3.100.99 \(\int \frac {-5+(8 e x-8 x^2) \log (x)+(8 e x-12 x^2) \log ^2(x)}{-5 x+(4 e x^2-4 x^3) \log ^2(x)} \, dx\) [9999]

Optimal. Leaf size=19 \[ \log \left (x+\frac {4}{5} x^2 (-e+x) \log ^2(x)\right ) \]

[Out]

ln(x+4/5*ln(x)^2*x^2*(x-exp(1)))

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Rubi [F]
time = 5.97, 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 {-5+\left (8 e x-8 x^2\right ) \log (x)+\left (8 e x-12 x^2\right ) \log ^2(x)}{-5 x+\left (4 e x^2-4 x^3\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-5 + (8*E*x - 8*x^2)*Log[x] + (8*E*x - 12*x^2)*Log[x]^2)/(-5*x + (4*E*x^2 - 4*x^3)*Log[x]^2),x]

[Out]

Log[E - x] + 2*Log[x] - 5*Defer[Int][1/((E - x)*(-5 + 4*E*x*Log[x]^2 - 4*x^2*Log[x]^2)), x] + 8*E*Defer[Int][L
og[x]/(-5 + 4*E*x*Log[x]^2 - 4*x^2*Log[x]^2), x] - 24*Defer[Int][(x*Log[x])/(-5 + 4*E*x*Log[x]^2 - 4*x^2*Log[x
]^2), x] - 5*Defer[Int][1/(x*(5 - 4*E*x*Log[x]^2 + 4*x^2*Log[x]^2)), x] - 16*Defer[Int][(x*Log[x])/(5 - 4*E*x*
Log[x]^2 + 4*x^2*Log[x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 e-3 x}{(e-x) x}+\frac {5 e-10 x+8 e^2 x \log (x)-16 e x^2 \log (x)+8 x^3 \log (x)}{x (-e+x) \left (5-4 e x \log ^2(x)+4 x^2 \log ^2(x)\right )}\right ) \, dx\\ &=\int \frac {2 e-3 x}{(e-x) x} \, dx+\int \frac {5 e-10 x+8 e^2 x \log (x)-16 e x^2 \log (x)+8 x^3 \log (x)}{x (-e+x) \left (5-4 e x \log ^2(x)+4 x^2 \log ^2(x)\right )} \, dx\\ &=\int \left (\frac {2}{x}+\frac {1}{-e+x}\right ) \, dx+\int \frac {-5 (e-2 x)-8 (e-x)^2 x \log (x)}{(e-x) x \left (5+4 x (-e+x) \log ^2(x)\right )} \, dx\\ &=\log (e-x)+2 \log (x)+\int \left (\frac {5 e-10 x+8 e^2 x \log (x)-16 e x^2 \log (x)+8 x^3 \log (x)}{e (e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )}+\frac {5 e-10 x+8 e^2 x \log (x)-16 e x^2 \log (x)+8 x^3 \log (x)}{e x \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )}\right ) \, dx\\ &=\log (e-x)+2 \log (x)+\frac {\int \frac {5 e-10 x+8 e^2 x \log (x)-16 e x^2 \log (x)+8 x^3 \log (x)}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )} \, dx}{e}+\frac {\int \frac {5 e-10 x+8 e^2 x \log (x)-16 e x^2 \log (x)+8 x^3 \log (x)}{x \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )} \, dx}{e}\\ &=\log (e-x)+2 \log (x)+\frac {\int \frac {-5 (e-2 x)-8 (e-x)^2 x \log (x)}{(e-x) \left (5-4 (e-x) x \log ^2(x)\right )} \, dx}{e}+\frac {\int \frac {-5 (e-2 x)-8 (e-x)^2 x \log (x)}{x \left (5+4 x (-e+x) \log ^2(x)\right )} \, dx}{e}\\ &=\log (e-x)+2 \log (x)+\frac {\int \left (\frac {5 e}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )}-\frac {10 x}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )}+\frac {8 e^2 x \log (x)}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )}-\frac {16 e x^2 \log (x)}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )}+\frac {8 x^3 \log (x)}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )}\right ) \, dx}{e}+\frac {\int \left (\frac {8 e^2 \log (x)}{-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)}-\frac {16 e x \log (x)}{-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)}+\frac {10}{5-4 e x \log ^2(x)+4 x^2 \log ^2(x)}-\frac {5 e}{x \left (5-4 e x \log ^2(x)+4 x^2 \log ^2(x)\right )}-\frac {8 x^2 \log (x)}{5-4 e x \log ^2(x)+4 x^2 \log ^2(x)}\right ) \, dx}{e}\\ &=\log (e-x)+2 \log (x)+5 \int \frac {1}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )} \, dx-5 \int \frac {1}{x \left (5-4 e x \log ^2(x)+4 x^2 \log ^2(x)\right )} \, dx-16 \int \frac {x \log (x)}{-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)} \, dx-16 \int \frac {x^2 \log (x)}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )} \, dx+\frac {8 \int \frac {x^3 \log (x)}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )} \, dx}{e}-\frac {8 \int \frac {x^2 \log (x)}{5-4 e x \log ^2(x)+4 x^2 \log ^2(x)} \, dx}{e}-\frac {10 \int \frac {x}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )} \, dx}{e}+\frac {10 \int \frac {1}{5-4 e x \log ^2(x)+4 x^2 \log ^2(x)} \, dx}{e}+(8 e) \int \frac {\log (x)}{-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)} \, dx+(8 e) \int \frac {x \log (x)}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )} \, dx\\ &=\log (e-x)+2 \log (x)+5 \int \frac {1}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )} \, dx-5 \int \frac {1}{x \left (5-4 e x \log ^2(x)+4 x^2 \log ^2(x)\right )} \, dx-16 \int \frac {x \log (x)}{-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)} \, dx-16 \int \left (-\frac {e \log (x)}{-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)}+\frac {e^2 \log (x)}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )}+\frac {x \log (x)}{5-4 e x \log ^2(x)+4 x^2 \log ^2(x)}\right ) \, dx-\frac {8 \int \frac {x^2 \log (x)}{5-4 e x \log ^2(x)+4 x^2 \log ^2(x)} \, dx}{e}+\frac {8 \int \left (-\frac {e^2 \log (x)}{-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)}+\frac {e^3 \log (x)}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )}-\frac {e x \log (x)}{-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)}+\frac {x^2 \log (x)}{5-4 e x \log ^2(x)+4 x^2 \log ^2(x)}\right ) \, dx}{e}+\frac {10 \int \frac {1}{5-4 e x \log ^2(x)+4 x^2 \log ^2(x)} \, dx}{e}-\frac {10 \int \left (\frac {e}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )}+\frac {1}{5-4 e x \log ^2(x)+4 x^2 \log ^2(x)}\right ) \, dx}{e}+(8 e) \int \frac {\log (x)}{-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)} \, dx+(8 e) \int \left (-\frac {\log (x)}{-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)}+\frac {e \log (x)}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )}\right ) \, dx\\ &=\log (e-x)+2 \log (x)+5 \int \frac {1}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )} \, dx-5 \int \frac {1}{x \left (5-4 e x \log ^2(x)+4 x^2 \log ^2(x)\right )} \, dx-8 \int \frac {x \log (x)}{-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)} \, dx-10 \int \frac {1}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )} \, dx-16 \int \frac {x \log (x)}{-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)} \, dx-16 \int \frac {x \log (x)}{5-4 e x \log ^2(x)+4 x^2 \log ^2(x)} \, dx-(8 e) \int \frac {\log (x)}{-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)} \, dx+(16 e) \int \frac {\log (x)}{-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)} \, dx+2 \left (\left (8 e^2\right ) \int \frac {\log (x)}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )} \, dx\right )-\left (16 e^2\right ) \int \frac {\log (x)}{(e-x) \left (-5+4 e x \log ^2(x)-4 x^2 \log ^2(x)\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(41\) vs. \(2(19)=38\).
time = 0.09, size = 41, normalized size = 2.16 \begin {gather*} \log (e-x)+2 \log (x)-\log ((e-x) x)+\log \left (5-4 e x \log ^2(x)+4 x^2 \log ^2(x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + (8*E*x - 8*x^2)*Log[x] + (8*E*x - 12*x^2)*Log[x]^2)/(-5*x + (4*E*x^2 - 4*x^3)*Log[x]^2),x]

[Out]

Log[E - x] + 2*Log[x] - Log[(E - x)*x] + Log[5 - 4*E*x*Log[x]^2 + 4*x^2*Log[x]^2]

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Maple [A]
time = 0.21, size = 25, normalized size = 1.32

method	result	size
default	\(\ln \left (x \right )+\ln \left (4 \ln \left (x \right )^{2} x \,{\mathrm e}-4 x^{2} \ln \left (x \right )^{2}-5\right )\)	\(25\)
norman	\(\ln \left (x \right )+\ln \left (4 \ln \left (x \right )^{2} x \,{\mathrm e}-4 x^{2} \ln \left (x \right )^{2}-5\right )\)	\(25\)
risch	\(\ln \left (x -{\mathrm e}\right )+2 \ln \left (x \right )+\ln \left (\ln \left (x \right )^{2}-\frac {5}{4 x \left ({\mathrm e}-x \right )}\right )\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x*exp(1)-12*x^2)*ln(x)^2+(8*x*exp(1)-8*x^2)*ln(x)-5)/((4*x^2*exp(1)-4*x^3)*ln(x)^2-5*x),x,method=_RETU
RNVERBOSE)

[Out]

ln(x)+ln(4*ln(x)^2*x*exp(1)-4*x^2*ln(x)^2-5)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (18) = 36\).
time = 0.29, size = 43, normalized size = 2.26 \begin {gather*} \log \left (x - e\right ) + 2 \, \log \left (x\right ) + \log \left (\frac {4 \, {\left (x^{2} - x e\right )} \log \left (x\right )^{2} + 5}{4 \, {\left (x^{2} - x e\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x*exp(1)-12*x^2)*log(x)^2+(8*x*exp(1)-8*x^2)*log(x)-5)/((4*x^2*exp(1)-4*x^3)*log(x)^2-5*x),x, al
gorithm="maxima")

[Out]

log(x - e) + 2*log(x) + log(1/4*(4*(x^2 - x*e)*log(x)^2 + 5)/(x^2 - x*e))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (18) = 36\).
time = 0.40, size = 43, normalized size = 2.26 \begin {gather*} \log \left (x - e\right ) + 2 \, \log \left (x\right ) + \log \left (-\frac {4 \, {\left (x^{2} - x e\right )} \log \left (x\right )^{2} + 5}{x^{2} - x e}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x*exp(1)-12*x^2)*log(x)^2+(8*x*exp(1)-8*x^2)*log(x)-5)/((4*x^2*exp(1)-4*x^3)*log(x)^2-5*x),x, al
gorithm="fricas")

[Out]

log(x - e) + 2*log(x) + log(-(4*(x^2 - x*e)*log(x)^2 + 5)/(x^2 - x*e))

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Sympy [A]
time = 0.35, size = 31, normalized size = 1.63 \begin {gather*} 2 \log {\left (x \right )} + \log {\left (x - e \right )} + \log {\left (\log {\left (x \right )}^{2} + \frac {5}{4 x^{2} - 4 e x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x*exp(1)-12*x**2)*ln(x)**2+(8*x*exp(1)-8*x**2)*ln(x)-5)/((4*x**2*exp(1)-4*x**3)*ln(x)**2-5*x),x)

[Out]

2*log(x) + log(x - E) + log(log(x)**2 + 5/(4*x**2 - 4*E*x))

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Giac [A]
time = 0.51, size = 24, normalized size = 1.26 \begin {gather*} \log \left (-4 \, x^{2} \log \left (x\right )^{2} + 4 \, x e \log \left (x\right )^{2} - 5\right ) + \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x*exp(1)-12*x^2)*log(x)^2+(8*x*exp(1)-8*x^2)*log(x)-5)/((4*x^2*exp(1)-4*x^3)*log(x)^2-5*x),x, al
gorithm="giac")

[Out]

log(-4*x^2*log(x)^2 + 4*x*e*log(x)^2 - 5) + log(x)

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Mupad [B]
time = 7.60, size = 24, normalized size = 1.26 \begin {gather*} \ln \left (4\,x^2\,{\ln \left (x\right )}^2-4\,\mathrm {e}\,x\,{\ln \left (x\right )}^2+5\right )+\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(8*x*exp(1) - 8*x^2) + log(x)^2*(8*x*exp(1) - 12*x^2) - 5)/(5*x - log(x)^2*(4*x^2*exp(1) - 4*x^3)
),x)

[Out]

log(4*x^2*log(x)^2 - 4*x*exp(1)*log(x)^2 + 5) + log(x)

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