3.90.1 \(\int \frac {1-8 x-x^2+4 x^3+(-1+4 x) \log (1-4 x)}{-1+4 x-2 x^2+8 x^3-x^4+4 x^5+(2-8 x+2 x^2-8 x^3) \log (1-4 x)+(-1+4 x) \log ^2(1-4 x)} \, dx\)

Optimal. Leaf size=17 \[ \frac {x}{-1-x^2+\log (1-4 x)} \]

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Rubi [F]  time = 0.70, 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 {1-8 x-x^2+4 x^3+(-1+4 x) \log (1-4 x)}{-1+4 x-2 x^2+8 x^3-x^4+4 x^5+\left (2-8 x+2 x^2-8 x^3\right ) \log (1-4 x)+(-1+4 x) \log ^2(1-4 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 - 8*x - x^2 + 4*x^3 + (-1 + 4*x)*Log[1 - 4*x])/(-1 + 4*x - 2*x^2 + 8*x^3 - x^4 + 4*x^5 + (2 - 8*x + 2*x
^2 - 8*x^3)*Log[1 - 4*x] + (-1 + 4*x)*Log[1 - 4*x]^2),x]

[Out]

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+8 x+x^2-4 x^3-(-1+4 x) \log (1-4 x)}{(1-4 x) \left (1+x^2-\log (1-4 x)\right )^2} \, dx\\ &=\int \left (\frac {2 x \left (-2-x+4 x^2\right )}{(-1+4 x) \left (1+x^2-\log (1-4 x)\right )^2}+\frac {1}{-1-x^2+\log (1-4 x)}\right ) \, dx\\ &=2 \int \frac {x \left (-2-x+4 x^2\right )}{(-1+4 x) \left (1+x^2-\log (1-4 x)\right )^2} \, dx+\int \frac {1}{-1-x^2+\log (1-4 x)} \, dx\\ &=2 \int \left (-\frac {1}{2 \left (1+x^2-\log (1-4 x)\right )^2}+\frac {x^2}{\left (1+x^2-\log (1-4 x)\right )^2}-\frac {1}{2 (-1+4 x) \left (1+x^2-\log (1-4 x)\right )^2}\right ) \, dx+\int \frac {1}{-1-x^2+\log (1-4 x)} \, dx\\ &=2 \int \frac {x^2}{\left (1+x^2-\log (1-4 x)\right )^2} \, dx-\int \frac {1}{\left (1+x^2-\log (1-4 x)\right )^2} \, dx-\int \frac {1}{(-1+4 x) \left (1+x^2-\log (1-4 x)\right )^2} \, dx+\int \frac {1}{-1-x^2+\log (1-4 x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.34, size = 17, normalized size = 1.00 \begin {gather*} \frac {x}{-1-x^2+\log (1-4 x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 8*x - x^2 + 4*x^3 + (-1 + 4*x)*Log[1 - 4*x])/(-1 + 4*x - 2*x^2 + 8*x^3 - x^4 + 4*x^5 + (2 - 8*x
+ 2*x^2 - 8*x^3)*Log[1 - 4*x] + (-1 + 4*x)*Log[1 - 4*x]^2),x]

[Out]

x/(-1 - x^2 + Log[1 - 4*x])

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fricas [A]  time = 0.44, size = 18, normalized size = 1.06 \begin {gather*} -\frac {x}{x^{2} - \log \left (-4 \, x + 1\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-1)*log(-4*x+1)+4*x^3-x^2-8*x+1)/((4*x-1)*log(-4*x+1)^2+(-8*x^3+2*x^2-8*x+2)*log(-4*x+1)+4*x^5-
x^4+8*x^3-2*x^2+4*x-1),x, algorithm="fricas")

[Out]

-x/(x^2 - log(-4*x + 1) + 1)

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giac [A]  time = 0.21, size = 18, normalized size = 1.06 \begin {gather*} -\frac {x}{x^{2} - \log \left (-4 \, x + 1\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-1)*log(-4*x+1)+4*x^3-x^2-8*x+1)/((4*x-1)*log(-4*x+1)^2+(-8*x^3+2*x^2-8*x+2)*log(-4*x+1)+4*x^5-
x^4+8*x^3-2*x^2+4*x-1),x, algorithm="giac")

[Out]

-x/(x^2 - log(-4*x + 1) + 1)

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maple [A]  time = 0.07, size = 19, normalized size = 1.12

method	result	size
norman	\(-\frac {x}{x^{2}-\ln \left (-4 x +1\right )+1}\)	\(19\)
risch	\(-\frac {x}{x^{2}-\ln \left (-4 x +1\right )+1}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x-1)*ln(-4*x+1)+4*x^3-x^2-8*x+1)/((4*x-1)*ln(-4*x+1)^2+(-8*x^3+2*x^2-8*x+2)*ln(-4*x+1)+4*x^5-x^4+8*x^3
-2*x^2+4*x-1),x,method=_RETURNVERBOSE)

[Out]

-x/(x^2-ln(-4*x+1)+1)

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maxima [A]  time = 0.41, size = 18, normalized size = 1.06 \begin {gather*} -\frac {x}{x^{2} - \log \left (-4 \, x + 1\right ) + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-1)*log(-4*x+1)+4*x^3-x^2-8*x+1)/((4*x-1)*log(-4*x+1)^2+(-8*x^3+2*x^2-8*x+2)*log(-4*x+1)+4*x^5-
x^4+8*x^3-2*x^2+4*x-1),x, algorithm="maxima")

[Out]

-x/(x^2 - log(-4*x + 1) + 1)

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mupad [B]  time = 5.54, size = 18, normalized size = 1.06 \begin {gather*} -\frac {x}{x^2-\ln \left (1-4\,x\right )+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(1 - 4*x)*(4*x - 1) - 8*x - x^2 + 4*x^3 + 1)/(4*x + log(1 - 4*x)^2*(4*x - 1) - log(1 - 4*x)*(8*x - 2*x
^2 + 8*x^3 - 2) - 2*x^2 + 8*x^3 - x^4 + 4*x^5 - 1),x)

[Out]

-x/(x^2 - log(1 - 4*x) + 1)

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sympy [A]  time = 0.13, size = 12, normalized size = 0.71 \begin {gather*} \frac {x}{- x^{2} + \log {\left (1 - 4 x \right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x-1)*ln(-4*x+1)+4*x**3-x**2-8*x+1)/((4*x-1)*ln(-4*x+1)**2+(-8*x**3+2*x**2-8*x+2)*ln(-4*x+1)+4*x*
*5-x**4+8*x**3-2*x**2+4*x-1),x)

[Out]

x/(-x**2 + log(1 - 4*x) - 1)

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