3.61.54 \(\int \frac {-160 x+68 x^2-8 x^3-4 x \log (7)+(-72 x+16 x^2) \log (7) \log (\frac {4}{x})-8 x \log ^2(7) \log ^2(\frac {4}{x})}{16-8 x+x^2+(8-2 x) \log (7) \log (\frac {4}{x})+\log ^2(7) \log ^2(\frac {4}{x})} \, dx\)

Optimal. Leaf size=27 \[ 5-x^2 \left (4+\frac {4}{4-x+\log (7) \log \left (\frac {4}{x}\right )}\right ) \]

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Rubi [F]  time = 0.61, 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 {-160 x+68 x^2-8 x^3-4 x \log (7)+\left (-72 x+16 x^2\right ) \log (7) \log \left (\frac {4}{x}\right )-8 x \log ^2(7) \log ^2\left (\frac {4}{x}\right )}{16-8 x+x^2+(8-2 x) \log (7) \log \left (\frac {4}{x}\right )+\log ^2(7) \log ^2\left (\frac {4}{x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-160*x + 68*x^2 - 8*x^3 - 4*x*Log[7] + (-72*x + 16*x^2)*Log[7]*Log[4/x] - 8*x*Log[7]^2*Log[4/x]^2)/(16 -
8*x + x^2 + (8 - 2*x)*Log[7]*Log[4/x] + Log[7]^2*Log[4/x]^2),x]

[Out]

-4*x^2 - 4*Log[7]*Defer[Int][x/(-4 + x - Log[7]*Log[4/x])^2, x] - 4*Defer[Int][x^2/(-4 + x - Log[7]*Log[4/x])^
2, x] + 8*Defer[Int][x/(-4 + x - Log[7]*Log[4/x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {68 x^2-8 x^3+x (-160-4 \log (7))+\left (-72 x+16 x^2\right ) \log (7) \log \left (\frac {4}{x}\right )-8 x \log ^2(7) \log ^2\left (\frac {4}{x}\right )}{16-8 x+x^2+(8-2 x) \log (7) \log \left (\frac {4}{x}\right )+\log ^2(7) \log ^2\left (\frac {4}{x}\right )} \, dx\\ &=\int \frac {4 x \left (17 x-2 x^2-40 \left (1+\frac {\log (7)}{40}\right )-2 (9-2 x) \log (7) \log \left (\frac {4}{x}\right )-2 \log ^2(7) \log ^2\left (\frac {4}{x}\right )\right )}{\left (4-x+\log (7) \log \left (\frac {4}{x}\right )\right )^2} \, dx\\ &=4 \int \frac {x \left (17 x-2 x^2-40 \left (1+\frac {\log (7)}{40}\right )-2 (9-2 x) \log (7) \log \left (\frac {4}{x}\right )-2 \log ^2(7) \log ^2\left (\frac {4}{x}\right )\right )}{\left (4-x+\log (7) \log \left (\frac {4}{x}\right )\right )^2} \, dx\\ &=4 \int \left (-2 x-\frac {x (x+\log (7))}{\left (-4+x-\log (7) \log \left (\frac {4}{x}\right )\right )^2}+\frac {2 x}{-4+x-\log (7) \log \left (\frac {4}{x}\right )}\right ) \, dx\\ &=-4 x^2-4 \int \frac {x (x+\log (7))}{\left (-4+x-\log (7) \log \left (\frac {4}{x}\right )\right )^2} \, dx+8 \int \frac {x}{-4+x-\log (7) \log \left (\frac {4}{x}\right )} \, dx\\ &=-4 x^2-4 \int \left (\frac {x^2}{\left (-4+x-\log (7) \log \left (\frac {4}{x}\right )\right )^2}+\frac {x \log (7)}{\left (-4+x-\log (7) \log \left (\frac {4}{x}\right )\right )^2}\right ) \, dx+8 \int \frac {x}{-4+x-\log (7) \log \left (\frac {4}{x}\right )} \, dx\\ &=-4 x^2-4 \int \frac {x^2}{\left (-4+x-\log (7) \log \left (\frac {4}{x}\right )\right )^2} \, dx+8 \int \frac {x}{-4+x-\log (7) \log \left (\frac {4}{x}\right )} \, dx-(4 \log (7)) \int \frac {x}{\left (-4+x-\log (7) \log \left (\frac {4}{x}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.29, size = 26, normalized size = 0.96 \begin {gather*} -4 \left (x^2+\frac {x^2}{4-x+\log (7) \log \left (\frac {4}{x}\right )}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-160*x + 68*x^2 - 8*x^3 - 4*x*Log[7] + (-72*x + 16*x^2)*Log[7]*Log[4/x] - 8*x*Log[7]^2*Log[4/x]^2)/
(16 - 8*x + x^2 + (8 - 2*x)*Log[7]*Log[4/x] + Log[7]^2*Log[4/x]^2),x]

[Out]

-4*(x^2 + x^2/(4 - x + Log[7]*Log[4/x]))

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fricas [A]  time = 0.53, size = 41, normalized size = 1.52 \begin {gather*} -\frac {4 \, {\left (x^{2} \log \relax (7) \log \left (\frac {4}{x}\right ) - x^{3} + 5 \, x^{2}\right )}}{\log \relax (7) \log \left (\frac {4}{x}\right ) - x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x*log(7)^2*log(4/x)^2+(16*x^2-72*x)*log(7)*log(4/x)-4*x*log(7)-8*x^3+68*x^2-160*x)/(log(7)^2*log
(4/x)^2+(-2*x+8)*log(7)*log(4/x)+x^2-8*x+16),x, algorithm="fricas")

[Out]

-4*(x^2*log(7)*log(4/x) - x^3 + 5*x^2)/(log(7)*log(4/x) - x + 4)

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giac [A]  time = 0.15, size = 33, normalized size = 1.22 \begin {gather*} -4 \, x^{2} - \frac {4}{\frac {\log \relax (7) \log \left (\frac {4}{x}\right )}{x^{2}} - \frac {1}{x} + \frac {4}{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x*log(7)^2*log(4/x)^2+(16*x^2-72*x)*log(7)*log(4/x)-4*x*log(7)-8*x^3+68*x^2-160*x)/(log(7)^2*log
(4/x)^2+(-2*x+8)*log(7)*log(4/x)+x^2-8*x+16),x, algorithm="giac")

[Out]

-4*x^2 - 4/(log(7)*log(4/x)/x^2 - 1/x + 4/x^2)

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maple [A]  time = 0.19, size = 28, normalized size = 1.04

method	result	size
risch	\(-4 x^{2}-\frac {4 x^{2}}{4+\ln \relax (7) \ln \left (\frac {4}{x}\right )-x}\)	\(28\)
norman	\(\frac {-20 x^{2}+4 x^{3}-4 \ln \relax (7) \ln \left (\frac {4}{x}\right ) x^{2}}{4+\ln \relax (7) \ln \left (\frac {4}{x}\right )-x}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-8*x*ln(7)^2*ln(4/x)^2+(16*x^2-72*x)*ln(7)*ln(4/x)-4*x*ln(7)-8*x^3+68*x^2-160*x)/(ln(7)^2*ln(4/x)^2+(-2*x
+8)*ln(7)*ln(4/x)+x^2-8*x+16),x,method=_RETURNVERBOSE)

[Out]

-4*x^2-4*x^2/(4+ln(7)*ln(4/x)-x)

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maxima [A]  time = 0.48, size = 46, normalized size = 1.70 \begin {gather*} \frac {4 \, {\left (x^{2} \log \relax (7) \log \relax (x) - {\left (2 \, \log \relax (7) \log \relax (2) + 5\right )} x^{2} + x^{3}\right )}}{2 \, \log \relax (7) \log \relax (2) - \log \relax (7) \log \relax (x) - x + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x*log(7)^2*log(4/x)^2+(16*x^2-72*x)*log(7)*log(4/x)-4*x*log(7)-8*x^3+68*x^2-160*x)/(log(7)^2*log
(4/x)^2+(-2*x+8)*log(7)*log(4/x)+x^2-8*x+16),x, algorithm="maxima")

[Out]

4*(x^2*log(7)*log(x) - (2*log(7)*log(2) + 5)*x^2 + x^3)/(2*log(7)*log(2) - log(7)*log(x) - x + 4)

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mupad [B]  time = 4.54, size = 85, normalized size = 3.15 \begin {gather*} 8\,x-\frac {\frac {8\,x^2\,\ln \left (\frac {4}{x}\right )}{x+\ln \relax (7)}+\frac {4\,x\,\left (8\,x+x\,\ln \relax (7)-x^2\right )}{\ln \relax (7)\,\left (x+\ln \relax (7)\right )}}{\ln \left (\frac {4}{x}\right )-\frac {x-4}{\ln \relax (7)}}+\frac {8\,{\ln \relax (7)}^2}{x+\ln \relax (7)}-4\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(160*x + 4*x*log(7) - 68*x^2 + 8*x^3 + 8*x*log(7)^2*log(4/x)^2 + log(7)*log(4/x)*(72*x - 16*x^2))/(log(7)
^2*log(4/x)^2 - 8*x + x^2 - log(7)*log(4/x)*(2*x - 8) + 16),x)

[Out]

8*x - ((8*x^2*log(4/x))/(x + log(7)) + (4*x*(8*x + x*log(7) - x^2))/(log(7)*(x + log(7))))/(log(4/x) - (x - 4)
/log(7)) + (8*log(7)^2)/(x + log(7)) - 4*x^2

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sympy [A]  time = 0.16, size = 22, normalized size = 0.81 \begin {gather*} - 4 x^{2} - \frac {4 x^{2}}{- x + \log {\relax (7 )} \log {\left (\frac {4}{x} \right )} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-8*x*ln(7)**2*ln(4/x)**2+(16*x**2-72*x)*ln(7)*ln(4/x)-4*x*ln(7)-8*x**3+68*x**2-160*x)/(ln(7)**2*ln(
4/x)**2+(-2*x+8)*ln(7)*ln(4/x)+x**2-8*x+16),x)

[Out]

-4*x**2 - 4*x**2/(-x + log(7)*log(4/x) + 4)

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