3.53.43 \(\int \frac {3+2 x+(-9-12 x+(3+4 x) \log (x)) \log (3-\log (x))}{-6+2 \log (x)} \, dx\)

Optimal. Leaf size=14 \[ x \left (\frac {3}{2}+x\right ) \log (3-\log (x)) \]

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Rubi [F]  time = 0.22, 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 {3+2 x+(-9-12 x+(3+4 x) \log (x)) \log (3-\log (x))}{-6+2 \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(3 + 2*x + (-9 - 12*x + (3 + 4*x)*Log[x])*Log[3 - Log[x]])/(-6 + 2*Log[x]),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3-2 x-(-9-12 x+(3+4 x) \log (x)) \log (3-\log (x))}{2 (3-\log (x))} \, dx\\ &=\frac {1}{2} \int \frac {-3-2 x-(-9-12 x+(3+4 x) \log (x)) \log (3-\log (x))}{3-\log (x)} \, dx\\ &=\frac {1}{2} \int \left (\frac {3+2 x}{-3+\log (x)}+(3+4 x) \log (3-\log (x))\right ) \, dx\\ &=\frac {1}{2} \int \frac {3+2 x}{-3+\log (x)} \, dx+\frac {1}{2} \int (3+4 x) \log (3-\log (x)) \, dx\\ &=\frac {1}{2} \int \left (\frac {3}{-3+\log (x)}+\frac {2 x}{-3+\log (x)}\right ) \, dx+\frac {1}{2} \int (3 \log (3-\log (x))+4 x \log (3-\log (x))) \, dx\\ &=\frac {3}{2} \int \frac {1}{-3+\log (x)} \, dx+\frac {3}{2} \int \log (3-\log (x)) \, dx+2 \int x \log (3-\log (x)) \, dx+\int \frac {x}{-3+\log (x)} \, dx\\ &=\frac {3}{2} \int \log (3-\log (x)) \, dx+\frac {3}{2} \operatorname {Subst}\left (\int \frac {e^x}{-3+x} \, dx,x,\log (x)\right )+2 \int x \log (3-\log (x)) \, dx+\operatorname {Subst}\left (\int \frac {e^{2 x}}{-3+x} \, dx,x,\log (x)\right )\\ &=e^6 \text {Ei}(-2 (3-\log (x)))+\frac {3}{2} e^3 \text {Ei}(-3+\log (x))+\frac {3}{2} \int \log (3-\log (x)) \, dx+2 \int x \log (3-\log (x)) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 17, normalized size = 1.21 \begin {gather*} \frac {1}{2} x (3+2 x) \log (3-\log (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x + (-9 - 12*x + (3 + 4*x)*Log[x])*Log[3 - Log[x]])/(-6 + 2*Log[x]),x]

[Out]

(x*(3 + 2*x)*Log[3 - Log[x]])/2

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fricas [A]  time = 0.48, size = 18, normalized size = 1.29 \begin {gather*} \frac {1}{2} \, {\left (2 \, x^{2} + 3 \, x\right )} \log \left (-\log \relax (x) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3+4*x)*log(x)-12*x-9)*log(3-log(x))+2*x+3)/(2*log(x)-6),x, algorithm="fricas")

[Out]

1/2*(2*x^2 + 3*x)*log(-log(x) + 3)

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giac [A]  time = 0.57, size = 22, normalized size = 1.57 \begin {gather*} x^{2} \log \left (-\log \relax (x) + 3\right ) + \frac {3}{2} \, x \log \left (-\log \relax (x) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3+4*x)*log(x)-12*x-9)*log(3-log(x))+2*x+3)/(2*log(x)-6),x, algorithm="giac")

[Out]

x^2*log(-log(x) + 3) + 3/2*x*log(-log(x) + 3)

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maple [A]  time = 0.05, size = 16, normalized size = 1.14

method	result	size
risch	\(\left (x^{2}+\frac {3}{2} x \right ) \ln \left (3-\ln \relax (x )\right )\)	\(16\)
norman	\(x^{2} \ln \left (3-\ln \relax (x )\right )+\frac {3 \ln \left (3-\ln \relax (x )\right ) x}{2}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((3+4*x)*ln(x)-12*x-9)*ln(3-ln(x))+2*x+3)/(2*ln(x)-6),x,method=_RETURNVERBOSE)

[Out]

(x^2+3/2*x)*ln(3-ln(x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {3}{2} \, e^{3} E_{1}\left (-\log \relax (x) + 3\right ) - e^{6} E_{1}\left (-2 \, \log \relax (x) + 6\right ) + \frac {1}{2} \, {\left (2 \, x^{2} + 3 \, x\right )} \log \left (-\log \relax (x) + 3\right ) - \frac {1}{2} \, \int \frac {2 \, x + 3}{\log \relax (x) - 3}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3+4*x)*log(x)-12*x-9)*log(3-log(x))+2*x+3)/(2*log(x)-6),x, algorithm="maxima")

[Out]

-3/2*e^3*exp_integral_e(1, -log(x) + 3) - e^6*exp_integral_e(1, -2*log(x) + 6) + 1/2*(2*x^2 + 3*x)*log(-log(x)
+ 3) - 1/2*integrate((2*x + 3)/(log(x) - 3), x)

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mupad [B]  time = 3.62, size = 15, normalized size = 1.07 \begin {gather*} \frac {x\,\ln \left (3-\ln \relax (x)\right )\,\left (2\,x+3\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - log(3 - log(x))*(12*x - log(x)*(4*x + 3) + 9) + 3)/(2*log(x) - 6),x)

[Out]

(x*log(3 - log(x))*(2*x + 3))/2

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sympy [A]  time = 0.33, size = 14, normalized size = 1.00 \begin {gather*} \left (x^{2} + \frac {3 x}{2}\right ) \log {\left (3 - \log {\relax (x )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((3+4*x)*ln(x)-12*x-9)*ln(3-ln(x))+2*x+3)/(2*ln(x)-6),x)

[Out]

(x**2 + 3*x/2)*log(3 - log(x))

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