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

Optimal. Leaf size=24 \[ \left (4 x+9 x^3\right ) \log \left (-16-\frac {e^4}{x}+x+\log (x)\right ) \]

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Rubi [F]  time = 0.77, 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 {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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.06, size = 23, normalized size = 0.96 \begin {gather*} x \left (4+9 x^2\right ) \log \left (-16-\frac {e^4}{x}+x+\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[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]]

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fricas [A]  time = 0.51, size = 30, normalized size = 1.25 \begin {gather*} {\left (9 \, x^{3} + 4 \, x\right )} \log \left (\frac {x^{2} + x \log \relax (x) - 16 \, x - e^{4}}{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [B]  time = 0.27, size = 53, normalized size = 2.21 \begin {gather*} 9 \, x^{3} \log \left (x^{2} + x \log \relax (x) - 16 \, x - e^{4}\right ) - 9 \, x^{3} \log \relax (x) + 4 \, x \log \left (x^{2} + x \log \relax (x) - 16 \, x - e^{4}\right ) - 4 \, x \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [C]  time = 0.28, size = 467, normalized size = 19.46

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

Verification of antiderivative is not currently implemented for this CAS.

[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*x^3+4*x)*ln(exp(4)-x^2+(-ln(x)+16)*x)-9*x^3*ln(x)-4*x*ln(x)+9/2*I*Pi*x^3*csgn(I/x*(exp(4)-x^2+(-ln(x)+16)*x
))^3+9/2*I*Pi*x^3*csgn(I/x)*csgn(I/x*(exp(4)-x^2+(-ln(x)+16)*x))^2-9*I*Pi*x^3*csgn(I/x*(exp(4)-x^2+(-ln(x)+16)
*x))^2-2*I*Pi*x*csgn(I/x)*csgn(I*(exp(4)-x^2+(-ln(x)+16)*x))*csgn(I/x*(exp(4)-x^2+(-ln(x)+16)*x))-4*I*Pi*x*csg
n(I/x*(exp(4)-x^2+(-ln(x)+16)*x))^2-9/2*I*Pi*x^3*csgn(I/x)*csgn(I*(exp(4)-x^2+(-ln(x)+16)*x))*csgn(I/x*(exp(4)
-x^2+(-ln(x)+16)*x))+2*I*Pi*x*csgn(I/x)*csgn(I/x*(exp(4)-x^2+(-ln(x)+16)*x))^2+4*I*Pi*x+9*I*Pi*x^3+2*I*Pi*x*cs
gn(I/x*(exp(4)-x^2+(-ln(x)+16)*x))^3+9/2*I*Pi*x^3*csgn(I*(exp(4)-x^2+(-ln(x)+16)*x))*csgn(I/x*(exp(4)-x^2+(-ln
(x)+16)*x))^2+2*I*Pi*x*csgn(I*(exp(4)-x^2+(-ln(x)+16)*x))*csgn(I/x*(exp(4)-x^2+(-ln(x)+16)*x))^2

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maxima [A]  time = 0.42, size = 40, normalized size = 1.67 \begin {gather*} {\left (9 \, x^{3} + 4 \, x\right )} \log \left (x^{2} + x \log \relax (x) - 16 \, x - e^{4}\right ) - {\left (9 \, x^{3} + 4 \, x\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 4.35, size = 31, normalized size = 1.29 \begin {gather*} x\,\ln \left (-\frac {16\,x+{\mathrm {e}}^4-x\,\ln \relax (x)-x^2}{x}\right )\,\left (9\,x^2+4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.46, size = 26, normalized size = 1.08 \begin {gather*} \left (9 x^{3} + 4 x\right ) \log {\left (\frac {x^{2} + x \log {\relax (x )} - 16 x - e^{4}}{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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