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

Optimal. Leaf size=28 \[ e^{\frac {1}{4} x \left (4-\left (-e^{4+4 x}+x\right ) \log (\log (x+\log (x)))\right )} \]

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Rubi [F]  time = 20.38, 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 {\exp \left (\frac {1}{4} \left (4 x+\left (e^{4+4 x} x-x^2\right ) \log (\log (x+\log (x)))\right )\right ) \left (-x-x^2+e^{4+4 x} (1+x)+(4 x+4 \log (x)) \log (x+\log (x))+\left (-2 x^2+e^{4+4 x} \left (x+4 x^2\right )+\left (-2 x+e^{4+4 x} (1+4 x)\right ) \log (x)\right ) \log (x+\log (x)) \log (\log (x+\log (x)))\right )}{(4 x+4 \log (x)) \log (x+\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.23, size = 26, normalized size = 0.93 \begin {gather*} e^x \log ^{\frac {1}{4} \left (e^{4+4 x}-x\right ) x}(x+\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^x*Log[x + Log[x]]^(((E^(4 + 4*x) - x)*x)/4)

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fricas [A]  time = 0.53, size = 24, normalized size = 0.86 \begin {gather*} e^{\left (-\frac {1}{4} \, {\left (x^{2} - x e^{\left (4 \, x + 4\right )}\right )} \log \left (\log \left (x + \log \relax (x)\right )\right ) + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {undef} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

undef

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maple [A]  time = 0.08, size = 23, normalized size = 0.82

method	result	size
risch	\(\ln \left (x +\ln \relax (x )\right )^{\frac {\left ({\mathrm e}^{4 x +4}-x \right ) x}{4}} {\mathrm e}^{x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((4*x+1)*exp(x+1)^4-2*x)*ln(x)+(4*x^2+x)*exp(x+1)^4-2*x^2)*ln(x+ln(x))*ln(ln(x+ln(x)))+(4*x+4*ln(x))*ln(
x+ln(x))+(x+1)*exp(x+1)^4-x^2-x)*exp(1/4*(x*exp(x+1)^4-x^2)*ln(ln(x+ln(x)))+x)/(4*x+4*ln(x))/ln(x+ln(x)),x,met
hod=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.51, size = 29, normalized size = 1.04 \begin {gather*} e^{\left (-\frac {1}{4} \, x^{2} \log \left (\log \left (x + \log \relax (x)\right )\right ) + \frac {1}{4} \, x e^{\left (4 \, x + 4\right )} \log \left (\log \left (x + \log \relax (x)\right )\right ) + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 5.79, size = 24, normalized size = 0.86 \begin {gather*} {\ln \left (x+\ln \relax (x)\right )}^{\frac {x\,{\mathrm {e}}^{4\,x+4}}{4}-\frac {x^2}{4}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(x + log(x))^((x*exp(4*x + 4))/4 - x^2/4)*exp(x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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