3.89.3 \(\int \frac {e^x (1-x-2 \log (4 x)+\log ^2(4 x)) (6+3 x^2+(-6+6 x) \log (4 x)-3 x \log ^2(4 x))}{x-x^2-2 x \log (4 x)+x \log ^2(4 x)} \, dx\)

Optimal. Leaf size=19 \[ -3 e^x \left (-x+(1-\log (4 x))^2\right ) \]

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Rubi [F]  time = 1.17, 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 \left (1-x-2 \log (4 x)+\log ^2(4 x)\right ) \left (6+3 x^2+(-6+6 x) \log (4 x)-3 x \log ^2(4 x)\right )}{x-x^2-2 x \log (4 x)+x \log ^2(4 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^x*(1 - x - 2*Log[4*x] + Log[4*x]^2)*(6 + 3*x^2 + (-6 + 6*x)*Log[4*x] - 3*x*Log[4*x]^2))/(x - x^2 - 2*x*
Log[4*x] + x*Log[4*x]^2),x]

[Out]

-3*E^x + 3*E^x*x + 6*x*HypergeometricPFQ[{1, 1, 1}, {2, 2, 2}, x] + 3*Log[-x]^2 + 6*EulerGamma*Log[x] + 6*(Exp
IntegralE[1, -x] + ExpIntegralEi[x])*Log[x] + 6*E^x*Log[4*x] - 6*ExpIntegralEi[x]*Log[4*x] - 3*Defer[Int][E^x*
Log[4*x]^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3 e^x \left (2+x^2+2 (-1+x) \log (4 x)-x \log ^2(4 x)\right )}{x} \, dx\\ &=3 \int \frac {e^x \left (2+x^2+2 (-1+x) \log (4 x)-x \log ^2(4 x)\right )}{x} \, dx\\ &=3 \int \left (\frac {e^x \left (2+x^2\right )}{x}+\frac {2 e^x (-1+x) \log (4 x)}{x}-e^x \log ^2(4 x)\right ) \, dx\\ &=3 \int \frac {e^x \left (2+x^2\right )}{x} \, dx-3 \int e^x \log ^2(4 x) \, dx+6 \int \frac {e^x (-1+x) \log (4 x)}{x} \, dx\\ &=6 e^x \log (4 x)-6 \text {Ei}(x) \log (4 x)+3 \int \left (\frac {2 e^x}{x}+e^x x\right ) \, dx-3 \int e^x \log ^2(4 x) \, dx-6 \int \frac {e^x-\text {Ei}(x)}{x} \, dx\\ &=6 e^x \log (4 x)-6 \text {Ei}(x) \log (4 x)+3 \int e^x x \, dx-3 \int e^x \log ^2(4 x) \, dx+6 \int \frac {e^x}{x} \, dx-6 \int \left (\frac {e^x}{x}-\frac {\text {Ei}(x)}{x}\right ) \, dx\\ &=3 e^x x+6 \text {Ei}(x)+6 e^x \log (4 x)-6 \text {Ei}(x) \log (4 x)-3 \int e^x \, dx-3 \int e^x \log ^2(4 x) \, dx-6 \int \frac {e^x}{x} \, dx+6 \int \frac {\text {Ei}(x)}{x} \, dx\\ &=-3 e^x+3 e^x x+6 (E_1(-x)+\text {Ei}(x)) \log (x)+6 e^x \log (4 x)-6 \text {Ei}(x) \log (4 x)-3 \int e^x \log ^2(4 x) \, dx-6 \int \frac {E_1(-x)}{x} \, dx\\ &=-3 e^x+3 e^x x+6 x \, _3F_3(1,1,1;2,2,2;x)+3 \log ^2(-x)+6 \gamma \log (x)+6 (E_1(-x)+\text {Ei}(x)) \log (x)+6 e^x \log (4 x)-6 \text {Ei}(x) \log (4 x)-3 \int e^x \log ^2(4 x) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 22, normalized size = 1.16 \begin {gather*} 3 e^x \left (-1+x+2 \log (4 x)-\log ^2(4 x)\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^x*(1 - x - 2*Log[4*x] + Log[4*x]^2)*(6 + 3*x^2 + (-6 + 6*x)*Log[4*x] - 3*x*Log[4*x]^2))/(x - x^2
- 2*x*Log[4*x] + x*Log[4*x]^2),x]

[Out]

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

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fricas [A]  time = 0.60, size = 23, normalized size = 1.21 \begin {gather*} -3 \, e^{\left (x + \log \left (\log \left (4 \, x\right )^{2} - x - 2 \, \log \left (4 \, x\right ) + 1\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(4*x)^2+(6*x-6)*log(4*x)+3*x^2+6)*exp(log(log(4*x)^2-2*log(4*x)-x+1)+x)/(x*log(4*x)^2-2*x*l
og(4*x)-x^2+x),x, algorithm="fricas")

[Out]

-3*e^(x + log(log(4*x)^2 - x - 2*log(4*x) + 1))

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giac [B]  time = 0.18, size = 46, normalized size = 2.42 \begin {gather*} -12 \, e^{x} \log \relax (2)^{2} - 12 \, e^{x} \log \relax (2) \log \relax (x) - 3 \, e^{x} \log \relax (x)^{2} + 3 \, x e^{x} + 12 \, e^{x} \log \relax (2) + 6 \, e^{x} \log \relax (x) - 3 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(4*x)^2+(6*x-6)*log(4*x)+3*x^2+6)*exp(log(log(4*x)^2-2*log(4*x)-x+1)+x)/(x*log(4*x)^2-2*x*l
og(4*x)-x^2+x),x, algorithm="giac")

[Out]

-12*e^x*log(2)^2 - 12*e^x*log(2)*log(x) - 3*e^x*log(x)^2 + 3*x*e^x + 12*e^x*log(2) + 6*e^x*log(x) - 3*e^x

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maple [A]  time = 0.04, size = 22, normalized size = 1.16

method	result	size
risch	\(-3 \left (\ln \left (4 x \right )^{2}-2 \ln \left (4 x \right )-x +1\right ) {\mathrm e}^{x}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*x*ln(4*x)^2+(6*x-6)*ln(4*x)+3*x^2+6)*exp(ln(ln(4*x)^2-2*ln(4*x)-x+1)+x)/(x*ln(4*x)^2-2*x*ln(4*x)-x^2+x
),x,method=_RETURNVERBOSE)

[Out]

-3*(ln(4*x)^2-2*ln(4*x)-x+1)*exp(x)

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maxima [A]  time = 0.49, size = 33, normalized size = 1.74 \begin {gather*} -3 \, {\left (4 \, \log \relax (2)^{2} + 2 \, {\left (2 \, \log \relax (2) - 1\right )} \log \relax (x) + \log \relax (x)^{2} - x - 4 \, \log \relax (2) + 1\right )} e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*log(4*x)^2+(6*x-6)*log(4*x)+3*x^2+6)*exp(log(log(4*x)^2-2*log(4*x)-x+1)+x)/(x*log(4*x)^2-2*x*l
og(4*x)-x^2+x),x, algorithm="maxima")

[Out]

-3*(4*log(2)^2 + 2*(2*log(2) - 1)*log(x) + log(x)^2 - x - 4*log(2) + 1)*e^x

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mupad [B]  time = 7.07, size = 46, normalized size = 2.42 \begin {gather*} 6\,{\mathrm {e}}^x\,\ln \relax (x)-12\,{\mathrm {e}}^x\,{\ln \relax (2)}^2-3\,{\mathrm {e}}^x-3\,{\mathrm {e}}^x\,{\ln \relax (x)}^2+12\,{\mathrm {e}}^x\,\ln \relax (2)+3\,x\,{\mathrm {e}}^x-12\,{\mathrm {e}}^x\,\ln \relax (2)\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + log(log(4*x)^2 - 2*log(4*x) - x + 1))*(3*x^2 - 3*x*log(4*x)^2 + log(4*x)*(6*x - 6) + 6))/(x - 2*x
*log(4*x) + x*log(4*x)^2 - x^2),x)

[Out]

6*exp(x)*log(x) - 12*exp(x)*log(2)^2 - 3*exp(x) - 3*exp(x)*log(x)^2 + 12*exp(x)*log(2) + 3*x*exp(x) - 12*exp(x
)*log(2)*log(x)

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sympy [A]  time = 0.40, size = 22, normalized size = 1.16 \begin {gather*} \left (3 x - 3 \log {\left (4 x \right )}^{2} + 6 \log {\left (4 x \right )} - 3\right ) e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*x*ln(4*x)**2+(6*x-6)*ln(4*x)+3*x**2+6)*exp(ln(ln(4*x)**2-2*ln(4*x)-x+1)+x)/(x*ln(4*x)**2-2*x*ln(
4*x)-x**2+x),x)

[Out]

(3*x - 3*log(4*x)**2 + 6*log(4*x) - 3)*exp(x)

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