3.46.71 \(\int \frac {e^{\frac {1+e^{12} x}{x}} (x-x^2-4 x^5)+(e^{\frac {1+e^{12} x}{x}} (x-2 x^2+x^4-2 x^5)+e^{\frac {1+e^{12} x}{x}} (-1+2 x) \log (x)) \log (x+x^4-\log (x))}{-x-x^4+\log (x)} \, dx\)

Optimal. Leaf size=23 \[ e^{e^{12}+\frac {1}{x}} x^2 \log \left (x+x^4-\log (x)\right ) \]

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Rubi [F]  time = 1.20, 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 {e^{\frac {1+e^{12} x}{x}} \left (x-x^2-4 x^5\right )+\left (e^{\frac {1+e^{12} x}{x}} \left (x-2 x^2+x^4-2 x^5\right )+e^{\frac {1+e^{12} x}{x}} (-1+2 x) \log (x)\right ) \log \left (x+x^4-\log (x)\right )}{-x-x^4+\log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((1 + E^12*x)/x)*(x - x^2 - 4*x^5) + (E^((1 + E^12*x)/x)*(x - 2*x^2 + x^4 - 2*x^5) + E^((1 + E^12*x)/x)
*(-1 + 2*x)*Log[x])*Log[x + x^4 - Log[x]])/(-x - x^4 + Log[x]),x]

[Out]

-Defer[Int][(E^(E^12 + x^(-1))*x)/(x + x^4 - Log[x]), x] + Defer[Int][(E^(E^12 + x^(-1))*x^2)/(x + x^4 - Log[x
]), x] + 4*Defer[Int][(E^(E^12 + x^(-1))*x^5)/(x + x^4 - Log[x]), x] - Defer[Int][E^(E^12 + x^(-1))*Log[x + x^
4 - Log[x]], x] + 2*Defer[Int][E^(E^12 + x^(-1))*x*Log[x + x^4 - Log[x]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{e^{12}+\frac {1}{x}} \left (x \left (-1+x+4 x^4\right )+(-1+2 x) \left (x+x^4-\log (x)\right ) \log \left (x+x^4-\log (x)\right )\right )}{x+x^4-\log (x)} \, dx\\ &=\int \left (\frac {e^{e^{12}+\frac {1}{x}} x \left (-1+x+4 x^4\right )}{x+x^4-\log (x)}+e^{e^{12}+\frac {1}{x}} (-1+2 x) \log \left (x+x^4-\log (x)\right )\right ) \, dx\\ &=\int \frac {e^{e^{12}+\frac {1}{x}} x \left (-1+x+4 x^4\right )}{x+x^4-\log (x)} \, dx+\int e^{e^{12}+\frac {1}{x}} (-1+2 x) \log \left (x+x^4-\log (x)\right ) \, dx\\ &=\int \left (-\frac {e^{e^{12}+\frac {1}{x}} x}{x+x^4-\log (x)}+\frac {e^{e^{12}+\frac {1}{x}} x^2}{x+x^4-\log (x)}+\frac {4 e^{e^{12}+\frac {1}{x}} x^5}{x+x^4-\log (x)}\right ) \, dx+\int \left (-e^{e^{12}+\frac {1}{x}} \log \left (x+x^4-\log (x)\right )+2 e^{e^{12}+\frac {1}{x}} x \log \left (x+x^4-\log (x)\right )\right ) \, dx\\ &=2 \int e^{e^{12}+\frac {1}{x}} x \log \left (x+x^4-\log (x)\right ) \, dx+4 \int \frac {e^{e^{12}+\frac {1}{x}} x^5}{x+x^4-\log (x)} \, dx-\int \frac {e^{e^{12}+\frac {1}{x}} x}{x+x^4-\log (x)} \, dx+\int \frac {e^{e^{12}+\frac {1}{x}} x^2}{x+x^4-\log (x)} \, dx-\int e^{e^{12}+\frac {1}{x}} \log \left (x+x^4-\log (x)\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.27, size = 23, normalized size = 1.00 \begin {gather*} e^{e^{12}+\frac {1}{x}} x^2 \log \left (x+x^4-\log (x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((1 + E^12*x)/x)*(x - x^2 - 4*x^5) + (E^((1 + E^12*x)/x)*(x - 2*x^2 + x^4 - 2*x^5) + E^((1 + E^12
*x)/x)*(-1 + 2*x)*Log[x])*Log[x + x^4 - Log[x]])/(-x - x^4 + Log[x]),x]

[Out]

E^(E^12 + x^(-1))*x^2*Log[x + x^4 - Log[x]]

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fricas [A]  time = 0.64, size = 25, normalized size = 1.09 \begin {gather*} x^{2} e^{\left (\frac {x e^{12} + 1}{x}\right )} \log \left (x^{4} + x - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x-1)*exp((x*exp(6)^2+1)/x)*log(x)+(-2*x^5+x^4-2*x^2+x)*exp((x*exp(6)^2+1)/x))*log(-log(x)+x^4+x
)+(-4*x^5-x^2+x)*exp((x*exp(6)^2+1)/x))/(log(x)-x^4-x),x, algorithm="fricas")

[Out]

x^2*e^((x*e^12 + 1)/x)*log(x^4 + x - log(x))

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giac [A]  time = 0.33, size = 25, normalized size = 1.09 \begin {gather*} x^{2} e^{\left (\frac {x e^{12} + 1}{x}\right )} \log \left (x^{4} + x - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x-1)*exp((x*exp(6)^2+1)/x)*log(x)+(-2*x^5+x^4-2*x^2+x)*exp((x*exp(6)^2+1)/x))*log(-log(x)+x^4+x
)+(-4*x^5-x^2+x)*exp((x*exp(6)^2+1)/x))/(log(x)-x^4-x),x, algorithm="giac")

[Out]

x^2*e^((x*e^12 + 1)/x)*log(x^4 + x - log(x))

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maple [A]  time = 0.03, size = 26, normalized size = 1.13

method	result	size
risch	\(x^{2} {\mathrm e}^{\frac {x \,{\mathrm e}^{12}+1}{x}} \ln \left (-\ln \relax (x )+x^{4}+x \right )\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((2*x-1)*exp((x*exp(6)^2+1)/x)*ln(x)+(-2*x^5+x^4-2*x^2+x)*exp((x*exp(6)^2+1)/x))*ln(-ln(x)+x^4+x)+(-4*x^5
-x^2+x)*exp((x*exp(6)^2+1)/x))/(ln(x)-x^4-x),x,method=_RETURNVERBOSE)

[Out]

x^2*exp((x*exp(12)+1)/x)*ln(-ln(x)+x^4+x)

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maxima [A]  time = 0.43, size = 21, normalized size = 0.91 \begin {gather*} x^{2} e^{\left (\frac {1}{x} + e^{12}\right )} \log \left (x^{4} + x - \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x-1)*exp((x*exp(6)^2+1)/x)*log(x)+(-2*x^5+x^4-2*x^2+x)*exp((x*exp(6)^2+1)/x))*log(-log(x)+x^4+x
)+(-4*x^5-x^2+x)*exp((x*exp(6)^2+1)/x))/(log(x)-x^4-x),x, algorithm="maxima")

[Out]

x^2*e^(1/x + e^12)*log(x^4 + x - log(x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} -\int \frac {\ln \left (x-\ln \relax (x)+x^4\right )\,\left ({\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{12}+1}{x}}\,\left (-2\,x^5+x^4-2\,x^2+x\right )+{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{12}+1}{x}}\,\ln \relax (x)\,\left (2\,x-1\right )\right )-{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^{12}+1}{x}}\,\left (4\,x^5+x^2-x\right )}{x-\ln \relax (x)+x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x - log(x) + x^4)*(exp((x*exp(12) + 1)/x)*(x - 2*x^2 + x^4 - 2*x^5) + exp((x*exp(12) + 1)/x)*log(x)*
(2*x - 1)) - exp((x*exp(12) + 1)/x)*(x^2 - x + 4*x^5))/(x - log(x) + x^4),x)

[Out]

-int((log(x - log(x) + x^4)*(exp((x*exp(12) + 1)/x)*(x - 2*x^2 + x^4 - 2*x^5) + exp((x*exp(12) + 1)/x)*log(x)*
(2*x - 1)) - exp((x*exp(12) + 1)/x)*(x^2 - x + 4*x^5))/(x - log(x) + x^4), x)

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sympy [A]  time = 2.39, size = 22, normalized size = 0.96 \begin {gather*} x^{2} e^{\frac {x e^{12} + 1}{x}} \log {\left (x^{4} + x - \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((2*x-1)*exp((x*exp(6)**2+1)/x)*ln(x)+(-2*x**5+x**4-2*x**2+x)*exp((x*exp(6)**2+1)/x))*ln(-ln(x)+x**
4+x)+(-4*x**5-x**2+x)*exp((x*exp(6)**2+1)/x))/(ln(x)-x**4-x),x)

[Out]

x**2*exp((x*exp(12) + 1)/x)*log(x**4 + x - log(x))

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