\(\int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 (4+9 x^2)+(-64 x+4 x^2-432 x^3+27 x^4+e^4 (-4-27 x^2)+(4 x+27 x^3) \log (x)) \log (\frac {-e^4-16 x+x^2+x \log (x)}{x})}{-e^4-16 x+x^2+x \log (x)} \, dx\) [6588]

Optimal result

Integrand size = 113, antiderivative size = 24 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx=\left (4 x+9 x^3\right ) \log \left (-16-\frac {e^4}{x}+x+\log (x)\right ) \]

[Out]

ln(x-exp(4)/x-16+ln(x))*(9*x^3+4*x)

Rubi [F]

\[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx=\int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx \]

[In]

Int[(4*x + 4*x^2 + 9*x^3 + 9*x^4 + E^4*(4 + 9*x^2) + (-64*x + 4*x^2 - 432*x^3 + 27*x^4 + E^4*(-4 - 27*x^2) + (
4*x + 27*x^3)*Log[x])*Log[(-E^4 - 16*x + x^2 + x*Log[x])/x])/(-E^4 - 16*x + x^2 + x*Log[x]),x]

[Out]

-4*E^4*Defer[Int][(E^4 + 16*x - x^2 - x*Log[x])^(-1), x] + 4*Defer[Int][x/(-E^4 - 16*x + x^2 + x*Log[x]), x] +
4*Defer[Int][x^2/(-E^4 - 16*x + x^2 + x*Log[x]), x] + 9*E^4*Defer[Int][x^2/(-E^4 - 16*x + x^2 + x*Log[x]), x]
+ 9*Defer[Int][x^3/(-E^4 - 16*x + x^2 + x*Log[x]), x] + 9*Defer[Int][x^4/(-E^4 - 16*x + x^2 + x*Log[x]), x] +
4*Defer[Int][Log[-16 - E^4/x + x + Log[x]], x] + 27*Defer[Int][x^2*Log[-16 - E^4/x + x + Log[x]], x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {e^4 \left (4+9 x^2\right )}{e^4+16 x-x^2-x \log (x)}+\frac {4 x}{-e^4-16 x+x^2+x \log (x)}+\frac {4 x^2}{-e^4-16 x+x^2+x \log (x)}+\frac {9 x^3}{-e^4-16 x+x^2+x \log (x)}+\frac {9 x^4}{-e^4-16 x+x^2+x \log (x)}+\left (4+27 x^2\right ) \log \left (-16-\frac {e^4}{x}+x+\log (x)\right )\right ) \, dx \\ & = 4 \int \frac {x}{-e^4-16 x+x^2+x \log (x)} \, dx+4 \int \frac {x^2}{-e^4-16 x+x^2+x \log (x)} \, dx+9 \int \frac {x^3}{-e^4-16 x+x^2+x \log (x)} \, dx+9 \int \frac {x^4}{-e^4-16 x+x^2+x \log (x)} \, dx-e^4 \int \frac {4+9 x^2}{e^4+16 x-x^2-x \log (x)} \, dx+\int \left (4+27 x^2\right ) \log \left (-16-\frac {e^4}{x}+x+\log (x)\right ) \, dx \\ & = 4 \int \frac {x}{-e^4-16 x+x^2+x \log (x)} \, dx+4 \int \frac {x^2}{-e^4-16 x+x^2+x \log (x)} \, dx+9 \int \frac {x^3}{-e^4-16 x+x^2+x \log (x)} \, dx+9 \int \frac {x^4}{-e^4-16 x+x^2+x \log (x)} \, dx-e^4 \int \left (\frac {4}{e^4+16 x-x^2-x \log (x)}-\frac {9 x^2}{-e^4-16 x+x^2+x \log (x)}\right ) \, dx+\int \left (4 \log \left (-16-\frac {e^4}{x}+x+\log (x)\right )+27 x^2 \log \left (-16-\frac {e^4}{x}+x+\log (x)\right )\right ) \, dx \\ & = 4 \int \frac {x}{-e^4-16 x+x^2+x \log (x)} \, dx+4 \int \frac {x^2}{-e^4-16 x+x^2+x \log (x)} \, dx+4 \int \log \left (-16-\frac {e^4}{x}+x+\log (x)\right ) \, dx+9 \int \frac {x^3}{-e^4-16 x+x^2+x \log (x)} \, dx+9 \int \frac {x^4}{-e^4-16 x+x^2+x \log (x)} \, dx+27 \int x^2 \log \left (-16-\frac {e^4}{x}+x+\log (x)\right ) \, dx-\left (4 e^4\right ) \int \frac {1}{e^4+16 x-x^2-x \log (x)} \, dx+\left (9 e^4\right ) \int \frac {x^2}{-e^4-16 x+x^2+x \log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx=x \left (4+9 x^2\right ) \log \left (-16-\frac {e^4}{x}+x+\log (x)\right ) \]

[In]

Integrate[(4*x + 4*x^2 + 9*x^3 + 9*x^4 + E^4*(4 + 9*x^2) + (-64*x + 4*x^2 - 432*x^3 + 27*x^4 + E^4*(-4 - 27*x^
2) + (4*x + 27*x^3)*Log[x])*Log[(-E^4 - 16*x + x^2 + x*Log[x])/x])/(-E^4 - 16*x + x^2 + x*Log[x]),x]

[Out]

x*(4 + 9*x^2)*Log[-16 - E^4/x + x + Log[x]]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(49\) vs. \(2(23)=46\).

Time = 6.96 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.08

method	result	size
parallelrisch	\(9 \ln \left (\frac {x \ln \left (x \right )-{\mathrm e}^{4}+x^{2}-16 x}{x}\right ) x^{3}+4 x \ln \left (\frac {x \ln \left (x \right )-{\mathrm e}^{4}+x^{2}-16 x}{x}\right )\)	\(50\)
risch	\(\left (9 x^{3}+4 x \right ) \ln \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )-9 x^{3} \ln \left (x \right )-4 x \ln \left (x \right )+2 i \pi x {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-2 i \pi x \,\operatorname {csgn}\left (i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )-4 i \pi x {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{2}+\frac {9 i \pi \,x^{3} {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{2} \operatorname {csgn}\left (\frac {i}{x}\right )}{2}+\frac {9 i \pi \,x^{3} {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{3}}{2}+2 i \pi x {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{3}-9 i \pi \,x^{3} {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{2}+9 i \pi \,x^{3}+\frac {9 i \pi \,x^{3} \operatorname {csgn}\left (i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{2}}{2}+4 i \pi x +2 i \pi x \,\operatorname {csgn}\left (i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )\right ) {\operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right )}^{2}-\frac {9 i \pi \,x^{3} \operatorname {csgn}\left (i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )\right ) \operatorname {csgn}\left (\frac {i \left ({\mathrm e}^{4}-x^{2}+\left (-\ln \left (x \right )+16\right ) x \right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right )}{2}\)	\(467\)

[In]

int((((27*x^3+4*x)*ln(x)+(-27*x^2-4)*exp(4)+27*x^4-432*x^3+4*x^2-64*x)*ln((x*ln(x)-exp(4)+x^2-16*x)/x)+(9*x^2+
4)*exp(4)+9*x^4+9*x^3+4*x^2+4*x)/(x*ln(x)-exp(4)+x^2-16*x),x,method=_RETURNVERBOSE)

[Out]

9*ln((x*ln(x)-exp(4)+x^2-16*x)/x)*x^3+4*x*ln((x*ln(x)-exp(4)+x^2-16*x)/x)

Fricas [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.25 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx={\left (9 \, x^{3} + 4 \, x\right )} \log \left (\frac {x^{2} + x \log \left (x\right ) - 16 \, x - e^{4}}{x}\right ) \]

[In]

integrate((((27*x^3+4*x)*log(x)+(-27*x^2-4)*exp(4)+27*x^4-432*x^3+4*x^2-64*x)*log((x*log(x)-exp(4)+x^2-16*x)/x
)+(9*x^2+4)*exp(4)+9*x^4+9*x^3+4*x^2+4*x)/(x*log(x)-exp(4)+x^2-16*x),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx=\left (9 x^{3} + 4 x\right ) \log {\left (\frac {x^{2} + x \log {\left (x \right )} - 16 x - e^{4}}{x} \right )} \]

[In]

integrate((((27*x**3+4*x)*ln(x)+(-27*x**2-4)*exp(4)+27*x**4-432*x**3+4*x**2-64*x)*ln((x*ln(x)-exp(4)+x**2-16*x
)/x)+(9*x**2+4)*exp(4)+9*x**4+9*x**3+4*x**2+4*x)/(x*ln(x)-exp(4)+x**2-16*x),x)

[Out]

(9*x**3 + 4*x)*log((x**2 + x*log(x) - 16*x - exp(4))/x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.67 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx={\left (9 \, x^{3} + 4 \, x\right )} \log \left (x^{2} + x \log \left (x\right ) - 16 \, x - e^{4}\right ) - {\left (9 \, x^{3} + 4 \, x\right )} \log \left (x\right ) \]

[In]

integrate((((27*x^3+4*x)*log(x)+(-27*x^2-4)*exp(4)+27*x^4-432*x^3+4*x^2-64*x)*log((x*log(x)-exp(4)+x^2-16*x)/x
)+(9*x^2+4)*exp(4)+9*x^4+9*x^3+4*x^2+4*x)/(x*log(x)-exp(4)+x^2-16*x),x, algorithm="maxima")

[Out]

(9*x^3 + 4*x)*log(x^2 + x*log(x) - 16*x - e^4) - (9*x^3 + 4*x)*log(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (23) = 46\).

Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.21 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx=9 \, x^{3} \log \left (x^{2} + x \log \left (x\right ) - 16 \, x - e^{4}\right ) - 9 \, x^{3} \log \left (x\right ) + 4 \, x \log \left (x^{2} + x \log \left (x\right ) - 16 \, x - e^{4}\right ) - 4 \, x \log \left (x\right ) \]

[In]

integrate((((27*x^3+4*x)*log(x)+(-27*x^2-4)*exp(4)+27*x^4-432*x^3+4*x^2-64*x)*log((x*log(x)-exp(4)+x^2-16*x)/x
)+(9*x^2+4)*exp(4)+9*x^4+9*x^3+4*x^2+4*x)/(x*log(x)-exp(4)+x^2-16*x),x, algorithm="giac")

[Out]

9*x^3*log(x^2 + x*log(x) - 16*x - e^4) - 9*x^3*log(x) + 4*x*log(x^2 + x*log(x) - 16*x - e^4) - 4*x*log(x)

Mupad [B] (verification not implemented)

Time = 11.56 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \frac {4 x+4 x^2+9 x^3+9 x^4+e^4 \left (4+9 x^2\right )+\left (-64 x+4 x^2-432 x^3+27 x^4+e^4 \left (-4-27 x^2\right )+\left (4 x+27 x^3\right ) \log (x)\right ) \log \left (\frac {-e^4-16 x+x^2+x \log (x)}{x}\right )}{-e^4-16 x+x^2+x \log (x)} \, dx=x\,\ln \left (-\frac {16\,x+{\mathrm {e}}^4-x\,\ln \left (x\right )-x^2}{x}\right )\,\left (9\,x^2+4\right ) \]

[In]

int(-(4*x + exp(4)*(9*x^2 + 4) - log(-(16*x + exp(4) - x*log(x) - x^2)/x)*(64*x + exp(4)*(27*x^2 + 4) - log(x)
*(4*x + 27*x^3) - 4*x^2 + 432*x^3 - 27*x^4) + 4*x^2 + 9*x^3 + 9*x^4)/(16*x + exp(4) - x*log(x) - x^2),x)

[Out]

x*log(-(16*x + exp(4) - x*log(x) - x^2)/x)*(9*x^2 + 4)