3.16.87 \(\int \frac {x-21 x^2+20 x^3+e^5 (-20 x+20 x^2)+x \log (4 x)+(-2 x+2 x^2+e^5 (-2+2 x)+(-e^5-x) \log (4 x)) \log (e^5+x)}{20 x^3-40 x^4+20 x^5+e^5 (20 x^2-40 x^3+20 x^4)+(40 x^3-40 x^4+e^5 (40 x^2-40 x^3)) \log (4 x)+(20 e^5 x^2+20 x^3) \log ^2(4 x)} \, dx\)

Optimal. Leaf size=27 \[ \frac {1+\frac {\log \left (e^5+x\right )}{20 x}}{1-x+\log (4 x)} \]

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Rubi [F]  time = 2.79, 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 {x-21 x^2+20 x^3+e^5 \left (-20 x+20 x^2\right )+x \log (4 x)+\left (-2 x+2 x^2+e^5 (-2+2 x)+\left (-e^5-x\right ) \log (4 x)\right ) \log \left (e^5+x\right )}{20 x^3-40 x^4+20 x^5+e^5 \left (20 x^2-40 x^3+20 x^4\right )+\left (40 x^3-40 x^4+e^5 \left (40 x^2-40 x^3\right )\right ) \log (4 x)+\left (20 e^5 x^2+20 x^3\right ) \log ^2(4 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(x - 21*x^2 + 20*x^3 + E^5*(-20*x + 20*x^2) + x*Log[4*x] + (-2*x + 2*x^2 + E^5*(-2 + 2*x) + (-E^5 - x)*Log
[4*x])*Log[E^5 + x])/(20*x^3 - 40*x^4 + 20*x^5 + E^5*(20*x^2 - 40*x^3 + 20*x^4) + (40*x^3 - 40*x^4 + E^5*(40*x
^2 - 40*x^3))*Log[4*x] + (20*E^5*x^2 + 20*x^3)*Log[4*x]^2),x]

[Out]

(1 - x + Log[4*x])^(-1) - Defer[Int][1/(x*(-1 + x - Log[4*x])), x]/(20*E^5) + Defer[Int][1/((E^5 + x)*(-1 + x
- Log[4*x])), x]/(20*E^5) - Defer[Int][Log[E^5 + x]/(x^2*(-1 + x - Log[4*x])^2), x]/10 + Defer[Int][Log[E^5 +
x]/(x*(-1 + x - Log[4*x])^2), x]/10 - Defer[Int][(Log[4*x]*Log[E^5 + x])/(x^2*(-1 + x - Log[4*x])^2), x]/20

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\log (4 x) \left (x-\left (e^5+x\right ) \log \left (e^5+x\right )\right )+(-1+x) \left (x \left (-1+20 e^5+20 x\right )+2 \left (e^5+x\right ) \log \left (e^5+x\right )\right )}{20 x^2 \left (e^5+x\right ) (1-x+\log (4 x))^2} \, dx\\ &=\frac {1}{20} \int \frac {\log (4 x) \left (x-\left (e^5+x\right ) \log \left (e^5+x\right )\right )+(-1+x) \left (x \left (-1+20 e^5+20 x\right )+2 \left (e^5+x\right ) \log \left (e^5+x\right )\right )}{x^2 \left (e^5+x\right ) (1-x+\log (4 x))^2} \, dx\\ &=\frac {1}{20} \int \left (\frac {1-20 e^5-21 \left (1-\frac {20 e^5}{21}\right ) x+20 x^2+\log (4 x)}{x \left (e^5+x\right ) (1-x+\log (4 x))^2}+\frac {(-2+2 x-\log (4 x)) \log \left (e^5+x\right )}{x^2 (-1+x-\log (4 x))^2}\right ) \, dx\\ &=\frac {1}{20} \int \frac {1-20 e^5-21 \left (1-\frac {20 e^5}{21}\right ) x+20 x^2+\log (4 x)}{x \left (e^5+x\right ) (1-x+\log (4 x))^2} \, dx+\frac {1}{20} \int \frac {(-2+2 x-\log (4 x)) \log \left (e^5+x\right )}{x^2 (-1+x-\log (4 x))^2} \, dx\\ &=\frac {1}{20} \int \left (\frac {20 (-1+x)}{x (-1+x-\log (4 x))^2}-\frac {1}{x \left (e^5+x\right ) (-1+x-\log (4 x))}\right ) \, dx+\frac {1}{20} \int \left (-\frac {2 \log \left (e^5+x\right )}{x^2 (-1+x-\log (4 x))^2}+\frac {2 \log \left (e^5+x\right )}{x (-1+x-\log (4 x))^2}-\frac {\log (4 x) \log \left (e^5+x\right )}{x^2 (-1+x-\log (4 x))^2}\right ) \, dx\\ &=-\left (\frac {1}{20} \int \frac {1}{x \left (e^5+x\right ) (-1+x-\log (4 x))} \, dx\right )-\frac {1}{20} \int \frac {\log (4 x) \log \left (e^5+x\right )}{x^2 (-1+x-\log (4 x))^2} \, dx-\frac {1}{10} \int \frac {\log \left (e^5+x\right )}{x^2 (-1+x-\log (4 x))^2} \, dx+\frac {1}{10} \int \frac {\log \left (e^5+x\right )}{x (-1+x-\log (4 x))^2} \, dx+\int \frac {-1+x}{x (-1+x-\log (4 x))^2} \, dx\\ &=\frac {1}{1-x+\log (4 x)}-\frac {1}{20} \int \left (\frac {1}{e^5 x (-1+x-\log (4 x))}-\frac {1}{e^5 \left (e^5+x\right ) (-1+x-\log (4 x))}\right ) \, dx-\frac {1}{20} \int \frac {\log (4 x) \log \left (e^5+x\right )}{x^2 (-1+x-\log (4 x))^2} \, dx-\frac {1}{10} \int \frac {\log \left (e^5+x\right )}{x^2 (-1+x-\log (4 x))^2} \, dx+\frac {1}{10} \int \frac {\log \left (e^5+x\right )}{x (-1+x-\log (4 x))^2} \, dx\\ &=\frac {1}{1-x+\log (4 x)}-\frac {1}{20} \int \frac {\log (4 x) \log \left (e^5+x\right )}{x^2 (-1+x-\log (4 x))^2} \, dx-\frac {1}{10} \int \frac {\log \left (e^5+x\right )}{x^2 (-1+x-\log (4 x))^2} \, dx+\frac {1}{10} \int \frac {\log \left (e^5+x\right )}{x (-1+x-\log (4 x))^2} \, dx-\frac {\int \frac {1}{x (-1+x-\log (4 x))} \, dx}{20 e^5}+\frac {\int \frac {1}{\left (e^5+x\right ) (-1+x-\log (4 x))} \, dx}{20 e^5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.80, size = 29, normalized size = 1.07 \begin {gather*} \frac {20 x+\log \left (e^5+x\right )}{20 \left (x-x^2+x \log (4 x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x - 21*x^2 + 20*x^3 + E^5*(-20*x + 20*x^2) + x*Log[4*x] + (-2*x + 2*x^2 + E^5*(-2 + 2*x) + (-E^5 -
x)*Log[4*x])*Log[E^5 + x])/(20*x^3 - 40*x^4 + 20*x^5 + E^5*(20*x^2 - 40*x^3 + 20*x^4) + (40*x^3 - 40*x^4 + E^5
*(40*x^2 - 40*x^3))*Log[4*x] + (20*E^5*x^2 + 20*x^3)*Log[4*x]^2),x]

[Out]

(20*x + Log[E^5 + x])/(20*(x - x^2 + x*Log[4*x]))

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fricas [A]  time = 0.83, size = 27, normalized size = 1.00 \begin {gather*} -\frac {20 \, x + \log \left (x + e^{5}\right )}{20 \, {\left (x^{2} - x \log \left (4 \, x\right ) - x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(5)-x)*log(4*x)+(2*x-2)*exp(5)+2*x^2-2*x)*log(exp(5)+x)+x*log(4*x)+(20*x^2-20*x)*exp(5)+20*x^
3-21*x^2+x)/((20*x^2*exp(5)+20*x^3)*log(4*x)^2+((-40*x^3+40*x^2)*exp(5)-40*x^4+40*x^3)*log(4*x)+(20*x^4-40*x^3
+20*x^2)*exp(5)+20*x^5-40*x^4+20*x^3),x, algorithm="fricas")

[Out]

-1/20*(20*x + log(x + e^5))/(x^2 - x*log(4*x) - x)

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giac [A]  time = 0.34, size = 27, normalized size = 1.00 \begin {gather*} -\frac {20 \, x + \log \left (x + e^{5}\right )}{20 \, {\left (x^{2} - x \log \left (4 \, x\right ) - x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(5)-x)*log(4*x)+(2*x-2)*exp(5)+2*x^2-2*x)*log(exp(5)+x)+x*log(4*x)+(20*x^2-20*x)*exp(5)+20*x^
3-21*x^2+x)/((20*x^2*exp(5)+20*x^3)*log(4*x)^2+((-40*x^3+40*x^2)*exp(5)-40*x^4+40*x^3)*log(4*x)+(20*x^4-40*x^3
+20*x^2)*exp(5)+20*x^5-40*x^4+20*x^3),x, algorithm="giac")

[Out]

-1/20*(20*x + log(x + e^5))/(x^2 - x*log(4*x) - x)

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maple [A]  time = 0.47, size = 36, normalized size = 1.33

method	result	size
risch	\(-\frac {\ln \left ({\mathrm e}^{5}+x \right )}{20 x \left (x -\ln \left (4 x \right )-1\right )}-\frac {1}{x -\ln \left (4 x \right )-1}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-exp(5)-x)*ln(4*x)+(2*x-2)*exp(5)+2*x^2-2*x)*ln(exp(5)+x)+x*ln(4*x)+(20*x^2-20*x)*exp(5)+20*x^3-21*x^2+
x)/((20*x^2*exp(5)+20*x^3)*ln(4*x)^2+((-40*x^3+40*x^2)*exp(5)-40*x^4+40*x^3)*ln(4*x)+(20*x^4-40*x^3+20*x^2)*ex
p(5)+20*x^5-40*x^4+20*x^3),x,method=_RETURNVERBOSE)

[Out]

-1/20/x/(x-ln(4*x)-1)*ln(exp(5)+x)-1/(x-ln(4*x)-1)

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maxima [A]  time = 0.99, size = 31, normalized size = 1.15 \begin {gather*} -\frac {20 \, x + \log \left (x + e^{5}\right )}{20 \, {\left (x^{2} - x {\left (2 \, \log \relax (2) + 1\right )} - x \log \relax (x)\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(5)-x)*log(4*x)+(2*x-2)*exp(5)+2*x^2-2*x)*log(exp(5)+x)+x*log(4*x)+(20*x^2-20*x)*exp(5)+20*x^
3-21*x^2+x)/((20*x^2*exp(5)+20*x^3)*log(4*x)^2+((-40*x^3+40*x^2)*exp(5)-40*x^4+40*x^3)*log(4*x)+(20*x^4-40*x^3
+20*x^2)*exp(5)+20*x^5-40*x^4+20*x^3),x, algorithm="maxima")

[Out]

-1/20*(20*x + log(x + e^5))/(x^2 - x*(2*log(2) + 1) - x*log(x))

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mupad [B]  time = 1.37, size = 25, normalized size = 0.93 \begin {gather*} \frac {20\,x+\ln \left (x+{\mathrm {e}}^5\right )}{20\,x\,\left (\ln \left (4\,x\right )-x+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + x*log(4*x) - exp(5)*(20*x - 20*x^2) - log(x + exp(5))*(2*x - 2*x^2 + log(4*x)*(x + exp(5)) - exp(5)*(
2*x - 2)) - 21*x^2 + 20*x^3)/(log(4*x)*(exp(5)*(40*x^2 - 40*x^3) + 40*x^3 - 40*x^4) + log(4*x)^2*(20*x^2*exp(5
) + 20*x^3) + exp(5)*(20*x^2 - 40*x^3 + 20*x^4) + 20*x^3 - 40*x^4 + 20*x^5),x)

[Out]

(20*x + log(x + exp(5)))/(20*x*(log(4*x) - x + 1))

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sympy [A]  time = 0.55, size = 32, normalized size = 1.19 \begin {gather*} - \frac {\log {\left (x + e^{5} \right )}}{20 x^{2} - 20 x \log {\left (4 x \right )} - 20 x} + \frac {1}{- x + \log {\left (4 x \right )} + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-exp(5)-x)*ln(4*x)+(2*x-2)*exp(5)+2*x**2-2*x)*ln(exp(5)+x)+x*ln(4*x)+(20*x**2-20*x)*exp(5)+20*x**
3-21*x**2+x)/((20*x**2*exp(5)+20*x**3)*ln(4*x)**2+((-40*x**3+40*x**2)*exp(5)-40*x**4+40*x**3)*ln(4*x)+(20*x**4
-40*x**3+20*x**2)*exp(5)+20*x**5-40*x**4+20*x**3),x)

[Out]

-log(x + exp(5))/(20*x**2 - 20*x*log(4*x) - 20*x) + 1/(-x + log(4*x) + 1)

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