log ^2\left (\frac {4 x-\log ^2(x)}{x}\right )} \, dx\\ &=\int \frac {-\left (\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)\right )-\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)-\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{2 x \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )} \, dx\\ &=\frac {1}{2} \int \frac {-\left (\left (-2 e x-2 e^{1+\frac {4}{x}} x\right ) \log (x)\right )-\left (e x+e^{1+\frac {4}{x}} x\right ) \log ^2(x)-\left (-4 e x^2+e^{1+\frac {4}{x}} \left (16 x-4 x^2\right )+\left (e^{1+\frac {4}{x}} (-4+x)+e x\right ) \log ^2(x)\right ) \log \left (\frac {4 x-\log ^2(x)}{x}\right )}{x \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )} \, dx\\ &=\frac {1}{2} \int \left (\frac {e \left (2 \log (x)-\log ^2(x)+4 x \log \left (4-\frac {\log ^2(x)}{x}\right )-\log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{\left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {e^{1+\frac {4}{x}} \left (2 x \log (x)-x \log ^2(x)-16 x \log \left (4-\frac {\log ^2(x)}{x}\right )+4 x^2 \log \left (4-\frac {\log ^2(x)}{x}\right )+4 \log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )-x \log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{x \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {e^{1+\frac {4}{x}} \left (2 x \log (x)-x \log ^2(x)-16 x \log \left (4-\frac {\log ^2(x)}{x}\right )+4 x^2 \log \left (4-\frac {\log ^2(x)}{x}\right )+4 \log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )-x \log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{x \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )} \, dx+\frac {1}{2} e \int \frac {2 \log (x)-\log ^2(x)+4 x \log \left (4-\frac {\log ^2(x)}{x}\right )-\log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )}{\left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )} \, dx\\ &=\frac {e^{1+\frac {4}{x}} x \left (4 x \log \left (4-\frac {\log ^2(x)}{x}\right )-\log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{2 \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {1}{2} e \int \left (\frac {(-2+\log (x)) \log (x)}{\left (-4 x+\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {1}{\log \left (4-\frac {\log ^2(x)}{x}\right )}\right ) \, dx\\ &=\frac {e^{1+\frac {4}{x}} x \left (4 x \log \left (4-\frac {\log ^2(x)}{x}\right )-\log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{2 \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {1}{2} e \int \frac {(-2+\log (x)) \log (x)}{\left (-4 x+\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )} \, dx+\frac {1}{2} e \int \frac {1}{\log \left (4-\frac {\log ^2(x)}{x}\right )} \, dx\\ &=\frac {e^{1+\frac {4}{x}} x \left (4 x \log \left (4-\frac {\log ^2(x)}{x}\right )-\log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{2 \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {1}{2} e \int \left (-\frac {2 \log (x)}{\left (-4 x+\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {\log ^2(x)}{\left (-4 x+\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}\right ) \, dx+\frac {1}{2} e \int \frac {1}{\log \left (4-\frac {\log ^2(x)}{x}\right )} \, dx\\ &=\frac {e^{1+\frac {4}{x}} x \left (4 x \log \left (4-\frac {\log ^2(x)}{x}\right )-\log ^2(x) \log \left (4-\frac {\log ^2(x)}{x}\right )\right )}{2 \left (4 x-\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )}+\frac {1}{2} e \int \frac {\log ^2(x)}{\left (-4 x+\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )} \, dx+\frac {1}{2} e \int \frac {1}{\log \left (4-\frac {\log ^2(x)}{x}\right )} \, dx-e \int \frac {\log (x)}{\left (-4 x+\log ^2(x)\right ) \log ^2\left (4-\frac {\log ^2(x)}{x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.08, size = 29, normalized size = 0.97 \begin {gather*} \frac {e \left (1+e^{4/x}\right ) x}{2 \log \left (4-\frac {\log ^2(x)}{x}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-2*E*x - 2*E^(1 + 4/x)*x)*Log[x] + (E*x + E^(1 + 4/x)*x)*Log[x]^2 + (-4*E*x^2 + E^(1 + 4/x)*(16*x
- 4*x^2) + (E^(1 + 4/x)*(-4 + x) + E*x)*Log[x]^2)*Log[(4*x - Log[x]^2)/x])/((-8*x^2 + 2*x*Log[x]^2)*Log[(4*x -
Log[x]^2)/x]^2),x]

[Out]

(E*(1 + E^(4/x))*x)/(2*Log[4 - Log[x]^2/x])

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 10.71, size = 166, normalized size = 5.53

method	result	size
risch	\(\frac {i \left ({\mathrm e}^{\frac {4}{x}}+1\right ) x \,{\mathrm e}}{\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{x}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{x}\right )^{2}+\pi \,\mathrm {csgn}\left (i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )\right ) \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{x}\right )^{2}-\pi \mathrm {csgn}\left (\frac {i \left (\frac {\ln \left (x \right )^{2}}{4}-x \right )}{x}\right )^{3}+4 i \ln \left (2\right )-2 i \ln \left (x \right )+2 i \ln \left (-\frac {\ln \left (x \right )^{2}}{4}+x \right )}\)	\(166\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((x-4)*exp(1)*exp(4/x)+x*exp(1))*ln(x)^2+(-4*x^2+16*x)*exp(1)*exp(4/x)-4*x^2*exp(1))*ln((-ln(x)^2+4*x)/x
)+(x*exp(1)*exp(4/x)+x*exp(1))*ln(x)^2+(-2*x*exp(1)*exp(4/x)-2*x*exp(1))*ln(x))/(2*x*ln(x)^2-8*x^2)/ln((-ln(x)
^2+4*x)/x)^2,x,method=_RETURNVERBOSE)

[Out]

I*(exp(4/x)+1)*x*exp(1)/(Pi*csgn(I/x)*csgn(I*(1/4*ln(x)^2-x))*csgn(I/x*(1/4*ln(x)^2-x))-Pi*csgn(I/x)*csgn(I/x*
(1/4*ln(x)^2-x))^2+Pi*csgn(I*(1/4*ln(x)^2-x))*csgn(I/x*(1/4*ln(x)^2-x))^2-Pi*csgn(I/x*(1/4*ln(x)^2-x))^3+4*I*l
n(2)-2*I*ln(x)+2*I*ln(-1/4*ln(x)^2+x))

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Maxima [A]
time = 0.32, size = 35, normalized size = 1.17 \begin {gather*} \frac {x e + x e^{\left (\frac {4}{x} + 1\right )}}{2 \, {\left (\log \left (-\log \left (x\right )^{2} + 4 \, x\right ) - \log \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-4)*exp(1)*exp(4/x)+x*exp(1))*log(x)^2+(-4*x^2+16*x)*exp(1)*exp(4/x)-4*x^2*exp(1))*log((-log(x)
^2+4*x)/x)+(x*exp(1)*exp(4/x)+x*exp(1))*log(x)^2+(-2*x*exp(1)*exp(4/x)-2*x*exp(1))*log(x))/(2*x*log(x)^2-8*x^2
)/log((-log(x)^2+4*x)/x)^2,x, algorithm="maxima")

[Out]

1/2*(x*e + x*e^(4/x + 1))/(log(-log(x)^2 + 4*x) - log(x))

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Fricas [A]
time = 0.36, size = 33, normalized size = 1.10 \begin {gather*} \frac {x e + x e^{\left (\frac {x + 4}{x}\right )}}{2 \, \log \left (-\frac {\log \left (x\right )^{2} - 4 \, x}{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-4)*exp(1)*exp(4/x)+x*exp(1))*log(x)^2+(-4*x^2+16*x)*exp(1)*exp(4/x)-4*x^2*exp(1))*log((-log(x)
^2+4*x)/x)+(x*exp(1)*exp(4/x)+x*exp(1))*log(x)^2+(-2*x*exp(1)*exp(4/x)-2*x*exp(1))*log(x))/(2*x*log(x)^2-8*x^2
)/log((-log(x)^2+4*x)/x)^2,x, algorithm="fricas")

[Out]

1/2*(x*e + x*e^((x + 4)/x))/log(-(log(x)^2 - 4*x)/x)

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Sympy [A]
time = 0.29, size = 41, normalized size = 1.37 \begin {gather*} \frac {e x e^{\frac {4}{x}}}{2 \log {\left (\frac {4 x - \log {\left (x \right )}^{2}}{x} \right )}} + \frac {e x}{2 \log {\left (\frac {4 x - \log {\left (x \right )}^{2}}{x} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-4)*exp(1)*exp(4/x)+x*exp(1))*ln(x)**2+(-4*x**2+16*x)*exp(1)*exp(4/x)-4*x**2*exp(1))*ln((-ln(x)
**2+4*x)/x)+(x*exp(1)*exp(4/x)+x*exp(1))*ln(x)**2+(-2*x*exp(1)*exp(4/x)-2*x*exp(1))*ln(x))/(2*x*ln(x)**2-8*x**
2)/ln((-ln(x)**2+4*x)/x)**2,x)

[Out]

E*x*exp(4/x)/(2*log((4*x - log(x)**2)/x)) + E*x/(2*log((4*x - log(x)**2)/x))

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Giac [A]
time = 0.51, size = 35, normalized size = 1.17 \begin {gather*} \frac {x e + x e^{\left (\frac {x + 4}{x}\right )}}{2 \, {\left (\log \left (-\log \left (x\right )^{2} + 4 \, x\right ) - \log \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-4)*exp(1)*exp(4/x)+x*exp(1))*log(x)^2+(-4*x^2+16*x)*exp(1)*exp(4/x)-4*x^2*exp(1))*log((-log(x)
^2+4*x)/x)+(x*exp(1)*exp(4/x)+x*exp(1))*log(x)^2+(-2*x*exp(1)*exp(4/x)-2*x*exp(1))*log(x))/(2*x*log(x)^2-8*x^2
)/log((-log(x)^2+4*x)/x)^2,x, algorithm="giac")

[Out]

1/2*(x*e + x*e^((x + 4)/x))/(log(-log(x)^2 + 4*x) - log(x))

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\ln \left (x\right )}^2\,\left (x\,\mathrm {e}+x\,\mathrm {e}\,{\mathrm {e}}^{4/x}\right )+\ln \left (\frac {4\,x-{\ln \left (x\right )}^2}{x}\right )\,\left ({\ln \left (x\right )}^2\,\left (x\,\mathrm {e}+\mathrm {e}\,{\mathrm {e}}^{4/x}\,\left (x-4\right )\right )-4\,x^2\,\mathrm {e}+\mathrm {e}\,{\mathrm {e}}^{4/x}\,\left (16\,x-4\,x^2\right )\right )-\ln \left (x\right )\,\left (2\,x\,\mathrm {e}+2\,x\,\mathrm {e}\,{\mathrm {e}}^{4/x}\right )}{{\ln \left (\frac {4\,x-{\ln \left (x\right )}^2}{x}\right )}^2\,\left (2\,x\,{\ln \left (x\right )}^2-8\,x^2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2*(x*exp(1) + x*exp(1)*exp(4/x)) + log((4*x - log(x)^2)/x)*(log(x)^2*(x*exp(1) + exp(1)*exp(4/x)*(
x - 4)) - 4*x^2*exp(1) + exp(1)*exp(4/x)*(16*x - 4*x^2)) - log(x)*(2*x*exp(1) + 2*x*exp(1)*exp(4/x)))/(log((4*
x - log(x)^2)/x)^2*(2*x*log(x)^2 - 8*x^2)),x)

[Out]

int((log(x)^2*(x*exp(1) + x*exp(1)*exp(4/x)) + log((4*x - log(x)^2)/x)*(log(x)^2*(x*exp(1) + exp(1)*exp(4/x)*(
x - 4)) - 4*x^2*exp(1) + exp(1)*exp(4/x)*(16*x - 4*x^2)) - log(x)*(2*x*exp(1) + 2*x*exp(1)*exp(4/x)))/(log((4*
x - log(x)^2)/x)^2*(2*x*log(x)^2 - 8*x^2)), x)

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