3.51.95 \(\int \frac {-2 x-x \log (15 x)+(2 x+2 x \log (15 x)) \log (x+x \log (15 x))}{(2+2 \log (15 x)) \log ^2(x+x \log (15 x))} \, dx\)

Optimal. Leaf size=18 \[ \frac {x^2}{2 \log (x+x \log (15 x))} \]

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Rubi [F]  time = 0.78, 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 {-2 x-x \log (15 x)+(2 x+2 x \log (15 x)) \log (x+x \log (15 x))}{(2+2 \log (15 x)) \log ^2(x+x \log (15 x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-2*x - x*Log[15*x] + (2*x + 2*x*Log[15*x])*Log[x + x*Log[15*x]])/((2 + 2*Log[15*x])*Log[x + x*Log[15*x]]^
2),x]

[Out]

Defer[Int][x/((-1 - Log[15*x])*Log[x + x*Log[15*x]]^2), x] + Defer[Int][(x*Log[15*x])/((-1 - Log[15*x])*Log[x
+ x*Log[15*x]]^2), x]/2 + Defer[Int][x/Log[x + x*Log[15*x]], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2 x-x \log (15 x)+(2 x+2 x \log (15 x)) \log (x+x \log (15 x))}{2 (1+\log (15 x)) \log ^2(x+x \log (15 x))} \, dx\\ &=\frac {1}{2} \int \frac {-2 x-x \log (15 x)+(2 x+2 x \log (15 x)) \log (x+x \log (15 x))}{(1+\log (15 x)) \log ^2(x+x \log (15 x))} \, dx\\ &=\frac {1}{2} \int \left (\frac {x (-2-\log (15 x))}{(1+\log (15 x)) \log ^2(x+x \log (15 x))}+\frac {2 x}{\log (x+x \log (15 x))}\right ) \, dx\\ &=\frac {1}{2} \int \frac {x (-2-\log (15 x))}{(1+\log (15 x)) \log ^2(x+x \log (15 x))} \, dx+\int \frac {x}{\log (x+x \log (15 x))} \, dx\\ &=\frac {1}{2} \int \left (\frac {2 x}{(-1-\log (15 x)) \log ^2(x+x \log (15 x))}+\frac {x \log (15 x)}{(-1-\log (15 x)) \log ^2(x+x \log (15 x))}\right ) \, dx+\int \frac {x}{\log (x+x \log (15 x))} \, dx\\ &=\frac {1}{2} \int \frac {x \log (15 x)}{(-1-\log (15 x)) \log ^2(x+x \log (15 x))} \, dx+\int \frac {x}{(-1-\log (15 x)) \log ^2(x+x \log (15 x))} \, dx+\int \frac {x}{\log (x+x \log (15 x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 18, normalized size = 1.00 \begin {gather*} \frac {x^2}{2 \log (x (1+\log (15 x)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2*x - x*Log[15*x] + (2*x + 2*x*Log[15*x])*Log[x + x*Log[15*x]])/((2 + 2*Log[15*x])*Log[x + x*Log[1
5*x]]^2),x]

[Out]

x^2/(2*Log[x*(1 + Log[15*x])])

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fricas [A]  time = 0.81, size = 16, normalized size = 0.89 \begin {gather*} \frac {x^{2}}{2 \, \log \left (x \log \left (15 \, x\right ) + x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(15*x)+2*x)*log(x*log(15*x)+x)-x*log(15*x)-2*x)/(2*log(15*x)+2)/log(x*log(15*x)+x)^2,x, alg
orithm="fricas")

[Out]

1/2*x^2/log(x*log(15*x) + x)

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giac [B]  time = 0.24, size = 164, normalized size = 9.11 \begin {gather*} \frac {x^{2} \log \left (15\right ) \log \left (15 \, x\right ) + x^{2} \log \left (15 \, x\right ) \log \relax (x) + x^{2} \log \left (15\right ) + 2 \, x^{2} \log \left (15 \, x\right ) + x^{2} \log \relax (x) + 2 \, x^{2}}{2 \, {\left (\log \left (15\right ) \log \left (15 \, x\right ) \log \relax (x) + \log \left (15 \, x\right ) \log \relax (x)^{2} + \log \left (15\right ) \log \left (15 \, x\right ) \log \left (\log \left (15 \, x\right ) + 1\right ) + \log \left (15 \, x\right ) \log \relax (x) \log \left (\log \left (15 \, x\right ) + 1\right ) + 2 \, \log \left (15\right ) \log \relax (x) + \log \left (15 \, x\right ) \log \relax (x) + 2 \, \log \relax (x)^{2} + 2 \, \log \left (15\right ) \log \left (\log \left (15 \, x\right ) + 1\right ) + \log \left (15 \, x\right ) \log \left (\log \left (15 \, x\right ) + 1\right ) + 2 \, \log \relax (x) \log \left (\log \left (15 \, x\right ) + 1\right ) + 2 \, \log \relax (x) + 2 \, \log \left (\log \left (15 \, x\right ) + 1\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(15*x)+2*x)*log(x*log(15*x)+x)-x*log(15*x)-2*x)/(2*log(15*x)+2)/log(x*log(15*x)+x)^2,x, alg
orithm="giac")

[Out]

1/2*(x^2*log(15)*log(15*x) + x^2*log(15*x)*log(x) + x^2*log(15) + 2*x^2*log(15*x) + x^2*log(x) + 2*x^2)/(log(1
5)*log(15*x)*log(x) + log(15*x)*log(x)^2 + log(15)*log(15*x)*log(log(15*x) + 1) + log(15*x)*log(x)*log(log(15*
x) + 1) + 2*log(15)*log(x) + log(15*x)*log(x) + 2*log(x)^2 + 2*log(15)*log(log(15*x) + 1) + log(15*x)*log(log(
15*x) + 1) + 2*log(x)*log(log(15*x) + 1) + 2*log(x) + 2*log(log(15*x) + 1))

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maple [A]  time = 0.04, size = 17, normalized size = 0.94

method	result	size
norman	\(\frac {x^{2}}{2 \ln \left (x \ln \left (15 x \right )+x \right )}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x*ln(15*x)+2*x)*ln(x*ln(15*x)+x)-x*ln(15*x)-2*x)/(2*ln(15*x)+2)/ln(x*ln(15*x)+x)^2,x,method=_RETURNVER
BOSE)

[Out]

1/2*x^2/ln(x*ln(15*x)+x)

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maxima [A]  time = 0.46, size = 19, normalized size = 1.06 \begin {gather*} \frac {x^{2}}{2 \, {\left (\log \relax (x) + \log \left (\log \relax (5) + \log \relax (3) + \log \relax (x) + 1\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*log(15*x)+2*x)*log(x*log(15*x)+x)-x*log(15*x)-2*x)/(2*log(15*x)+2)/log(x*log(15*x)+x)^2,x, alg
orithm="maxima")

[Out]

1/2*x^2/(log(x) + log(log(5) + log(3) + log(x) + 1))

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mupad [B]  time = 3.55, size = 16, normalized size = 0.89 \begin {gather*} \frac {x^2}{2\,\ln \left (x+x\,\ln \left (15\,x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - log(x + x*log(15*x))*(2*x + 2*x*log(15*x)) + x*log(15*x))/(log(x + x*log(15*x))^2*(2*log(15*x) + 2
)),x)

[Out]

x^2/(2*log(x + x*log(15*x)))

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sympy [A]  time = 0.26, size = 14, normalized size = 0.78 \begin {gather*} \frac {x^{2}}{2 \log {\left (x \log {\left (15 x \right )} + x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x*ln(15*x)+2*x)*ln(x*ln(15*x)+x)-x*ln(15*x)-2*x)/(2*ln(15*x)+2)/ln(x*ln(15*x)+x)**2,x)

[Out]

x**2/(2*log(x*log(15*x) + x))

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