3.78.68 \(\int \frac {2-e^x-x-x^2+(64 x^2+64 e^x x^2+128 x^3) \log ^3(-2+e^x+x+x^2)}{-2 x^2+e^x x^2+x^3+x^4} \, dx\)

Optimal. Leaf size=18 \[ \frac {1}{x}+16 \log ^4\left (-2+e^x+x+x^2\right ) \]

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Rubi [F]  time = 0.87, 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 {2-e^x-x-x^2+\left (64 x^2+64 e^x x^2+128 x^3\right ) \log ^3\left (-2+e^x+x+x^2\right )}{-2 x^2+e^x x^2+x^3+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 - E^x - x - x^2 + (64*x^2 + 64*E^x*x^2 + 128*x^3)*Log[-2 + E^x + x + x^2]^3)/(-2*x^2 + E^x*x^2 + x^3 +
x^4),x]

[Out]

x^(-1) + 64*Defer[Int][Log[-2 + E^x + x + x^2]^3, x] + 192*Defer[Int][Log[-2 + E^x + x + x^2]^3/(-2 + E^x + x
+ x^2), x] + 64*Defer[Int][(x*Log[-2 + E^x + x + x^2]^3)/(-2 + E^x + x + x^2), x] - 64*Defer[Int][(x^2*Log[-2
+ E^x + x + x^2]^3)/(-2 + E^x + x + x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {64 \left (-3-x+x^2\right ) \log ^3\left (-2+e^x+x+x^2\right )}{-2+e^x+x+x^2}+\frac {-1+64 x^2 \log ^3\left (-2+e^x+x+x^2\right )}{x^2}\right ) \, dx\\ &=-\left (64 \int \frac {\left (-3-x+x^2\right ) \log ^3\left (-2+e^x+x+x^2\right )}{-2+e^x+x+x^2} \, dx\right )+\int \frac {-1+64 x^2 \log ^3\left (-2+e^x+x+x^2\right )}{x^2} \, dx\\ &=-\left (64 \int \left (-\frac {3 \log ^3\left (-2+e^x+x+x^2\right )}{-2+e^x+x+x^2}-\frac {x \log ^3\left (-2+e^x+x+x^2\right )}{-2+e^x+x+x^2}+\frac {x^2 \log ^3\left (-2+e^x+x+x^2\right )}{-2+e^x+x+x^2}\right ) \, dx\right )+\int \left (-\frac {1}{x^2}+64 \log ^3\left (-2+e^x+x+x^2\right )\right ) \, dx\\ &=\frac {1}{x}+64 \int \log ^3\left (-2+e^x+x+x^2\right ) \, dx+64 \int \frac {x \log ^3\left (-2+e^x+x+x^2\right )}{-2+e^x+x+x^2} \, dx-64 \int \frac {x^2 \log ^3\left (-2+e^x+x+x^2\right )}{-2+e^x+x+x^2} \, dx+192 \int \frac {\log ^3\left (-2+e^x+x+x^2\right )}{-2+e^x+x+x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 18, normalized size = 1.00 \begin {gather*} \frac {1}{x}+16 \log ^4\left (-2+e^x+x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - E^x - x - x^2 + (64*x^2 + 64*E^x*x^2 + 128*x^3)*Log[-2 + E^x + x + x^2]^3)/(-2*x^2 + E^x*x^2 +
x^3 + x^4),x]

[Out]

x^(-1) + 16*Log[-2 + E^x + x + x^2]^4

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fricas [A]  time = 1.15, size = 20, normalized size = 1.11 \begin {gather*} \frac {16 \, x \log \left (x^{2} + x + e^{x} - 2\right )^{4} + 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*exp(x)*x^2+128*x^3+64*x^2)*log(exp(x)+x^2+x-2)^3-exp(x)-x^2-x+2)/(exp(x)*x^2+x^4+x^3-2*x^2),x,
algorithm="fricas")

[Out]

(16*x*log(x^2 + x + e^x - 2)^4 + 1)/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {64 \, {\left (2 \, x^{3} + x^{2} e^{x} + x^{2}\right )} \log \left (x^{2} + x + e^{x} - 2\right )^{3} - x^{2} - x - e^{x} + 2}{x^{4} + x^{3} + x^{2} e^{x} - 2 \, x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*exp(x)*x^2+128*x^3+64*x^2)*log(exp(x)+x^2+x-2)^3-exp(x)-x^2-x+2)/(exp(x)*x^2+x^4+x^3-2*x^2),x,
algorithm="giac")

[Out]

integrate((64*(2*x^3 + x^2*e^x + x^2)*log(x^2 + x + e^x - 2)^3 - x^2 - x - e^x + 2)/(x^4 + x^3 + x^2*e^x - 2*x
^2), x)

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maple [A]  time = 0.02, size = 18, normalized size = 1.00

method	result	size
risch	\(16 \ln \left ({\mathrm e}^{x}+x^{2}+x -2\right )^{4}+\frac {1}{x}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((64*exp(x)*x^2+128*x^3+64*x^2)*ln(exp(x)+x^2+x-2)^3-exp(x)-x^2-x+2)/(exp(x)*x^2+x^4+x^3-2*x^2),x,method=_
RETURNVERBOSE)

[Out]

16*ln(exp(x)+x^2+x-2)^4+1/x

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maxima [A]  time = 0.40, size = 20, normalized size = 1.11 \begin {gather*} \frac {16 \, x \log \left (x^{2} + x + e^{x} - 2\right )^{4} + 1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*exp(x)*x^2+128*x^3+64*x^2)*log(exp(x)+x^2+x-2)^3-exp(x)-x^2-x+2)/(exp(x)*x^2+x^4+x^3-2*x^2),x,
algorithm="maxima")

[Out]

(16*x*log(x^2 + x + e^x - 2)^4 + 1)/x

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mupad [B]  time = 0.24, size = 17, normalized size = 0.94 \begin {gather*} 16\,{\ln \left (x+{\mathrm {e}}^x+x^2-2\right )}^4+\frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x + exp(x) - log(x + exp(x) + x^2 - 2)^3*(64*x^2*exp(x) + 64*x^2 + 128*x^3) + x^2 - 2)/(x^2*exp(x) - 2*x
^2 + x^3 + x^4),x)

[Out]

16*log(x + exp(x) + x^2 - 2)^4 + 1/x

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sympy [A]  time = 0.43, size = 17, normalized size = 0.94 \begin {gather*} 16 \log {\left (x^{2} + x + e^{x} - 2 \right )}^{4} + \frac {1}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((64*exp(x)*x**2+128*x**3+64*x**2)*ln(exp(x)+x**2+x-2)**3-exp(x)-x**2-x+2)/(exp(x)*x**2+x**4+x**3-2*
x**2),x)

[Out]

16*log(x**2 + x + exp(x) - 2)**4 + 1/x

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