3.35.42 \(\int \frac {e^x (-12-12 x) \log (4)+e^x (-12+12 x+12 x^2) \log (4) \log (x)+e^x (-3+3 x+3 x^2) \log ^2(x)}{x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=25 \[ \frac {3 e^x (2+2 x) \left (\frac {1}{2}+\frac {2 \log (4)}{\log (x)}\right )}{x} \]

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Rubi [F]  time = 0.71, 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^x (-12-12 x) \log (4)+e^x \left (-12+12 x+12 x^2\right ) \log (4) \log (x)+e^x \left (-3+3 x+3 x^2\right ) \log ^2(x)}{x^2 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(-12 - 12*x)*Log[4] + E^x*(-12 + 12*x + 12*x^2)*Log[4]*Log[x] + E^x*(-3 + 3*x + 3*x^2)*Log[x]^2)/(x^2
*Log[x]^2),x]

[Out]

3*E^x + (3*E^x)/x - 12*Log[4]*Defer[Int][E^x/(x^2*Log[x]^2), x] - 12*Log[4]*Defer[Int][E^x/(x*Log[x]^2), x] +
12*Log[4]*Defer[Int][E^x/Log[x], x] - 12*Log[4]*Defer[Int][E^x/(x^2*Log[x]), x] + 12*Log[4]*Defer[Int][E^x/(x*
Log[x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (3 e^x-\frac {3 e^x}{x^2}+\frac {3 e^x}{x}-\frac {12 e^x \log (4)}{x^2 \log ^2(x)}-\frac {12 e^x \log (4)}{x \log ^2(x)}+\frac {12 e^x \log (4)}{\log (x)}-\frac {12 e^x \log (4)}{x^2 \log (x)}+\frac {12 e^x \log (4)}{x \log (x)}\right ) \, dx\\ &=3 \int e^x \, dx-3 \int \frac {e^x}{x^2} \, dx+3 \int \frac {e^x}{x} \, dx-(12 \log (4)) \int \frac {e^x}{x^2 \log ^2(x)} \, dx-(12 \log (4)) \int \frac {e^x}{x \log ^2(x)} \, dx+(12 \log (4)) \int \frac {e^x}{\log (x)} \, dx-(12 \log (4)) \int \frac {e^x}{x^2 \log (x)} \, dx+(12 \log (4)) \int \frac {e^x}{x \log (x)} \, dx\\ &=3 e^x+\frac {3 e^x}{x}+3 \text {Ei}(x)-3 \int \frac {e^x}{x} \, dx-(12 \log (4)) \int \frac {e^x}{x^2 \log ^2(x)} \, dx-(12 \log (4)) \int \frac {e^x}{x \log ^2(x)} \, dx+(12 \log (4)) \int \frac {e^x}{\log (x)} \, dx-(12 \log (4)) \int \frac {e^x}{x^2 \log (x)} \, dx+(12 \log (4)) \int \frac {e^x}{x \log (x)} \, dx\\ &=3 e^x+\frac {3 e^x}{x}-(12 \log (4)) \int \frac {e^x}{x^2 \log ^2(x)} \, dx-(12 \log (4)) \int \frac {e^x}{x \log ^2(x)} \, dx+(12 \log (4)) \int \frac {e^x}{\log (x)} \, dx-(12 \log (4)) \int \frac {e^x}{x^2 \log (x)} \, dx+(12 \log (4)) \int \frac {e^x}{x \log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 19, normalized size = 0.76 \begin {gather*} \frac {3 e^x (1+x) \log (256 x)}{x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(-12 - 12*x)*Log[4] + E^x*(-12 + 12*x + 12*x^2)*Log[4]*Log[x] + E^x*(-3 + 3*x + 3*x^2)*Log[x]^2
)/(x^2*Log[x]^2),x]

[Out]

(3*E^x*(1 + x)*Log[256*x])/(x*Log[x])

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fricas [A]  time = 0.55, size = 27, normalized size = 1.08 \begin {gather*} \frac {3 \, {\left (8 \, {\left (x + 1\right )} e^{x} \log \relax (2) + {\left (x + 1\right )} e^{x} \log \relax (x)\right )}}{x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+3*x-3)*exp(x)*log(x)^2+2*(12*x^2+12*x-12)*log(2)*exp(x)*log(x)+2*(-12*x-12)*log(2)*exp(x))/x
^2/log(x)^2,x, algorithm="fricas")

[Out]

3*(8*(x + 1)*e^x*log(2) + (x + 1)*e^x*log(x))/(x*log(x))

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giac [A]  time = 0.20, size = 34, normalized size = 1.36 \begin {gather*} \frac {3 \, {\left (8 \, x e^{x} \log \relax (2) + x e^{x} \log \relax (x) + 8 \, e^{x} \log \relax (2) + e^{x} \log \relax (x)\right )}}{x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+3*x-3)*exp(x)*log(x)^2+2*(12*x^2+12*x-12)*log(2)*exp(x)*log(x)+2*(-12*x-12)*log(2)*exp(x))/x
^2/log(x)^2,x, algorithm="giac")

[Out]

3*(8*x*e^x*log(2) + x*e^x*log(x) + 8*e^x*log(2) + e^x*log(x))/(x*log(x))

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maple [A]  time = 0.08, size = 28, normalized size = 1.12

method	result	size
risch	\(\frac {3 \left (x +1\right ) {\mathrm e}^{x}}{x}+\frac {24 \ln \relax (2) {\mathrm e}^{x} \left (x +1\right )}{x \ln \relax (x )}\)	\(28\)
norman	\(\frac {24 \,{\mathrm e}^{x} \ln \relax (2)+3 \,{\mathrm e}^{x} \ln \relax (x )+3 x \,{\mathrm e}^{x} \ln \relax (x )+24 x \ln \relax (2) {\mathrm e}^{x}}{x \ln \relax (x )}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2+3*x-3)*exp(x)*ln(x)^2+2*(12*x^2+12*x-12)*ln(2)*exp(x)*ln(x)+2*(-12*x-12)*ln(2)*exp(x))/x^2/ln(x)^2
,x,method=_RETURNVERBOSE)

[Out]

3*(x+1)/x*exp(x)+24/x*ln(2)*exp(x)*(x+1)/ln(x)

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maxima [C]  time = 0.70, size = 34, normalized size = 1.36 \begin {gather*} \frac {24 \, {\left (x \log \relax (2) + \log \relax (2)\right )} e^{x}}{x \log \relax (x)} + 3 \, {\rm Ei}\relax (x) + 3 \, e^{x} - 3 \, \Gamma \left (-1, -x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2+3*x-3)*exp(x)*log(x)^2+2*(12*x^2+12*x-12)*log(2)*exp(x)*log(x)+2*(-12*x-12)*log(2)*exp(x))/x
^2/log(x)^2,x, algorithm="maxima")

[Out]

24*(x*log(2) + log(2))*e^x/(x*log(x)) + 3*Ei(x) + 3*e^x - 3*gamma(-1, -x)

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mupad [B]  time = 2.28, size = 35, normalized size = 1.40 \begin {gather*} 3\,{\mathrm {e}}^x+\frac {3\,{\mathrm {e}}^x}{x}+\frac {24\,{\mathrm {e}}^x\,\ln \relax (2)}{\ln \relax (x)}+\frac {24\,{\mathrm {e}}^x\,\ln \relax (2)}{x\,\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)*log(x)^2*(3*x + 3*x^2 - 3) - 2*exp(x)*log(2)*(12*x + 12) + 2*exp(x)*log(2)*log(x)*(12*x + 12*x^2 -
12))/(x^2*log(x)^2),x)

[Out]

3*exp(x) + (3*exp(x))/x + (24*exp(x)*log(2))/log(x) + (24*exp(x)*log(2))/(x*log(x))

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sympy [A]  time = 0.31, size = 31, normalized size = 1.24 \begin {gather*} \frac {\left (3 x \log {\relax (x )} + 24 x \log {\relax (2 )} + 3 \log {\relax (x )} + 24 \log {\relax (2 )}\right ) e^{x}}{x \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**2+3*x-3)*exp(x)*ln(x)**2+2*(12*x**2+12*x-12)*ln(2)*exp(x)*ln(x)+2*(-12*x-12)*ln(2)*exp(x))/x*
*2/ln(x)**2,x)

[Out]

(3*x*log(x) + 24*x*log(2) + 3*log(x) + 24*log(2))*exp(x)/(x*log(x))

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