3.14.62 \(\int \frac {4 x+8 x^2+(8+8 x) \log (5 x)+(-4-4 x+4 x^2-\log (25)+8 x \log (5 x)+4 \log ^2(5 x)) \log (\frac {1}{4} (-4-4 x+4 x^2-\log (25)+8 x \log (5 x)+4 \log ^2(5 x)))}{-4-4 x+4 x^2-\log (25)+8 x \log (5 x)+4 \log ^2(5 x)} \, dx\) [1362]

Optimal. Leaf size=22 \[ x \log \left (-1-x-\frac {\log (25)}{4}+(x+\log (5 x))^2\right ) \]

[Out]

ln(-1+(ln(5*x)+x)^2-x-1/2*ln(5))*x

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Rubi [F]
time = 0.79, 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 {4 x+8 x^2+(8+8 x) \log (5 x)+\left (-4-4 x+4 x^2-\log (25)+8 x \log (5 x)+4 \log ^2(5 x)\right ) \log \left (\frac {1}{4} \left (-4-4 x+4 x^2-\log (25)+8 x \log (5 x)+4 \log ^2(5 x)\right )\right )}{-4-4 x+4 x^2-\log (25)+8 x \log (5 x)+4 \log ^2(5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4*x + 8*x^2 + (8 + 8*x)*Log[5*x] + (-4 - 4*x + 4*x^2 - Log[25] + 8*x*Log[5*x] + 4*Log[5*x]^2)*Log[(-4 - 4
*x + 4*x^2 - Log[25] + 8*x*Log[5*x] + 4*Log[5*x]^2)/4])/(-4 - 4*x + 4*x^2 - Log[25] + 8*x*Log[5*x] + 4*Log[5*x
]^2),x]

[Out]

2*Defer[Int][x/(-2*x + 2*x^2 - 2*(1 + Log[5]/2) + 4*x*Log[5*x] + 2*Log[5*x]^2), x] + 4*Defer[Int][x^2/(-2*x +
2*x^2 - 2*(1 + Log[5]/2) + 4*x*Log[5*x] + 2*Log[5*x]^2), x] + 4*Defer[Int][Log[5*x]/(-2*x + 2*x^2 - 2*(1 + Log
[5]/2) + 4*x*Log[5*x] + 2*Log[5*x]^2), x] + 4*Defer[Int][(x*Log[5*x])/(-2*x + 2*x^2 - 2*(1 + Log[5]/2) + 4*x*L
og[5*x] + 2*Log[5*x]^2), x] + Defer[Int][Log[-1 + x^2 - ((1 - 2*Log[5])*Log[5])/2 + x*(-1 + Log[25]) + (2*x +
Log[25])*Log[x] + Log[x]^2], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-4 x-8 x^2-(8+8 x) \log (5 x)-\left (-4-4 x+4 x^2-\log (25)+8 x \log (5 x)+4 \log ^2(5 x)\right ) \log \left (\frac {1}{4} \left (-4-4 x+4 x^2-\log (25)+8 x \log (5 x)+4 \log ^2(5 x)\right )\right )}{4 x-4 x^2+4 \left (1+\frac {\log (5)}{2}\right )-8 x \log (5 x)-4 \log ^2(5 x)} \, dx\\ &=\int \left (\frac {2 \left (-x-2 x^2-2 \log (5 x)-2 x \log (5 x)\right )}{2 x-2 x^2+2 \left (1+\frac {\log (5)}{2}\right )-4 x \log (5 x)-2 \log ^2(5 x)}+\log \left (-1+x^2-\frac {1}{2} (1-2 \log (5)) \log (5)+x (-1+\log (25))+(2 x+\log (25)) \log (x)+\log ^2(x)\right )\right ) \, dx\\ &=2 \int \frac {-x-2 x^2-2 \log (5 x)-2 x \log (5 x)}{2 x-2 x^2+2 \left (1+\frac {\log (5)}{2}\right )-4 x \log (5 x)-2 \log ^2(5 x)} \, dx+\int \log \left (-1+x^2-\frac {1}{2} (1-2 \log (5)) \log (5)+x (-1+\log (25))+(2 x+\log (25)) \log (x)+\log ^2(x)\right ) \, dx\\ &=2 \int \left (\frac {x}{-2 x+2 x^2-2 \left (1+\frac {\log (5)}{2}\right )+4 x \log (5 x)+2 \log ^2(5 x)}+\frac {2 x^2}{-2 x+2 x^2-2 \left (1+\frac {\log (5)}{2}\right )+4 x \log (5 x)+2 \log ^2(5 x)}+\frac {2 \log (5 x)}{-2 x+2 x^2-2 \left (1+\frac {\log (5)}{2}\right )+4 x \log (5 x)+2 \log ^2(5 x)}+\frac {2 x \log (5 x)}{-2 x+2 x^2-2 \left (1+\frac {\log (5)}{2}\right )+4 x \log (5 x)+2 \log ^2(5 x)}\right ) \, dx+\int \log \left (-1+x^2-\frac {1}{2} (1-2 \log (5)) \log (5)+x (-1+\log (25))+(2 x+\log (25)) \log (x)+\log ^2(x)\right ) \, dx\\ &=2 \int \frac {x}{-2 x+2 x^2-2 \left (1+\frac {\log (5)}{2}\right )+4 x \log (5 x)+2 \log ^2(5 x)} \, dx+4 \int \frac {x^2}{-2 x+2 x^2-2 \left (1+\frac {\log (5)}{2}\right )+4 x \log (5 x)+2 \log ^2(5 x)} \, dx+4 \int \frac {\log (5 x)}{-2 x+2 x^2-2 \left (1+\frac {\log (5)}{2}\right )+4 x \log (5 x)+2 \log ^2(5 x)} \, dx+4 \int \frac {x \log (5 x)}{-2 x+2 x^2-2 \left (1+\frac {\log (5)}{2}\right )+4 x \log (5 x)+2 \log ^2(5 x)} \, dx+\int \log \left (-1+x^2-\frac {1}{2} (1-2 \log (5)) \log (5)+x (-1+\log (25))+(2 x+\log (25)) \log (x)+\log ^2(x)\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]
time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 x+8 x^2+(8+8 x) \log (5 x)+\left (-4-4 x+4 x^2-\log (25)+8 x \log (5 x)+4 \log ^2(5 x)\right ) \log \left (\frac {1}{4} \left (-4-4 x+4 x^2-\log (25)+8 x \log (5 x)+4 \log ^2(5 x)\right )\right )}{-4-4 x+4 x^2-\log (25)+8 x \log (5 x)+4 \log ^2(5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(4*x + 8*x^2 + (8 + 8*x)*Log[5*x] + (-4 - 4*x + 4*x^2 - Log[25] + 8*x*Log[5*x] + 4*Log[5*x]^2)*Log[(
-4 - 4*x + 4*x^2 - Log[25] + 8*x*Log[5*x] + 4*Log[5*x]^2)/4])/(-4 - 4*x + 4*x^2 - Log[25] + 8*x*Log[5*x] + 4*L
og[5*x]^2),x]

[Out]

Integrate[(4*x + 8*x^2 + (8 + 8*x)*Log[5*x] + (-4 - 4*x + 4*x^2 - Log[25] + 8*x*Log[5*x] + 4*Log[5*x]^2)*Log[(
-4 - 4*x + 4*x^2 - Log[25] + 8*x*Log[5*x] + 4*Log[5*x]^2)/4])/(-4 - 4*x + 4*x^2 - Log[25] + 8*x*Log[5*x] + 4*L
og[5*x]^2), x]

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Maple [A]
time = 1.54, size = 39, normalized size = 1.77

method	result	size
risch	\(\ln \left (\ln \left (5 x \right )^{2}+2 x \ln \left (5 x \right )-\frac {\ln \left (5\right )}{2}+x^{2}-x -1\right ) x\)	\(29\)
default	\(-x \ln \left (2\right )+x \ln \left (2 \ln \left (5 x \right )^{2}+4 x \ln \left (5 x \right )-\ln \left (5\right )+2 x^{2}-2 x -2\right )\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*ln(5*x)^2+8*x*ln(5*x)-2*ln(5)+4*x^2-4*x-4)*ln(ln(5*x)^2+2*x*ln(5*x)-1/2*ln(5)+x^2-x-1)+(8*x+8)*ln(5*x)
+8*x^2+4*x)/(4*ln(5*x)^2+8*x*ln(5*x)-2*ln(5)+4*x^2-4*x-4),x,method=_RETURNVERBOSE)

[Out]

-x*ln(2)+x*ln(2*ln(5*x)^2+4*x*ln(5*x)-ln(5)+2*x^2-2*x-2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (20) = 40\).
time = 0.49, size = 49, normalized size = 2.23 \begin {gather*} -x \log \left (2\right ) + x \log \left (2 \, x^{2} + 2 \, x {\left (2 \, \log \left (5\right ) - 1\right )} + 2 \, \log \left (5\right )^{2} + 4 \, {\left (x + \log \left (5\right )\right )} \log \left (x\right ) + 2 \, \log \left (x\right )^{2} - \log \left (5\right ) - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(5*x)^2+8*x*log(5*x)-2*log(5)+4*x^2-4*x-4)*log(log(5*x)^2+2*x*log(5*x)-1/2*log(5)+x^2-x-1)+(8
*x+8)*log(5*x)+8*x^2+4*x)/(4*log(5*x)^2+8*x*log(5*x)-2*log(5)+4*x^2-4*x-4),x, algorithm="maxima")

[Out]

-x*log(2) + x*log(2*x^2 + 2*x*(2*log(5) - 1) + 2*log(5)^2 + 4*(x + log(5))*log(x) + 2*log(x)^2 - log(5) - 2)

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Fricas [A]
time = 0.35, size = 28, normalized size = 1.27 \begin {gather*} x \log \left (x^{2} + 2 \, x \log \left (5 \, x\right ) + \log \left (5 \, x\right )^{2} - x - \frac {1}{2} \, \log \left (5\right ) - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(5*x)^2+8*x*log(5*x)-2*log(5)+4*x^2-4*x-4)*log(log(5*x)^2+2*x*log(5*x)-1/2*log(5)+x^2-x-1)+(8
*x+8)*log(5*x)+8*x^2+4*x)/(4*log(5*x)^2+8*x*log(5*x)-2*log(5)+4*x^2-4*x-4),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.23, size = 29, normalized size = 1.32 \begin {gather*} x \log {\left (x^{2} + 2 x \log {\left (5 x \right )} - x + \log {\left (5 x \right )}^{2} - 1 - \frac {\log {\left (5 \right )}}{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*ln(5*x)**2+8*x*ln(5*x)-2*ln(5)+4*x**2-4*x-4)*ln(ln(5*x)**2+2*x*ln(5*x)-1/2*ln(5)+x**2-x-1)+(8*x+
8)*ln(5*x)+8*x**2+4*x)/(4*ln(5*x)**2+8*x*ln(5*x)-2*ln(5)+4*x**2-4*x-4),x)

[Out]

x*log(x**2 + 2*x*log(5*x) - x + log(5*x)**2 - 1 - log(5)/2)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (20) = 40\).
time = 0.50, size = 51, normalized size = 2.32 \begin {gather*} -x \log \left (2\right ) + x \log \left (2 \, x^{2} + 4 \, x \log \left (5\right ) + 2 \, \log \left (5\right )^{2} + 4 \, x \log \left (x\right ) + 4 \, \log \left (5\right ) \log \left (x\right ) + 2 \, \log \left (x\right )^{2} - 2 \, x - \log \left (5\right ) - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*log(5*x)^2+8*x*log(5*x)-2*log(5)+4*x^2-4*x-4)*log(log(5*x)^2+2*x*log(5*x)-1/2*log(5)+x^2-x-1)+(8
*x+8)*log(5*x)+8*x^2+4*x)/(4*log(5*x)^2+8*x*log(5*x)-2*log(5)+4*x^2-4*x-4),x, algorithm="giac")

[Out]

-x*log(2) + x*log(2*x^2 + 4*x*log(5) + 2*log(5)^2 + 4*x*log(x) + 4*log(5)*log(x) + 2*log(x)^2 - 2*x - log(5) -
2)

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Mupad [B]
time = 1.35, size = 28, normalized size = 1.27 \begin {gather*} x\,\ln \left (x^2+2\,x\,\ln \left (5\,x\right )-x+{\ln \left (5\,x\right )}^2-\frac {\ln \left (5\right )}{2}-1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x - log(2*x*log(5*x) - log(5)/2 - x + log(5*x)^2 + x^2 - 1)*(4*x + 2*log(5) - 8*x*log(5*x) - 4*log(5*x
)^2 - 4*x^2 + 4) + 8*x^2 + log(5*x)*(8*x + 8))/(4*x + 2*log(5) - 8*x*log(5*x) - 4*log(5*x)^2 - 4*x^2 + 4),x)

[Out]

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

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