3.83.21 \(\int \frac {(-4+4 e^{4-x}) \log ^2(x)+e^{\frac {5 x^2}{4 \log (x)}} (-5 x+10 x \log (x))}{4 e^{\frac {5 x^2}{4 \log (x)}} \log ^2(x)+(-28-4 e^{4-x}-4 x) \log ^2(x)} \, dx\)

Optimal. Leaf size=26 \[ \log \left (7+e^{4-x}-e^{\frac {5 x^2}{4 \log (x)}}+x\right ) \]

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Rubi [F]  time = 5.25, 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 {\left (-4+4 e^{4-x}\right ) \log ^2(x)+e^{\frac {5 x^2}{4 \log (x)}} (-5 x+10 x \log (x))}{4 e^{\frac {5 x^2}{4 \log (x)}} \log ^2(x)+\left (-28-4 e^{4-x}-4 x\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-4 + 4*E^(4 - x))*Log[x]^2 + E^((5*x^2)/(4*Log[x]))*(-5*x + 10*x*Log[x]))/(4*E^((5*x^2)/(4*Log[x]))*Log[
x]^2 + (-28 - 4*E^(4 - x) - 4*x)*Log[x]^2),x]

[Out]

(5*x^2)/2 + (5*ExpIntegralEi[2*Log[x]])/2 - (5*ExpIntegralEi[2*Log[x]]*(1 - 2*Log[x]))/2 + (5*x^2*(1 - 2*Log[x
]))/(4*Log[x]) - 5*ExpIntegralEi[2*Log[x]]*Log[x] - Defer[Int][E^x/(-E^4 - 7*E^x + E^(x + (5*x^2)/(4*Log[x]))
- E^x*x), x] - E^4*Defer[Int][(E^4 + 7*E^x - E^(x + (5*x^2)/(4*Log[x])) + E^x*x)^(-1), x] + (5*E^4*Defer[Int][
x/((E^4 + 7*E^x - E^(x + (5*x^2)/(4*Log[x])) + E^x*x)*Log[x]^2), x])/4 + (35*Defer[Int][(E^x*x)/((E^4 + 7*E^x
- E^(x + (5*x^2)/(4*Log[x])) + E^x*x)*Log[x]^2), x])/4 + (5*Defer[Int][(E^x*x^2)/((E^4 + 7*E^x - E^(x + (5*x^2
)/(4*Log[x])) + E^x*x)*Log[x]^2), x])/4 - (5*E^4*Defer[Int][x/((E^4 + 7*E^x - E^(x + (5*x^2)/(4*Log[x])) + E^x
*x)*Log[x]), x])/2 - (35*Defer[Int][(E^x*x)/((E^4 + 7*E^x - E^(x + (5*x^2)/(4*Log[x])) + E^x*x)*Log[x]), x])/2
- (5*Defer[Int][(E^x*x^2)/((E^4 + 7*E^x - E^(x + (5*x^2)/(4*Log[x])) + E^x*x)*Log[x]), x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {5 x (-1+2 \log (x))}{4 \log ^2(x)}+\frac {-5 e^4 x-35 e^x x-5 e^x x^2+10 e^4 x \log (x)+70 e^x x \log (x)+10 e^x x^2 \log (x)+4 e^4 \log ^2(x)-4 e^x \log ^2(x)}{4 \left (-e^4-7 e^x+e^{x+\frac {5 x^2}{4 \log (x)}}-e^x x\right ) \log ^2(x)}\right ) \, dx\\ &=\frac {1}{4} \int \frac {-5 e^4 x-35 e^x x-5 e^x x^2+10 e^4 x \log (x)+70 e^x x \log (x)+10 e^x x^2 \log (x)+4 e^4 \log ^2(x)-4 e^x \log ^2(x)}{\left (-e^4-7 e^x+e^{x+\frac {5 x^2}{4 \log (x)}}-e^x x\right ) \log ^2(x)} \, dx+\frac {5}{4} \int \frac {x (-1+2 \log (x))}{\log ^2(x)} \, dx\\ &=-\frac {5}{2} \text {Ei}(2 \log (x)) (1-2 \log (x))+\frac {5 x^2 (1-2 \log (x))}{4 \log (x)}+\frac {1}{4} \int \left (-\frac {4 e^x}{-e^4-7 e^x+e^{x+\frac {5 x^2}{4 \log (x)}}-e^x x}-\frac {4 e^4}{e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x}+\frac {5 e^4 x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log ^2(x)}+\frac {35 e^x x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log ^2(x)}+\frac {5 e^x x^2}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log ^2(x)}-\frac {10 e^4 x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log (x)}-\frac {70 e^x x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log (x)}-\frac {10 e^x x^2}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log (x)}\right ) \, dx-\frac {5}{2} \int \left (\frac {2 \text {Ei}(2 \log (x))}{x}-\frac {x}{\log (x)}\right ) \, dx\\ &=-\frac {5}{2} \text {Ei}(2 \log (x)) (1-2 \log (x))+\frac {5 x^2 (1-2 \log (x))}{4 \log (x)}+\frac {5}{4} \int \frac {e^x x^2}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log ^2(x)} \, dx+\frac {5}{2} \int \frac {x}{\log (x)} \, dx-\frac {5}{2} \int \frac {e^x x^2}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log (x)} \, dx-5 \int \frac {\text {Ei}(2 \log (x))}{x} \, dx+\frac {35}{4} \int \frac {e^x x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log ^2(x)} \, dx-\frac {35}{2} \int \frac {e^x x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log (x)} \, dx-e^4 \int \frac {1}{e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x} \, dx+\frac {1}{4} \left (5 e^4\right ) \int \frac {x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log ^2(x)} \, dx-\frac {1}{2} \left (5 e^4\right ) \int \frac {x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log (x)} \, dx-\int \frac {e^x}{-e^4-7 e^x+e^{x+\frac {5 x^2}{4 \log (x)}}-e^x x} \, dx\\ &=-\frac {5}{2} \text {Ei}(2 \log (x)) (1-2 \log (x))+\frac {5 x^2 (1-2 \log (x))}{4 \log (x)}+\frac {5}{4} \int \frac {e^x x^2}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log ^2(x)} \, dx-\frac {5}{2} \int \frac {e^x x^2}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log (x)} \, dx+\frac {5}{2} \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )-5 \operatorname {Subst}(\int \text {Ei}(2 x) \, dx,x,\log (x))+\frac {35}{4} \int \frac {e^x x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log ^2(x)} \, dx-\frac {35}{2} \int \frac {e^x x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log (x)} \, dx-e^4 \int \frac {1}{e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x} \, dx+\frac {1}{4} \left (5 e^4\right ) \int \frac {x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log ^2(x)} \, dx-\frac {1}{2} \left (5 e^4\right ) \int \frac {x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log (x)} \, dx-\int \frac {e^x}{-e^4-7 e^x+e^{x+\frac {5 x^2}{4 \log (x)}}-e^x x} \, dx\\ &=\frac {5 x^2}{2}+\frac {5}{2} \text {Ei}(2 \log (x))-\frac {5}{2} \text {Ei}(2 \log (x)) (1-2 \log (x))+\frac {5 x^2 (1-2 \log (x))}{4 \log (x)}-5 \text {Ei}(2 \log (x)) \log (x)+\frac {5}{4} \int \frac {e^x x^2}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log ^2(x)} \, dx-\frac {5}{2} \int \frac {e^x x^2}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log (x)} \, dx+\frac {35}{4} \int \frac {e^x x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log ^2(x)} \, dx-\frac {35}{2} \int \frac {e^x x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log (x)} \, dx-e^4 \int \frac {1}{e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x} \, dx+\frac {1}{4} \left (5 e^4\right ) \int \frac {x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log ^2(x)} \, dx-\frac {1}{2} \left (5 e^4\right ) \int \frac {x}{\left (e^4+7 e^x-e^{x+\frac {5 x^2}{4 \log (x)}}+e^x x\right ) \log (x)} \, dx-\int \frac {e^x}{-e^4-7 e^x+e^{x+\frac {5 x^2}{4 \log (x)}}-e^x x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 40, normalized size = 1.54 \begin {gather*} \frac {1}{4} \left (-4 x+4 \log \left (-e^4+e^{x+\frac {5 x^2}{4 \log (x)}}-e^x (7+x)\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-4 + 4*E^(4 - x))*Log[x]^2 + E^((5*x^2)/(4*Log[x]))*(-5*x + 10*x*Log[x]))/(4*E^((5*x^2)/(4*Log[x])
)*Log[x]^2 + (-28 - 4*E^(4 - x) - 4*x)*Log[x]^2),x]

[Out]

(-4*x + 4*Log[-E^4 + E^(x + (5*x^2)/(4*Log[x])) - E^x*(7 + x)])/4

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fricas [A]  time = 0.55, size = 24, normalized size = 0.92 \begin {gather*} \log \left (-x - e^{\left (-x + 4\right )} + e^{\left (\frac {5 \, x^{2}}{4 \, \log \relax (x)}\right )} - 7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x*log(x)-5*x)*exp(5/4*x^2/log(x))+(4*exp(-x+4)-4)*log(x)^2)/(4*log(x)^2*exp(5/4*x^2/log(x))+(-4
*exp(-x+4)-4*x-28)*log(x)^2),x, algorithm="fricas")

[Out]

log(-x - e^(-x + 4) + e^(5/4*x^2/log(x)) - 7)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x*log(x)-5*x)*exp(5/4*x^2/log(x))+(4*exp(-x+4)-4)*log(x)^2)/(4*log(x)^2*exp(5/4*x^2/log(x))+(-4
*exp(-x+4)-4*x-28)*log(x)^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{62500,[0,17]%%%} / %%%{250000,[0,17]%%%} Error: Bad Argu
ment Value

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maple [A]  time = 0.04, size = 25, normalized size = 0.96

method	result	size
risch	\(\ln \left ({\mathrm e}^{\frac {5 x^{2}}{4 \ln \relax (x )}}-{\mathrm e}^{-x +4}-x -7\right )\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x*ln(x)-5*x)*exp(5/4*x^2/ln(x))+(4*exp(-x+4)-4)*ln(x)^2)/(4*ln(x)^2*exp(5/4*x^2/ln(x))+(-4*exp(-x+4)-
4*x-28)*ln(x)^2),x,method=_RETURNVERBOSE)

[Out]

ln(exp(5/4*x^2/ln(x))-exp(-x+4)-x-7)

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maxima [A]  time = 0.44, size = 30, normalized size = 1.15 \begin {gather*} \log \left (-{\left ({\left (x + 7\right )} e^{x} + e^{4} - e^{\left (x + \frac {5 \, x^{2}}{4 \, \log \relax (x)}\right )}\right )} e^{\left (-x\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x*log(x)-5*x)*exp(5/4*x^2/log(x))+(4*exp(-x+4)-4)*log(x)^2)/(4*log(x)^2*exp(5/4*x^2/log(x))+(-4
*exp(-x+4)-4*x-28)*log(x)^2),x, algorithm="maxima")

[Out]

log(-((x + 7)*e^x + e^4 - e^(x + 5/4*x^2/log(x)))*e^(-x))

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mupad [B]  time = 5.31, size = 22, normalized size = 0.85 \begin {gather*} \ln \left (x+{\mathrm {e}}^{4-x}-{\mathrm {e}}^{\frac {5\,x^2}{4\,\ln \relax (x)}}+7\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((5*x^2)/(4*log(x)))*(5*x - 10*x*log(x)) - log(x)^2*(4*exp(4 - x) - 4))/(4*exp((5*x^2)/(4*log(x)))*lo
g(x)^2 - log(x)^2*(4*x + 4*exp(4 - x) + 28)),x)

[Out]

log(x + exp(4 - x) - exp((5*x^2)/(4*log(x))) + 7)

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sympy [A]  time = 0.62, size = 20, normalized size = 0.77 \begin {gather*} \log {\left (x - e^{\frac {5 x^{2}}{4 \log {\relax (x )}}} + e^{4 - x} + 7 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x*ln(x)-5*x)*exp(5/4*x**2/ln(x))+(4*exp(-x+4)-4)*ln(x)**2)/(4*ln(x)**2*exp(5/4*x**2/ln(x))+(-4*
exp(-x+4)-4*x-28)*ln(x)**2),x)

[Out]

log(x - exp(5*x**2/(4*log(x))) + exp(4 - x) + 7)

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