3.60.14 \(\int \frac {-15+5 \log (x)+(4 x-2 x \log (x)) \log (3 x-x \log (x))+(6 x-2 x \log (x)) \log ^2(3 x-x \log (x))}{-3+\log (x)} \, dx\)

Optimal. Leaf size=20 \[ 5 x-x^2 \log ^2(x (3-\log (x))) \]

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Rubi [F]  time = 0.35, 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 {-15+5 \log (x)+(4 x-2 x \log (x)) \log (3 x-x \log (x))+(6 x-2 x \log (x)) \log ^2(3 x-x \log (x))}{-3+\log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-15 + 5*Log[x] + (4*x - 2*x*Log[x])*Log[3*x - x*Log[x]] + (6*x - 2*x*Log[x])*Log[3*x - x*Log[x]]^2)/(-3 +
Log[x]),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (5+\frac {2 x (-2+\log (x)) \log (3 x-x \log (x))}{3-\log (x)}-2 x \log ^2(3 x-x \log (x))\right ) \, dx\\ &=5 x+2 \int \frac {x (-2+\log (x)) \log (3 x-x \log (x))}{3-\log (x)} \, dx-2 \int x \log ^2(3 x-x \log (x)) \, dx\\ &=5 x-2 \int x \log ^2(3 x-x \log (x)) \, dx+2 \int \left (\frac {2 x \log (3 x-x \log (x))}{-3+\log (x)}+\frac {x \log (x) \log (3 x-x \log (x))}{3-\log (x)}\right ) \, dx\\ &=5 x+2 \int \frac {x \log (x) \log (3 x-x \log (x))}{3-\log (x)} \, dx-2 \int x \log ^2(3 x-x \log (x)) \, dx+4 \int \frac {x \log (3 x-x \log (x))}{-3+\log (x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 0.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-15+5 \log (x)+(4 x-2 x \log (x)) \log (3 x-x \log (x))+(6 x-2 x \log (x)) \log ^2(3 x-x \log (x))}{-3+\log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-15 + 5*Log[x] + (4*x - 2*x*Log[x])*Log[3*x - x*Log[x]] + (6*x - 2*x*Log[x])*Log[3*x - x*Log[x]]^2)
/(-3 + Log[x]),x]

[Out]

Integrate[(-15 + 5*Log[x] + (4*x - 2*x*Log[x])*Log[3*x - x*Log[x]] + (6*x - 2*x*Log[x])*Log[3*x - x*Log[x]]^2)
/(-3 + Log[x]), x]

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fricas [A]  time = 0.60, size = 21, normalized size = 1.05 \begin {gather*} -x^{2} \log \left (-x \log \relax (x) + 3 \, x\right )^{2} + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(x)+6*x)*log(-x*log(x)+3*x)^2+(-2*x*log(x)+4*x)*log(-x*log(x)+3*x)+5*log(x)-15)/(log(x)-3)
,x, algorithm="fricas")

[Out]

-x^2*log(-x*log(x) + 3*x)^2 + 5*x

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giac [B]  time = 0.20, size = 41, normalized size = 2.05 \begin {gather*} -x^{2} \log \relax (x)^{2} - 2 \, x^{2} \log \relax (x) \log \left (-\log \relax (x) + 3\right ) - x^{2} \log \left (-\log \relax (x) + 3\right )^{2} + 5 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(x)+6*x)*log(-x*log(x)+3*x)^2+(-2*x*log(x)+4*x)*log(-x*log(x)+3*x)+5*log(x)-15)/(log(x)-3)
,x, algorithm="giac")

[Out]

-x^2*log(x)^2 - 2*x^2*log(x)*log(-log(x) + 3) - x^2*log(-log(x) + 3)^2 + 5*x

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maple [A]  time = 0.18, size = 22, normalized size = 1.10

method	result	size
norman	\(5 x -x^{2} \ln \left (-x \ln \relax (x )+3 x \right )^{2}\)	\(22\)
risch	\(5 x +\pi ^{2} x^{2}-x^{2} \ln \relax (x )^{2}-\pi ^{2} x^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{4}-\pi ^{2} x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{4} \mathrm {csgn}\left (i x \right )+\frac {\pi ^{2} x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{5} \mathrm {csgn}\left (i x \right )}{2}+\frac {\pi ^{2} x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{4} \mathrm {csgn}\left (i x \right )^{2}}{4}+\pi ^{2} x^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{2}+\pi ^{2} x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{2} \mathrm {csgn}\left (i x \right )+\frac {\pi ^{2} x^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right )^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{4}}{4}+\frac {\pi ^{2} x^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{5}}{2}-2 i \pi \,x^{2} \ln \relax (x )+i \pi \,x^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i x \right ) \ln \relax (x )-x^{2} \ln \left (\ln \relax (x )-3\right )^{2}+\left (-i \pi \,x^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{2}+i \pi \,x^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i x \right )-i \pi \,x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{3}+2 i \pi \,x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{2}-i \pi \,x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{2} \mathrm {csgn}\left (i x \right )-2 i \pi \,x^{2}-2 x^{2} \ln \relax (x )\right ) \ln \left (\ln \relax (x )-3\right )+\frac {\pi ^{2} x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{6}}{4}+\pi ^{2} x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{4}-\pi ^{2} x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{5}+\pi ^{2} x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{3}-2 \pi ^{2} x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{2}-i \pi \,x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{3} \ln \relax (x )-\pi ^{2} x^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i x \right )+2 i \pi \,x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{2} \ln \relax (x )-\frac {\pi ^{2} x^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right )^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{3} \mathrm {csgn}\left (i x \right )}{2}+\frac {\pi ^{2} x^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right )^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{2} \mathrm {csgn}\left (i x \right )^{2}}{4}-\frac {\pi ^{2} x^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{3} \mathrm {csgn}\left (i x \right )^{2}}{2}+\pi ^{2} x^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{3} \mathrm {csgn}\left (i x \right )-i \pi \,x^{2} \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{2} \mathrm {csgn}\left (i x \right ) \ln \relax (x )-i \pi \,x^{2} \mathrm {csgn}\left (i \left (\ln \relax (x )-3\right )\right ) \mathrm {csgn}\left (i x \left (\ln \relax (x )-3\right )\right )^{2} \ln \relax (x )\)	\(765\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x*ln(x)+6*x)*ln(-x*ln(x)+3*x)^2+(-2*x*ln(x)+4*x)*ln(-x*ln(x)+3*x)+5*ln(x)-15)/(ln(x)-3),x,method=_RET
URNVERBOSE)

[Out]

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

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maxima [C]  time = 0.42, size = 76, normalized size = 3.80 \begin {gather*} -x^{2} \log \relax (x)^{2} - 2 \, x^{2} \log \relax (x) \log \left (-\log \relax (x) + 3\right ) - x^{2} \log \left (-\log \relax (x) + 3\right )^{2} - 5 \, e^{3} E_{1}\left (-\log \relax (x) + 3\right ) \log \relax (x) + 5 \, e^{3} E_{2}\left (-\log \relax (x) + 3\right ) + 15 \, e^{3} E_{1}\left (-\log \relax (x) + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*log(x)+6*x)*log(-x*log(x)+3*x)^2+(-2*x*log(x)+4*x)*log(-x*log(x)+3*x)+5*log(x)-15)/(log(x)-3)
,x, algorithm="maxima")

[Out]

-x^2*log(x)^2 - 2*x^2*log(x)*log(-log(x) + 3) - x^2*log(-log(x) + 3)^2 - 5*e^3*exp_integral_e(1, -log(x) + 3)*
log(x) + 5*e^3*exp_integral_e(2, -log(x) + 3) + 15*e^3*exp_integral_e(1, -log(x) + 3)

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mupad [B]  time = 4.37, size = 21, normalized size = 1.05 \begin {gather*} 5\,x-x^2\,{\ln \left (3\,x-x\,\ln \relax (x)\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*log(x) + log(3*x - x*log(x))^2*(6*x - 2*x*log(x)) + log(3*x - x*log(x))*(4*x - 2*x*log(x)) - 15)/(log(x
) - 3),x)

[Out]

5*x - x^2*log(3*x - x*log(x))^2

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sympy [A]  time = 0.33, size = 17, normalized size = 0.85 \begin {gather*} - x^{2} \log {\left (- x \log {\relax (x )} + 3 x \right )}^{2} + 5 x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x*ln(x)+6*x)*ln(-x*ln(x)+3*x)**2+(-2*x*ln(x)+4*x)*ln(-x*ln(x)+3*x)+5*ln(x)-15)/(ln(x)-3),x)

[Out]

-x**2*log(-x*log(x) + 3*x)**2 + 5*x

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