3.42.6 \(\int \frac {4095-2047 x+384 x^2-32 x^3+x^4+e^x (4095-2047 x+384 x^2-32 x^3+x^4)+(2047-2815 x+864 x^2-100 x^3+4 x^4+e^x (-2047+768 x-96 x^2+4 x^3)) \log (\frac {-1+e^x+x}{\log (3)})}{-1+e^x+x} \, dx\)

Optimal. Leaf size=23 \[ \left (-1+(8-x)^4+x\right ) \log \left (\frac {-1+e^x+x}{\log (3)}\right ) \]

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Rubi [F]  time = 1.22, 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 {4095-2047 x+384 x^2-32 x^3+x^4+e^x \left (4095-2047 x+384 x^2-32 x^3+x^4\right )+\left (2047-2815 x+864 x^2-100 x^3+4 x^4+e^x \left (-2047+768 x-96 x^2+4 x^3\right )\right ) \log \left (\frac {-1+e^x+x}{\log (3)}\right )}{-1+e^x+x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4095 - 2047*x + 384*x^2 - 32*x^3 + x^4 + E^x*(4095 - 2047*x + 384*x^2 - 32*x^3 + x^4) + (2047 - 2815*x +
864*x^2 - 100*x^3 + 4*x^4 + E^x*(-2047 + 768*x - 96*x^2 + 4*x^3))*Log[(-1 + E^x + x)/Log[3]])/(-1 + E^x + x),x
]

[Out]

4095*x - 2047*x*Log[-((1 - E^x - x)/Log[3])] + 384*x^2*Log[-((1 - E^x - x)/Log[3])] - 32*x^3*Log[-((1 - E^x -
x)/Log[3])] + x^4*Log[-((1 - E^x - x)/Log[3])] + 8190*Defer[Int][(-1 + E^x + x)^(-1), x] - 4095*Defer[Int][x/(
-1 + E^x + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {\left (1+e^x\right ) \left (4095-2047 x+384 x^2-32 x^3+x^4\right )}{-1+e^x+x}+\left (-2047+768 x-96 x^2+4 x^3\right ) \log \left (\frac {-1+e^x+x}{\log (3)}\right )\right ) \, dx\\ &=\int \frac {\left (1+e^x\right ) \left (4095-2047 x+384 x^2-32 x^3+x^4\right )}{-1+e^x+x} \, dx+\int \left (-2047+768 x-96 x^2+4 x^3\right ) \log \left (\frac {-1+e^x+x}{\log (3)}\right ) \, dx\\ &=-2047 x \log \left (-\frac {1-e^x-x}{\log (3)}\right )+384 x^2 \log \left (-\frac {1-e^x-x}{\log (3)}\right )-32 x^3 \log \left (-\frac {1-e^x-x}{\log (3)}\right )+x^4 \log \left (-\frac {1-e^x-x}{\log (3)}\right )-\int \frac {\left (1+e^x\right ) x \left (2047-384 x+32 x^2-x^3\right )}{1-e^x-x} \, dx+\int \left (4095-2047 x+384 x^2-32 x^3+x^4-\frac {-8190+8189 x-2815 x^2+448 x^3-34 x^4+x^5}{-1+e^x+x}\right ) \, dx\\ &=4095 x-\frac {2047 x^2}{2}+128 x^3-8 x^4+\frac {x^5}{5}-2047 x \log \left (-\frac {1-e^x-x}{\log (3)}\right )+384 x^2 \log \left (-\frac {1-e^x-x}{\log (3)}\right )-32 x^3 \log \left (-\frac {1-e^x-x}{\log (3)}\right )+x^4 \log \left (-\frac {1-e^x-x}{\log (3)}\right )-\int \frac {-8190+8189 x-2815 x^2+448 x^3-34 x^4+x^5}{-1+e^x+x} \, dx-\int \left (x \left (-2047+384 x-32 x^2+x^3\right )-\frac {x \left (4094-2815 x+448 x^2-34 x^3+x^4\right )}{-1+e^x+x}\right ) \, dx\\ &=4095 x-\frac {2047 x^2}{2}+128 x^3-8 x^4+\frac {x^5}{5}-2047 x \log \left (-\frac {1-e^x-x}{\log (3)}\right )+384 x^2 \log \left (-\frac {1-e^x-x}{\log (3)}\right )-32 x^3 \log \left (-\frac {1-e^x-x}{\log (3)}\right )+x^4 \log \left (-\frac {1-e^x-x}{\log (3)}\right )-\int x \left (-2047+384 x-32 x^2+x^3\right ) \, dx+\int \frac {x \left (4094-2815 x+448 x^2-34 x^3+x^4\right )}{-1+e^x+x} \, dx-\int \left (-\frac {8190}{-1+e^x+x}+\frac {8189 x}{-1+e^x+x}-\frac {2815 x^2}{-1+e^x+x}+\frac {448 x^3}{-1+e^x+x}-\frac {34 x^4}{-1+e^x+x}+\frac {x^5}{-1+e^x+x}\right ) \, dx\\ &=4095 x-\frac {2047 x^2}{2}+128 x^3-8 x^4+\frac {x^5}{5}-2047 x \log \left (-\frac {1-e^x-x}{\log (3)}\right )+384 x^2 \log \left (-\frac {1-e^x-x}{\log (3)}\right )-32 x^3 \log \left (-\frac {1-e^x-x}{\log (3)}\right )+x^4 \log \left (-\frac {1-e^x-x}{\log (3)}\right )+34 \int \frac {x^4}{-1+e^x+x} \, dx-448 \int \frac {x^3}{-1+e^x+x} \, dx+2815 \int \frac {x^2}{-1+e^x+x} \, dx-8189 \int \frac {x}{-1+e^x+x} \, dx+8190 \int \frac {1}{-1+e^x+x} \, dx-\int \frac {x^5}{-1+e^x+x} \, dx-\int \left (-2047 x+384 x^2-32 x^3+x^4\right ) \, dx+\int \left (\frac {4094 x}{-1+e^x+x}-\frac {2815 x^2}{-1+e^x+x}+\frac {448 x^3}{-1+e^x+x}-\frac {34 x^4}{-1+e^x+x}+\frac {x^5}{-1+e^x+x}\right ) \, dx\\ &=4095 x-2047 x \log \left (-\frac {1-e^x-x}{\log (3)}\right )+384 x^2 \log \left (-\frac {1-e^x-x}{\log (3)}\right )-32 x^3 \log \left (-\frac {1-e^x-x}{\log (3)}\right )+x^4 \log \left (-\frac {1-e^x-x}{\log (3)}\right )+4094 \int \frac {x}{-1+e^x+x} \, dx-8189 \int \frac {x}{-1+e^x+x} \, dx+8190 \int \frac {1}{-1+e^x+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 41, normalized size = 1.78 \begin {gather*} 4095 \log \left (1-e^x-x\right )+x \left (-2047+384 x-32 x^2+x^3\right ) \log \left (\frac {-1+e^x+x}{\log (3)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4095 - 2047*x + 384*x^2 - 32*x^3 + x^4 + E^x*(4095 - 2047*x + 384*x^2 - 32*x^3 + x^4) + (2047 - 281
5*x + 864*x^2 - 100*x^3 + 4*x^4 + E^x*(-2047 + 768*x - 96*x^2 + 4*x^3))*Log[(-1 + E^x + x)/Log[3]])/(-1 + E^x
+ x),x]

[Out]

4095*Log[1 - E^x - x] + x*(-2047 + 384*x - 32*x^2 + x^3)*Log[(-1 + E^x + x)/Log[3]]

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fricas [A]  time = 1.04, size = 30, normalized size = 1.30 \begin {gather*} {\left (x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2047 \, x + 4095\right )} \log \left (\frac {x + e^{x} - 1}{\log \relax (3)}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^3-96*x^2+768*x-2047)*exp(x)+4*x^4-100*x^3+864*x^2-2815*x+2047)*log((x+exp(x)-1)/log(3))+(x^4-
32*x^3+384*x^2-2047*x+4095)*exp(x)+x^4-32*x^3+384*x^2-2047*x+4095)/(x+exp(x)-1),x, algorithm="fricas")

[Out]

(x^4 - 32*x^3 + 384*x^2 - 2047*x + 4095)*log((x + e^x - 1)/log(3))

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giac [B]  time = 0.20, size = 80, normalized size = 3.48 \begin {gather*} x^{4} \log \left (x + e^{x} - 1\right ) - x^{4} \log \left (\log \relax (3)\right ) - 32 \, x^{3} \log \left (x + e^{x} - 1\right ) + 32 \, x^{3} \log \left (\log \relax (3)\right ) + 384 \, x^{2} \log \left (x + e^{x} - 1\right ) - 384 \, x^{2} \log \left (\log \relax (3)\right ) - 2047 \, x \log \left (x + e^{x} - 1\right ) + 2047 \, x \log \left (\log \relax (3)\right ) + 4095 \, \log \left (x + e^{x} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^3-96*x^2+768*x-2047)*exp(x)+4*x^4-100*x^3+864*x^2-2815*x+2047)*log((x+exp(x)-1)/log(3))+(x^4-
32*x^3+384*x^2-2047*x+4095)*exp(x)+x^4-32*x^3+384*x^2-2047*x+4095)/(x+exp(x)-1),x, algorithm="giac")

[Out]

x^4*log(x + e^x - 1) - x^4*log(log(3)) - 32*x^3*log(x + e^x - 1) + 32*x^3*log(log(3)) + 384*x^2*log(x + e^x -
1) - 384*x^2*log(log(3)) - 2047*x*log(x + e^x - 1) + 2047*x*log(log(3)) + 4095*log(x + e^x - 1)

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maple [A]  time = 0.05, size = 39, normalized size = 1.70

method	result	size
risch	\(\left (x^{4}-32 x^{3}+384 x^{2}-2047 x \right ) \ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \relax (3)}\right )+4095 \ln \left (x +{\mathrm e}^{x}-1\right )\)	\(39\)
norman	\(4095 \ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \relax (3)}\right )+\ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \relax (3)}\right ) x^{4}-2047 \ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \relax (3)}\right ) x +384 \ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \relax (3)}\right ) x^{2}-32 \ln \left (\frac {x +{\mathrm e}^{x}-1}{\ln \relax (3)}\right ) x^{3}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((4*x^3-96*x^2+768*x-2047)*exp(x)+4*x^4-100*x^3+864*x^2-2815*x+2047)*ln((x+exp(x)-1)/ln(3))+(x^4-32*x^3+3
84*x^2-2047*x+4095)*exp(x)+x^4-32*x^3+384*x^2-2047*x+4095)/(x+exp(x)-1),x,method=_RETURNVERBOSE)

[Out]

(x^4-32*x^3+384*x^2-2047*x)*ln((x+exp(x)-1)/ln(3))+4095*ln(x+exp(x)-1)

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maxima [B]  time = 0.49, size = 56, normalized size = 2.43 \begin {gather*} -x^{4} \log \left (\log \relax (3)\right ) + 32 \, x^{3} \log \left (\log \relax (3)\right ) - 384 \, x^{2} \log \left (\log \relax (3)\right ) + {\left (x^{4} - 32 \, x^{3} + 384 \, x^{2} - 2047 \, x + 4095\right )} \log \left (x + e^{x} - 1\right ) + 2047 \, x \log \left (\log \relax (3)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x^3-96*x^2+768*x-2047)*exp(x)+4*x^4-100*x^3+864*x^2-2815*x+2047)*log((x+exp(x)-1)/log(3))+(x^4-
32*x^3+384*x^2-2047*x+4095)*exp(x)+x^4-32*x^3+384*x^2-2047*x+4095)/(x+exp(x)-1),x, algorithm="maxima")

[Out]

-x^4*log(log(3)) + 32*x^3*log(log(3)) - 384*x^2*log(log(3)) + (x^4 - 32*x^3 + 384*x^2 - 2047*x + 4095)*log(x +
e^x - 1) + 2047*x*log(log(3))

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mupad [B]  time = 0.20, size = 41, normalized size = 1.78 \begin {gather*} 4095\,\ln \left (x+{\mathrm {e}}^x-1\right )-\ln \left (\frac {x+{\mathrm {e}}^x-1}{\ln \relax (3)}\right )\,\left (-x^4+32\,x^3-384\,x^2+2047\,x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((x + exp(x) - 1)/log(3))*(864*x^2 - 2815*x - 100*x^3 + 4*x^4 + exp(x)*(768*x - 96*x^2 + 4*x^3 - 2047)
+ 2047) - 2047*x + exp(x)*(384*x^2 - 2047*x - 32*x^3 + x^4 + 4095) + 384*x^2 - 32*x^3 + x^4 + 4095)/(x + exp(
x) - 1),x)

[Out]

4095*log(x + exp(x) - 1) - log((x + exp(x) - 1)/log(3))*(2047*x - 384*x^2 + 32*x^3 - x^4)

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sympy [A]  time = 0.39, size = 37, normalized size = 1.61 \begin {gather*} \left (x^{4} - 32 x^{3} + 384 x^{2} - 2047 x\right ) \log {\left (\frac {x + e^{x} - 1}{\log {\relax (3 )}} \right )} + 4095 \log {\left (x + e^{x} - 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((4*x**3-96*x**2+768*x-2047)*exp(x)+4*x**4-100*x**3+864*x**2-2815*x+2047)*ln((x+exp(x)-1)/ln(3))+(x
**4-32*x**3+384*x**2-2047*x+4095)*exp(x)+x**4-32*x**3+384*x**2-2047*x+4095)/(x+exp(x)-1),x)

[Out]

(x**4 - 32*x**3 + 384*x**2 - 2047*x)*log((x + exp(x) - 1)/log(3)) + 4095*log(x + exp(x) - 1)

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