3.74.10 \(\int \frac {-108+144 x^3+2 x^6+e^{-1+x} (-54+72 x^3+x^6)+(e^{-1+x} (9 x-24 x^4+16 x^7)+e^{-1+x} x^7 \log (x)) \log (\frac {9-24 x^3+16 x^6+x^6 \log (x)}{x^6})}{9 x-24 x^4+16 x^7+x^7 \log (x)} \, dx\)

Optimal. Leaf size=21 \[ \left (2+e^{-1+x}\right ) \log \left (\left (4-\frac {3}{x^3}\right )^2+\log (x)\right ) \]

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Rubi [F]  time = 1.98, 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 {-108+144 x^3+2 x^6+e^{-1+x} \left (-54+72 x^3+x^6\right )+\left (e^{-1+x} \left (9 x-24 x^4+16 x^7\right )+e^{-1+x} x^7 \log (x)\right ) \log \left (\frac {9-24 x^3+16 x^6+x^6 \log (x)}{x^6}\right )}{9 x-24 x^4+16 x^7+x^7 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-108 + 144*x^3 + 2*x^6 + E^(-1 + x)*(-54 + 72*x^3 + x^6) + (E^(-1 + x)*(9*x - 24*x^4 + 16*x^7) + E^(-1 +
x)*x^7*Log[x])*Log[(9 - 24*x^3 + 16*x^6 + x^6*Log[x])/x^6])/(9*x - 24*x^4 + 16*x^7 + x^7*Log[x]),x]

[Out]

-108*Defer[Int][1/(x*(9 - 24*x^3 + 16*x^6 + x^6*Log[x])), x] - (54*Defer[Int][E^x/(x*(9 - 24*x^3 + 16*x^6 + x^
6*Log[x])), x])/E + 144*Defer[Int][x^2/(9 - 24*x^3 + 16*x^6 + x^6*Log[x]), x] + (72*Defer[Int][(E^x*x^2)/(9 -
24*x^3 + 16*x^6 + x^6*Log[x]), x])/E + 2*Defer[Int][x^5/(9 - 24*x^3 + 16*x^6 + x^6*Log[x]), x] + Defer[Int][(E
^x*x^5)/(9 - 24*x^3 + 16*x^6 + x^6*Log[x]), x]/E + Defer[Int][E^x*Log[(3 - 4*x^3)^2/x^6 + Log[x]], x]/E

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\frac {\left (2 e+e^x\right ) \left (-54+72 x^3+x^6\right )}{x \left (\left (3-4 x^3\right )^2+x^6 \log (x)\right )}+e^x \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right )}{e} \, dx\\ &=\frac {\int \left (\frac {\left (2 e+e^x\right ) \left (-54+72 x^3+x^6\right )}{x \left (\left (3-4 x^3\right )^2+x^6 \log (x)\right )}+e^x \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right )\right ) \, dx}{e}\\ &=\frac {\int \frac {\left (2 e+e^x\right ) \left (-54+72 x^3+x^6\right )}{x \left (\left (3-4 x^3\right )^2+x^6 \log (x)\right )} \, dx}{e}+\frac {\int e^x \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right ) \, dx}{e}\\ &=\frac {\int \left (\frac {2 e \left (-54+72 x^3+x^6\right )}{x \left (9-24 x^3+16 x^6+x^6 \log (x)\right )}+\frac {e^x \left (-54+72 x^3+x^6\right )}{x \left (9-24 x^3+16 x^6+x^6 \log (x)\right )}\right ) \, dx}{e}+\frac {\int e^x \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right ) \, dx}{e}\\ &=2 \int \frac {-54+72 x^3+x^6}{x \left (9-24 x^3+16 x^6+x^6 \log (x)\right )} \, dx+\frac {\int \frac {e^x \left (-54+72 x^3+x^6\right )}{x \left (9-24 x^3+16 x^6+x^6 \log (x)\right )} \, dx}{e}+\frac {\int e^x \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right ) \, dx}{e}\\ &=2 \int \left (-\frac {54}{x \left (9-24 x^3+16 x^6+x^6 \log (x)\right )}+\frac {72 x^2}{9-24 x^3+16 x^6+x^6 \log (x)}+\frac {x^5}{9-24 x^3+16 x^6+x^6 \log (x)}\right ) \, dx+\frac {\int \left (-\frac {54 e^x}{x \left (9-24 x^3+16 x^6+x^6 \log (x)\right )}+\frac {72 e^x x^2}{9-24 x^3+16 x^6+x^6 \log (x)}+\frac {e^x x^5}{9-24 x^3+16 x^6+x^6 \log (x)}\right ) \, dx}{e}+\frac {\int e^x \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right ) \, dx}{e}\\ &=2 \int \frac {x^5}{9-24 x^3+16 x^6+x^6 \log (x)} \, dx-108 \int \frac {1}{x \left (9-24 x^3+16 x^6+x^6 \log (x)\right )} \, dx+144 \int \frac {x^2}{9-24 x^3+16 x^6+x^6 \log (x)} \, dx+\frac {\int \frac {e^x x^5}{9-24 x^3+16 x^6+x^6 \log (x)} \, dx}{e}+\frac {\int e^x \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right ) \, dx}{e}-\frac {54 \int \frac {e^x}{x \left (9-24 x^3+16 x^6+x^6 \log (x)\right )} \, dx}{e}+\frac {72 \int \frac {e^x x^2}{9-24 x^3+16 x^6+x^6 \log (x)} \, dx}{e}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 1.71, size = 53, normalized size = 2.52 \begin {gather*} \frac {-12 e \log (x)+e^x \log \left (\frac {\left (3-4 x^3\right )^2}{x^6}+\log (x)\right )+2 e \log \left (9-24 x^3+16 x^6+x^6 \log (x)\right )}{e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-108 + 144*x^3 + 2*x^6 + E^(-1 + x)*(-54 + 72*x^3 + x^6) + (E^(-1 + x)*(9*x - 24*x^4 + 16*x^7) + E^
(-1 + x)*x^7*Log[x])*Log[(9 - 24*x^3 + 16*x^6 + x^6*Log[x])/x^6])/(9*x - 24*x^4 + 16*x^7 + x^7*Log[x]),x]

[Out]

(-12*E*Log[x] + E^x*Log[(3 - 4*x^3)^2/x^6 + Log[x]] + 2*E*Log[9 - 24*x^3 + 16*x^6 + x^6*Log[x]])/E

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fricas [A]  time = 0.53, size = 30, normalized size = 1.43 \begin {gather*} {\left (e^{\left (x - 1\right )} + 2\right )} \log \left (\frac {x^{6} \log \relax (x) + 16 \, x^{6} - 24 \, x^{3} + 9}{x^{6}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^7*exp(x-1)*log(x)+(16*x^7-24*x^4+9*x)*exp(x-1))*log((x^6*log(x)+16*x^6-24*x^3+9)/x^6)+(x^6+72*x^
3-54)*exp(x-1)+2*x^6+144*x^3-108)/(x^7*log(x)+16*x^7-24*x^4+9*x),x, algorithm="fricas")

[Out]

(e^(x - 1) + 2)*log((x^6*log(x) + 16*x^6 - 24*x^3 + 9)/x^6)

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giac [B]  time = 0.28, size = 61, normalized size = 2.90 \begin {gather*} {\left (2 \, e \log \left (x^{6} \log \relax (x) + 16 \, x^{6} - 24 \, x^{3} + 9\right ) + e^{x} \log \left (x^{6} \log \relax (x) + 16 \, x^{6} - 24 \, x^{3} + 9\right ) - 12 \, e \log \relax (x) - 6 \, e^{x} \log \relax (x)\right )} e^{\left (-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^7*exp(x-1)*log(x)+(16*x^7-24*x^4+9*x)*exp(x-1))*log((x^6*log(x)+16*x^6-24*x^3+9)/x^6)+(x^6+72*x^
3-54)*exp(x-1)+2*x^6+144*x^3-108)/(x^7*log(x)+16*x^7-24*x^4+9*x),x, algorithm="giac")

[Out]

(2*e*log(x^6*log(x) + 16*x^6 - 24*x^3 + 9) + e^x*log(x^6*log(x) + 16*x^6 - 24*x^3 + 9) - 12*e*log(x) - 6*e^x*l
og(x))*e^(-1)

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maple [C]  time = 0.20, size = 652, normalized size = 31.05

method	result	size
risch	\(\frac {i \pi \mathrm {csgn}\left (i x^{2}\right )^{3} {\mathrm e}^{x -1}}{2}+\frac {i \pi \mathrm {csgn}\left (i x^{3}\right )^{3} {\mathrm e}^{x -1}}{2}+\frac {i \pi \mathrm {csgn}\left (i x^{4}\right )^{3} {\mathrm e}^{x -1}}{2}+\frac {i \pi \mathrm {csgn}\left (i x^{5}\right )^{3} {\mathrm e}^{x -1}}{2}+\frac {i \pi \mathrm {csgn}\left (i x^{6}\right )^{3} {\mathrm e}^{x -1}}{2}-\frac {i \pi \mathrm {csgn}\left (\frac {i \left (9+\left (16+\ln \relax (x )\right ) x^{6}-24 x^{3}\right )}{x^{6}}\right )^{3} {\mathrm e}^{x -1}}{2}-6 \,{\mathrm e}^{x -1} \ln \relax (x )+{\mathrm e}^{x -1} \ln \left (9+\left (16+\ln \relax (x )\right ) x^{6}-24 x^{3}\right )+2 \ln \left (\ln \relax (x )+\frac {16 x^{6}-24 x^{3}+9}{x^{6}}\right )-\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{x^{6}}\right ) \mathrm {csgn}\left (i \left (9+\left (16+\ln \relax (x )\right ) x^{6}-24 x^{3}\right )\right ) \mathrm {csgn}\left (\frac {i \left (9+\left (16+\ln \relax (x )\right ) x^{6}-24 x^{3}\right )}{x^{6}}\right ) {\mathrm e}^{x -1}}{2}+\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right ) \mathrm {csgn}\left (i x^{5}\right ) {\mathrm e}^{x -1}}{2}+\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right ) {\mathrm e}^{x -1}}{2}+\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right ) {\mathrm e}^{x -1}}{2}+\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{5}\right ) \mathrm {csgn}\left (i x^{6}\right ) {\mathrm e}^{x -1}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{3}\right )^{2} {\mathrm e}^{x -1}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{3}\right )^{2} {\mathrm e}^{x -1}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{5}\right ) \mathrm {csgn}\left (i x^{6}\right )^{2} {\mathrm e}^{x -1}}{2}+\frac {i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) {\mathrm e}^{x -1}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{3}\right ) \mathrm {csgn}\left (i x^{4}\right )^{2} {\mathrm e}^{x -1}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{5}\right )^{2} {\mathrm e}^{x -1}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{6}\right )^{2} {\mathrm e}^{x -1}}{2}-i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} {\mathrm e}^{x -1}+\frac {i \pi \,\mathrm {csgn}\left (\frac {i}{x^{6}}\right ) \mathrm {csgn}\left (\frac {i \left (9+\left (16+\ln \relax (x )\right ) x^{6}-24 x^{3}\right )}{x^{6}}\right )^{2} {\mathrm e}^{x -1}}{2}+\frac {i \pi \,\mathrm {csgn}\left (i \left (9+\left (16+\ln \relax (x )\right ) x^{6}-24 x^{3}\right )\right ) \mathrm {csgn}\left (\frac {i \left (9+\left (16+\ln \relax (x )\right ) x^{6}-24 x^{3}\right )}{x^{6}}\right )^{2} {\mathrm e}^{x -1}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{4}\right )^{2} {\mathrm e}^{x -1}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i x^{4}\right ) \mathrm {csgn}\left (i x^{5}\right )^{2} {\mathrm e}^{x -1}}{2}\)	\(652\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^7*exp(x-1)*ln(x)+(16*x^7-24*x^4+9*x)*exp(x-1))*ln((x^6*ln(x)+16*x^6-24*x^3+9)/x^6)+(x^6+72*x^3-54)*exp
(x-1)+2*x^6+144*x^3-108)/(x^7*ln(x)+16*x^7-24*x^4+9*x),x,method=_RETURNVERBOSE)

[Out]

1/2*I*Pi*csgn(I*x^2)^3*exp(x-1)+1/2*I*Pi*csgn(I*x^3)^3*exp(x-1)+1/2*I*Pi*csgn(I*x^4)^3*exp(x-1)+1/2*I*Pi*csgn(
I*x^5)^3*exp(x-1)+1/2*I*Pi*csgn(I*x^6)^3*exp(x-1)-1/2*I*Pi*csgn(I/x^6*(9+(16+ln(x))*x^6-24*x^3))^3*exp(x-1)-6*
exp(x-1)*ln(x)+exp(x-1)*ln(9+(16+ln(x))*x^6-24*x^3)-1/2*I*Pi*csgn(I/x^6)*csgn(I*(9+(16+ln(x))*x^6-24*x^3))*csg
n(I/x^6*(9+(16+ln(x))*x^6-24*x^3))*exp(x-1)+1/2*I*Pi*csgn(I*x)*csgn(I*x^4)*csgn(I*x^5)*exp(x-1)+1/2*I*Pi*csgn(
I*x)*csgn(I*x^2)*csgn(I*x^3)*exp(x-1)+1/2*I*Pi*csgn(I*x)*csgn(I*x^3)*csgn(I*x^4)*exp(x-1)+1/2*I*Pi*csgn(I*x)*c
sgn(I*x^5)*csgn(I*x^6)*exp(x-1)-1/2*I*Pi*csgn(I*x)*csgn(I*x^3)^2*exp(x-1)-1/2*I*Pi*csgn(I*x^2)*csgn(I*x^3)^2*e
xp(x-1)-1/2*I*Pi*csgn(I*x^5)*csgn(I*x^6)^2*exp(x-1)+1/2*I*Pi*csgn(I*x)^2*csgn(I*x^2)*exp(x-1)-1/2*I*Pi*csgn(I*
x^3)*csgn(I*x^4)^2*exp(x-1)-1/2*I*Pi*csgn(I*x)*csgn(I*x^5)^2*exp(x-1)-1/2*I*Pi*csgn(I*x)*csgn(I*x^6)^2*exp(x-1
)-I*Pi*csgn(I*x)*csgn(I*x^2)^2*exp(x-1)+1/2*I*Pi*csgn(I/x^6)*csgn(I/x^6*(9+(16+ln(x))*x^6-24*x^3))^2*exp(x-1)+
1/2*I*Pi*csgn(I*(9+(16+ln(x))*x^6-24*x^3))*csgn(I/x^6*(9+(16+ln(x))*x^6-24*x^3))^2*exp(x-1)+2*ln(ln(x)+(16*x^6
-24*x^3+9)/x^6)-1/2*I*Pi*csgn(I*x)*csgn(I*x^4)^2*exp(x-1)-1/2*I*Pi*csgn(I*x^4)*csgn(I*x^5)^2*exp(x-1)

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maxima [B]  time = 0.39, size = 58, normalized size = 2.76 \begin {gather*} {\left (e^{x} \log \left (x^{6} \log \relax (x) + 16 \, x^{6} - 24 \, x^{3} + 9\right ) - 6 \, e^{x} \log \relax (x)\right )} e^{\left (-1\right )} + 2 \, \log \left (\frac {x^{6} \log \relax (x) + 16 \, x^{6} - 24 \, x^{3} + 9}{x^{6}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^7*exp(x-1)*log(x)+(16*x^7-24*x^4+9*x)*exp(x-1))*log((x^6*log(x)+16*x^6-24*x^3+9)/x^6)+(x^6+72*x^
3-54)*exp(x-1)+2*x^6+144*x^3-108)/(x^7*log(x)+16*x^7-24*x^4+9*x),x, algorithm="maxima")

[Out]

(e^x*log(x^6*log(x) + 16*x^6 - 24*x^3 + 9) - 6*e^x*log(x))*e^(-1) + 2*log((x^6*log(x) + 16*x^6 - 24*x^3 + 9)/x
^6)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {{\mathrm {e}}^{x-1}\,\left (x^6+72\,x^3-54\right )+144\,x^3+2\,x^6+\ln \left (\frac {x^6\,\ln \relax (x)-24\,x^3+16\,x^6+9}{x^6}\right )\,\left ({\mathrm {e}}^{x-1}\,\left (16\,x^7-24\,x^4+9\,x\right )+x^7\,{\mathrm {e}}^{x-1}\,\ln \relax (x)\right )-108}{9\,x+x^7\,\ln \relax (x)-24\,x^4+16\,x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x - 1)*(72*x^3 + x^6 - 54) + 144*x^3 + 2*x^6 + log((x^6*log(x) - 24*x^3 + 16*x^6 + 9)/x^6)*(exp(x - 1
)*(9*x - 24*x^4 + 16*x^7) + x^7*exp(x - 1)*log(x)) - 108)/(9*x + x^7*log(x) - 24*x^4 + 16*x^7),x)

[Out]

int((exp(x - 1)*(72*x^3 + x^6 - 54) + 144*x^3 + 2*x^6 + log((x^6*log(x) - 24*x^3 + 16*x^6 + 9)/x^6)*(exp(x - 1
)*(9*x - 24*x^4 + 16*x^7) + x^7*exp(x - 1)*log(x)) - 108)/(9*x + x^7*log(x) - 24*x^4 + 16*x^7), x)

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sympy [B]  time = 9.50, size = 49, normalized size = 2.33 \begin {gather*} e^{x - 1} \log {\left (\frac {x^{6} \log {\relax (x )} + 16 x^{6} - 24 x^{3} + 9}{x^{6}} \right )} + 2 \log {\left (\log {\relax (x )} + \frac {16 x^{6} - 24 x^{3} + 9}{x^{6}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**7*exp(x-1)*ln(x)+(16*x**7-24*x**4+9*x)*exp(x-1))*ln((x**6*ln(x)+16*x**6-24*x**3+9)/x**6)+(x**6+
72*x**3-54)*exp(x-1)+2*x**6+144*x**3-108)/(x**7*ln(x)+16*x**7-24*x**4+9*x),x)

[Out]

exp(x - 1)*log((x**6*log(x) + 16*x**6 - 24*x**3 + 9)/x**6) + 2*log(log(x) + (16*x**6 - 24*x**3 + 9)/x**6)

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