\(\int \frac {-2 \log (x)-\log (\frac {5 x}{4})+(-x+x \log (x)) \log ^3(\frac {5 x}{4})}{x \log ^2(x) \log ^3(\frac {5 x}{4})} \, dx\) [1650]

Optimal result

Integrand size = 46, antiderivative size = 17 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=2+\frac {x+\frac {1}{\log ^2\left (\frac {5 x}{4}\right )}}{\log (x)} \]

[Out]

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

Rubi [F]

\[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=\int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx \]

[In]

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

[Out]

x/Log[x] - 2*Defer[Int][1/(x*Log[x]*Log[(5*x)/4]^3), x] - Defer[Int][1/(x*Log[x]^2*Log[(5*x)/4]^2), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-1+\log (x)}{\log ^2(x)}-\frac {2}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )}-\frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )}\right ) \, dx \\ & = -\left (2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx\right )+\int \frac {-1+\log (x)}{\log ^2(x)} \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx \\ & = -\left (2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx\right )+\int \left (-\frac {1}{\log ^2(x)}+\frac {1}{\log (x)}\right ) \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx \\ & = -\left (2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx\right )-\int \frac {1}{\log ^2(x)} \, dx+\int \frac {1}{\log (x)} \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx \\ & = \frac {x}{\log (x)}+\operatorname {LogIntegral}(x)-2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx-\int \frac {1}{\log (x)} \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx \\ & = \frac {x}{\log (x)}-2 \int \frac {1}{x \log (x) \log ^3\left (\frac {5 x}{4}\right )} \, dx-\int \frac {1}{x \log ^2(x) \log ^2\left (\frac {5 x}{4}\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=\frac {x+\frac {1}{\log ^2\left (\frac {5 x}{4}\right )}}{\log (x)} \]

[In]

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

[Out]

(x + Log[(5*x)/4]^(-2))/Log[x]

Maple [A] (verified)

Time = 0.71 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.41

method	result	size
parallelrisch	\(-\frac {-6-6 x \ln \left (\frac {5 x}{4}\right )^{2}}{6 \ln \left (x \right ) \ln \left (\frac {5 x}{4}\right )^{2}}\)	\(24\)
risch	\(\frac {4-16 x \ln \left (2\right ) \ln \left (5\right )+16 x \ln \left (2\right )^{2}+4 x \ln \left (x \right )^{2}+4 x \ln \left (5\right )^{2}+8 x \ln \left (5\right ) \ln \left (x \right )-16 x \ln \left (2\right ) \ln \left (x \right )}{\left (2 \ln \left (5\right )-4 \ln \left (2\right )+2 \ln \left (x \right )\right )^{2} \ln \left (x \right )}\)	\(65\)
parts	\(\text {Expression too large to display}\)	\(1911\)
default	\(\text {Expression too large to display}\)	\(2586\)

[In]

int(((x*ln(x)-x)*ln(5/4*x)^3-ln(5/4*x)-2*ln(x))/x/ln(x)^2/ln(5/4*x)^3,x,method=_RETURNVERBOSE)

[Out]

-1/6*(-6-6*x*ln(5/4*x)^2)/ln(x)/ln(5/4*x)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (15) = 30\).

Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.59 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=\frac {x \log \left (\frac {5}{4}\right )^{2} + 2 \, x \log \left (\frac {5}{4}\right ) \log \left (x\right ) + x \log \left (x\right )^{2} + 1}{\log \left (\frac {5}{4}\right )^{2} \log \left (x\right ) + 2 \, \log \left (\frac {5}{4}\right ) \log \left (x\right )^{2} + \log \left (x\right )^{3}} \]

[In]

integrate(((x*log(x)-x)*log(5/4*x)^3-log(5/4*x)-2*log(x))/x/log(x)^2/log(5/4*x)^3,x, algorithm="fricas")

[Out]

(x*log(5/4)^2 + 2*x*log(5/4)*log(x) + x*log(x)^2 + 1)/(log(5/4)^2*log(x) + 2*log(5/4)*log(x)^2 + log(x)^3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (15) = 30\).

Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 5.53 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=\frac {x \log {\left (x \right )}^{2} - 4 x \log {\left (2 \right )} \log {\left (5 \right )} + 4 x \log {\left (2 \right )}^{2} + x \log {\left (5 \right )}^{2} + \left (- 4 x \log {\left (2 \right )} + 2 x \log {\left (5 \right )}\right ) \log {\left (x \right )} + 1}{\log {\left (x \right )}^{3} + \left (- 4 \log {\left (2 \right )} + 2 \log {\left (5 \right )}\right ) \log {\left (x \right )}^{2} + \left (- 4 \log {\left (2 \right )} \log {\left (5 \right )} + 4 \log {\left (2 \right )}^{2} + \log {\left (5 \right )}^{2}\right ) \log {\left (x \right )}} \]

[In]

integrate(((x*ln(x)-x)*ln(5/4*x)**3-ln(5/4*x)-2*ln(x))/x/ln(x)**2/ln(5/4*x)**3,x)

[Out]

(x*log(x)**2 - 4*x*log(2)*log(5) + 4*x*log(2)**2 + x*log(5)**2 + (-4*x*log(2) + 2*x*log(5))*log(x) + 1)/(log(x
)**3 + (-4*log(2) + 2*log(5))*log(x)**2 + (-4*log(2)*log(5) + 4*log(2)**2 + log(5)**2)*log(x))

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.31 (sec) , antiderivative size = 186, normalized size of antiderivative = 10.94 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=-\frac {3 \, \log \left (5\right ) - 6 \, \log \left (2\right ) + 2 \, \log \left (x\right )}{\log \left (5\right )^{4} - 8 \, \log \left (5\right )^{3} \log \left (2\right ) + 24 \, \log \left (5\right )^{2} \log \left (2\right )^{2} - 32 \, \log \left (5\right ) \log \left (2\right )^{3} + 16 \, \log \left (2\right )^{4} + {\left (\log \left (5\right )^{2} - 4 \, \log \left (5\right ) \log \left (2\right ) + 4 \, \log \left (2\right )^{2}\right )} \log \left (x\right )^{2} + 2 \, {\left (\log \left (5\right )^{3} - 6 \, \log \left (5\right )^{2} \log \left (2\right ) + 12 \, \log \left (5\right ) \log \left (2\right )^{2} - 8 \, \log \left (2\right )^{3}\right )} \log \left (x\right )} + \frac {\log \left (5\right ) - 2 \, \log \left (2\right ) + 2 \, \log \left (x\right )}{{\left (\log \left (5\right )^{2} - 4 \, \log \left (5\right ) \log \left (2\right ) + 4 \, \log \left (2\right )^{2}\right )} \log \left (x\right )^{2} + {\left (\log \left (5\right )^{3} - 6 \, \log \left (5\right )^{2} \log \left (2\right ) + 12 \, \log \left (5\right ) \log \left (2\right )^{2} - 8 \, \log \left (2\right )^{3}\right )} \log \left (x\right )} + {\rm Ei}\left (\log \left (x\right )\right ) - \Gamma \left (-1, -\log \left (x\right )\right ) \]

[In]

integrate(((x*log(x)-x)*log(5/4*x)^3-log(5/4*x)-2*log(x))/x/log(x)^2/log(5/4*x)^3,x, algorithm="maxima")

[Out]

-(3*log(5) - 6*log(2) + 2*log(x))/(log(5)^4 - 8*log(5)^3*log(2) + 24*log(5)^2*log(2)^2 - 32*log(5)*log(2)^3 +
16*log(2)^4 + (log(5)^2 - 4*log(5)*log(2) + 4*log(2)^2)*log(x)^2 + 2*(log(5)^3 - 6*log(5)^2*log(2) + 12*log(5)
*log(2)^2 - 8*log(2)^3)*log(x)) + (log(5) - 2*log(2) + 2*log(x))/((log(5)^2 - 4*log(5)*log(2) + 4*log(2)^2)*lo
g(x)^2 + (log(5)^3 - 6*log(5)^2*log(2) + 12*log(5)*log(2)^2 - 8*log(2)^3)*log(x)) + Ei(log(x)) - gamma(-1, -lo
g(x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (15) = 30\).

Time = 0.28 (sec) , antiderivative size = 187, normalized size of antiderivative = 11.00 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=\frac {x \log \left (5\right )^{2} - 4 \, x \log \left (5\right ) \log \left (2\right ) + 4 \, x \log \left (2\right )^{2} + 1}{2 \, \log \left (5\right )^{2} \log \left (2\right ) - 8 \, \log \left (5\right ) \log \left (2\right )^{2} + 8 \, \log \left (2\right )^{3} + \log \left (5\right )^{2} \log \left (\frac {1}{4} \, x\right ) - 4 \, \log \left (5\right ) \log \left (2\right ) \log \left (\frac {1}{4} \, x\right ) + 4 \, \log \left (2\right )^{2} \log \left (\frac {1}{4} \, x\right )} - \frac {2 \, \log \left (5\right ) - 2 \, \log \left (2\right ) + \log \left (\frac {1}{4} \, x\right )}{\log \left (5\right )^{4} - 4 \, \log \left (5\right )^{3} \log \left (2\right ) + 4 \, \log \left (5\right )^{2} \log \left (2\right )^{2} + 2 \, \log \left (5\right )^{3} \log \left (\frac {1}{4} \, x\right ) - 8 \, \log \left (5\right )^{2} \log \left (2\right ) \log \left (\frac {1}{4} \, x\right ) + 8 \, \log \left (5\right ) \log \left (2\right )^{2} \log \left (\frac {1}{4} \, x\right ) + \log \left (5\right )^{2} \log \left (\frac {1}{4} \, x\right )^{2} - 4 \, \log \left (5\right ) \log \left (2\right ) \log \left (\frac {1}{4} \, x\right )^{2} + 4 \, \log \left (2\right )^{2} \log \left (\frac {1}{4} \, x\right )^{2}} \]

[In]

integrate(((x*log(x)-x)*log(5/4*x)^3-log(5/4*x)-2*log(x))/x/log(x)^2/log(5/4*x)^3,x, algorithm="giac")

[Out]

(x*log(5)^2 - 4*x*log(5)*log(2) + 4*x*log(2)^2 + 1)/(2*log(5)^2*log(2) - 8*log(5)*log(2)^2 + 8*log(2)^3 + log(
5)^2*log(1/4*x) - 4*log(5)*log(2)*log(1/4*x) + 4*log(2)^2*log(1/4*x)) - (2*log(5) - 2*log(2) + log(1/4*x))/(lo
g(5)^4 - 4*log(5)^3*log(2) + 4*log(5)^2*log(2)^2 + 2*log(5)^3*log(1/4*x) - 8*log(5)^2*log(2)*log(1/4*x) + 8*lo
g(5)*log(2)^2*log(1/4*x) + log(5)^2*log(1/4*x)^2 - 4*log(5)*log(2)*log(1/4*x)^2 + 4*log(2)^2*log(1/4*x)^2)

Mupad [B] (verification not implemented)

Time = 8.52 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.24 \[ \int \frac {-2 \log (x)-\log \left (\frac {5 x}{4}\right )+(-x+x \log (x)) \log ^3\left (\frac {5 x}{4}\right )}{x \log ^2(x) \log ^3\left (\frac {5 x}{4}\right )} \, dx=\frac {x\,{\ln \left (\frac {5\,x}{4}\right )}^2+1}{{\ln \left (\frac {5\,x}{4}\right )}^2\,\ln \left (x\right )} \]

[In]

int(-(log((5*x)/4) + 2*log(x) + log((5*x)/4)^3*(x - x*log(x)))/(x*log((5*x)/4)^3*log(x)^2),x)

[Out]

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