3.12.73 \(\int \frac {-x+(32+64 x+32 x^2) \log (x)+(32 x+32 x^2) \log ^2(x)}{-x^2+(16 x+32 x^2+16 x^3) \log ^2(x)} \, dx\)

Optimal. Leaf size=20 \[ 4+\log ^2(3)+\log \left (x-16 (1+x)^2 \log ^2(x)\right ) \]

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Rubi [F]  time = 12.83, 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 {-x+\left (32+64 x+32 x^2\right ) \log (x)+\left (32 x+32 x^2\right ) \log ^2(x)}{-x^2+\left (16 x+32 x^2+16 x^3\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-x + (32 + 64*x + 32*x^2)*Log[x] + (32*x + 32*x^2)*Log[x]^2)/(-x^2 + (16*x + 32*x^2 + 16*x^3)*Log[x]^2),x
]

[Out]

2*Log[1 + x] - 2*Defer[Int][1/((-1 - x)*(x - 16*(1 + x)^2*Log[x]^2)), x] + Defer[Int][(-x + 16*(1 + x)^2*Log[x
]^2)^(-1), x] + 64*Defer[Int][Log[x]/(-x + 16*(1 + x)^2*Log[x]^2), x] + 32*Defer[Int][Log[x]/(x*(-x + 16*(1 +
x)^2*Log[x]^2)), x] + 32*Defer[Int][(x*Log[x])/(-x + 16*(1 + x)^2*Log[x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {x-32 (1+x)^2 \log (x)-32 x (1+x) \log ^2(x)}{x \left (x-16 (1+x)^2 \log ^2(x)\right )} \, dx\\ &=\int \left (\frac {2}{1+x}+\frac {-x+x^2+32 \log (x)+96 x \log (x)+96 x^2 \log (x)+32 x^3 \log (x)}{x (1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}\right ) \, dx\\ &=2 \log (1+x)+\int \frac {-x+x^2+32 \log (x)+96 x \log (x)+96 x^2 \log (x)+32 x^3 \log (x)}{x (1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )} \, dx\\ &=2 \log (1+x)+\int \frac {-((-1+x) x)-32 (1+x)^3 \log (x)}{x (1+x) \left (x-16 (1+x)^2 \log ^2(x)\right )} \, dx\\ &=2 \log (1+x)+\int \left (\frac {x-x^2-32 \log (x)-96 x \log (x)-96 x^2 \log (x)-32 x^3 \log (x)}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}+\frac {-x+x^2+32 \log (x)+96 x \log (x)+96 x^2 \log (x)+32 x^3 \log (x)}{x \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}\right ) \, dx\\ &=2 \log (1+x)+\int \frac {x-x^2-32 \log (x)-96 x \log (x)-96 x^2 \log (x)-32 x^3 \log (x)}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )} \, dx+\int \frac {-x+x^2+32 \log (x)+96 x \log (x)+96 x^2 \log (x)+32 x^3 \log (x)}{x \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )} \, dx\\ &=2 \log (1+x)+\int \frac {-((-1+x) x)-32 (1+x)^3 \log (x)}{x \left (x-16 (1+x)^2 \log ^2(x)\right )} \, dx+\int \frac {(-1+x) x+32 (1+x)^3 \log (x)}{(1+x) \left (x-16 (1+x)^2 \log ^2(x)\right )} \, dx\\ &=2 \log (1+x)+\int \left (-\frac {1}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)}+\frac {x}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)}+\frac {96 \log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)}+\frac {32 \log (x)}{x \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}+\frac {96 x \log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)}+\frac {32 x^2 \log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)}\right ) \, dx+\int \left (\frac {x}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}-\frac {x^2}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}-\frac {32 \log (x)}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}-\frac {96 x \log (x)}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}-\frac {96 x^2 \log (x)}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}-\frac {32 x^3 \log (x)}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}\right ) \, dx\\ &=2 \log (1+x)+32 \int \frac {\log (x)}{x \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )} \, dx+32 \int \frac {x^2 \log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)} \, dx-32 \int \frac {\log (x)}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )} \, dx-32 \int \frac {x^3 \log (x)}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )} \, dx+96 \int \frac {\log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)} \, dx+96 \int \frac {x \log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)} \, dx-96 \int \frac {x \log (x)}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )} \, dx-96 \int \frac {x^2 \log (x)}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )} \, dx-\int \frac {1}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)} \, dx+\int \frac {x}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)} \, dx+\int \frac {x}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )} \, dx-\int \frac {x^2}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )} \, dx\\ &=2 \log (1+x)-32 \int \frac {\log (x)}{(-1-x) \left (x-16 (1+x)^2 \log ^2(x)\right )} \, dx-32 \int \frac {x^3 \log (x)}{(-1-x) \left (x-16 (1+x)^2 \log ^2(x)\right )} \, dx+32 \int \frac {\log (x)}{x \left (-x+16 (1+x)^2 \log ^2(x)\right )} \, dx+32 \int \frac {x^2 \log (x)}{-x+16 (1+x)^2 \log ^2(x)} \, dx-96 \int \frac {x \log (x)}{(-1-x) \left (x-16 (1+x)^2 \log ^2(x)\right )} \, dx-96 \int \frac {x^2 \log (x)}{(-1-x) \left (x-16 (1+x)^2 \log ^2(x)\right )} \, dx+96 \int \frac {\log (x)}{-x+16 (1+x)^2 \log ^2(x)} \, dx+96 \int \frac {x \log (x)}{-x+16 (1+x)^2 \log ^2(x)} \, dx+\int \frac {x}{(-1-x) \left (x-16 (1+x)^2 \log ^2(x)\right )} \, dx-\int \frac {x^2}{(-1-x) \left (x-16 (1+x)^2 \log ^2(x)\right )} \, dx-\int \frac {1}{-x+16 (1+x)^2 \log ^2(x)} \, dx+\int \frac {x}{-x+16 (1+x)^2 \log ^2(x)} \, dx\\ &=2 \log (1+x)-32 \int \frac {\log (x)}{(-1-x) \left (x-16 (1+x)^2 \log ^2(x)\right )} \, dx+32 \int \frac {\log (x)}{x \left (-x+16 (1+x)^2 \log ^2(x)\right )} \, dx+32 \int \frac {x^2 \log (x)}{-x+16 (1+x)^2 \log ^2(x)} \, dx-32 \int \left (\frac {\log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)}-\frac {x \log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)}+\frac {x^2 \log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)}-\frac {\log (x)}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}\right ) \, dx+96 \int \frac {\log (x)}{-x+16 (1+x)^2 \log ^2(x)} \, dx+96 \int \frac {x \log (x)}{-x+16 (1+x)^2 \log ^2(x)} \, dx-96 \int \left (\frac {\log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)}-\frac {\log (x)}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}\right ) \, dx-96 \int \left (-\frac {\log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)}+\frac {x \log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)}+\frac {\log (x)}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}\right ) \, dx-\int \frac {1}{-x+16 (1+x)^2 \log ^2(x)} \, dx+\int \frac {x}{-x+16 (1+x)^2 \log ^2(x)} \, dx+\int \left (\frac {1}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)}-\frac {1}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}\right ) \, dx-\int \left (-\frac {1}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)}+\frac {x}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)}+\frac {1}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )}\right ) \, dx\\ &=2 \log (1+x)-32 \int \frac {\log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)} \, dx+32 \int \frac {x \log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)} \, dx-32 \int \frac {x^2 \log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)} \, dx+32 \int \frac {\log (x)}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )} \, dx-32 \int \frac {\log (x)}{(-1-x) \left (x-16 (1+x)^2 \log ^2(x)\right )} \, dx+32 \int \frac {\log (x)}{x \left (-x+16 (1+x)^2 \log ^2(x)\right )} \, dx+32 \int \frac {x^2 \log (x)}{-x+16 (1+x)^2 \log ^2(x)} \, dx-96 \int \frac {x \log (x)}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)} \, dx+96 \int \frac {\log (x)}{-x+16 (1+x)^2 \log ^2(x)} \, dx+96 \int \frac {x \log (x)}{-x+16 (1+x)^2 \log ^2(x)} \, dx+2 \int \frac {1}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)} \, dx-\int \frac {x}{-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)} \, dx-2 \int \frac {1}{(1+x) \left (-x+16 \log ^2(x)+32 x \log ^2(x)+16 x^2 \log ^2(x)\right )} \, dx-\int \frac {1}{-x+16 (1+x)^2 \log ^2(x)} \, dx+\int \frac {x}{-x+16 (1+x)^2 \log ^2(x)} \, dx\\ &=2 \log (1+x)-32 \int \frac {\log (x)}{-x+16 (1+x)^2 \log ^2(x)} \, dx+32 \int \frac {\log (x)}{x \left (-x+16 (1+x)^2 \log ^2(x)\right )} \, dx+32 \int \frac {x \log (x)}{-x+16 (1+x)^2 \log ^2(x)} \, dx+96 \int \frac {\log (x)}{-x+16 (1+x)^2 \log ^2(x)} \, dx-2 \int \frac {1}{(-1-x) \left (x-16 (1+x)^2 \log ^2(x)\right )} \, dx+\int \frac {1}{-x+16 (1+x)^2 \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [F]  time = 2.03, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-x+\left (32+64 x+32 x^2\right ) \log (x)+\left (32 x+32 x^2\right ) \log ^2(x)}{-x^2+\left (16 x+32 x^2+16 x^3\right ) \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-x + (32 + 64*x + 32*x^2)*Log[x] + (32*x + 32*x^2)*Log[x]^2)/(-x^2 + (16*x + 32*x^2 + 16*x^3)*Log[x
]^2),x]

[Out]

Integrate[(-x + (32 + 64*x + 32*x^2)*Log[x] + (32*x + 32*x^2)*Log[x]^2)/(-x^2 + (16*x + 32*x^2 + 16*x^3)*Log[x
]^2), x]

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fricas [A]  time = 0.72, size = 37, normalized size = 1.85 \begin {gather*} 2 \, \log \left (x + 1\right ) + \log \left (\frac {16 \, {\left (x^{2} + 2 \, x + 1\right )} \log \relax (x)^{2} - x}{x^{2} + 2 \, x + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^2+32*x)*log(x)^2+(32*x^2+64*x+32)*log(x)-x)/((16*x^3+32*x^2+16*x)*log(x)^2-x^2),x, algorithm=
"fricas")

[Out]

2*log(x + 1) + log((16*(x^2 + 2*x + 1)*log(x)^2 - x)/(x^2 + 2*x + 1))

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giac [A]  time = 0.40, size = 27, normalized size = 1.35 \begin {gather*} \log \left (16 \, x^{2} \log \relax (x)^{2} + 32 \, x \log \relax (x)^{2} + 16 \, \log \relax (x)^{2} - x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^2+32*x)*log(x)^2+(32*x^2+64*x+32)*log(x)-x)/((16*x^3+32*x^2+16*x)*log(x)^2-x^2),x, algorithm=
"giac")

[Out]

log(16*x^2*log(x)^2 + 32*x*log(x)^2 + 16*log(x)^2 - x)

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maple [A]  time = 0.03, size = 27, normalized size = 1.35

method	result	size
risch	\(2 \ln \left (x +1\right )+\ln \left (\ln \relax (x )^{2}-\frac {x}{16 \left (x^{2}+2 x +1\right )}\right )\)	\(27\)
norman	\(\ln \left (16 x^{2} \ln \relax (x )^{2}+32 x \ln \relax (x )^{2}+16 \ln \relax (x )^{2}-x \right )\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((32*x^2+32*x)*ln(x)^2+(32*x^2+64*x+32)*ln(x)-x)/((16*x^3+32*x^2+16*x)*ln(x)^2-x^2),x,method=_RETURNVERBOS
E)

[Out]

2*ln(x+1)+ln(ln(x)^2-1/16*x/(x^2+2*x+1))

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maxima [A]  time = 0.47, size = 38, normalized size = 1.90 \begin {gather*} 2 \, \log \left (x + 1\right ) + \log \left (\frac {16 \, {\left (x^{2} + 2 \, x + 1\right )} \log \relax (x)^{2} - x}{16 \, {\left (x^{2} + 2 \, x + 1\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x^2+32*x)*log(x)^2+(32*x^2+64*x+32)*log(x)-x)/((16*x^3+32*x^2+16*x)*log(x)^2-x^2),x, algorithm=
"maxima")

[Out]

2*log(x + 1) + log(1/16*(16*(x^2 + 2*x + 1)*log(x)^2 - x)/(x^2 + 2*x + 1))

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mupad [B]  time = 0.81, size = 27, normalized size = 1.35 \begin {gather*} \ln \left (16\,x^2\,{\ln \relax (x)}^2+32\,x\,{\ln \relax (x)}^2-x+16\,{\ln \relax (x)}^2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)^2*(32*x + 32*x^2) - x + log(x)*(64*x + 32*x^2 + 32))/(log(x)^2*(16*x + 32*x^2 + 16*x^3) - x^2),x)

[Out]

log(32*x*log(x)^2 - x + 16*log(x)^2 + 16*x^2*log(x)^2)

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sympy [A]  time = 0.40, size = 24, normalized size = 1.20 \begin {gather*} 2 \log {\left (x + 1 \right )} + \log {\left (- \frac {x}{16 x^{2} + 32 x + 16} + \log {\relax (x )}^{2} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((32*x**2+32*x)*ln(x)**2+(32*x**2+64*x+32)*ln(x)-x)/((16*x**3+32*x**2+16*x)*ln(x)**2-x**2),x)

[Out]

2*log(x + 1) + log(-x/(16*x**2 + 32*x + 16) + log(x)**2)

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