3.38.96 \(\int \frac {-4+4 x+60 x^2-448 x^3+(-12 x-64 x^2) \log (x)+(32+136 x-336 x^2+(-8-48 x) \log (x)) \log (4-7 x-\log (x))}{-4 x^5-25 x^6-8 x^7+112 x^8+(x^5+8 x^6+16 x^7) \log (x)+(-8 x^4-50 x^5-16 x^6+224 x^7+(2 x^4+16 x^5+32 x^6) \log (x)) \log (4-7 x-\log (x))+(-4 x^3-25 x^4-8 x^5+112 x^6+(x^3+8 x^4+16 x^5) \log (x)) \log ^2(4-7 x-\log (x))} \, dx\)

Optimal. Leaf size=28 \[ \frac {4}{x \left (x+4 x^2\right ) (x+\log (4-7 x-\log (x)))} \]

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Rubi [F]  time = 8.19, 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+4 x+60 x^2-448 x^3+\left (-12 x-64 x^2\right ) \log (x)+\left (32+136 x-336 x^2+(-8-48 x) \log (x)\right ) \log (4-7 x-\log (x))}{-4 x^5-25 x^6-8 x^7+112 x^8+\left (x^5+8 x^6+16 x^7\right ) \log (x)+\left (-8 x^4-50 x^5-16 x^6+224 x^7+\left (2 x^4+16 x^5+32 x^6\right ) \log (x)\right ) \log (4-7 x-\log (x))+\left (-4 x^3-25 x^4-8 x^5+112 x^6+\left (x^3+8 x^4+16 x^5\right ) \log (x)\right ) \log ^2(4-7 x-\log (x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-4 + 4*x + 60*x^2 - 448*x^3 + (-12*x - 64*x^2)*Log[x] + (32 + 136*x - 336*x^2 + (-8 - 48*x)*Log[x])*Log[4
- 7*x - Log[x]])/(-4*x^5 - 25*x^6 - 8*x^7 + 112*x^8 + (x^5 + 8*x^6 + 16*x^7)*Log[x] + (-8*x^4 - 50*x^5 - 16*x
^6 + 224*x^7 + (2*x^4 + 16*x^5 + 32*x^6)*Log[x])*Log[4 - 7*x - Log[x]] + (-4*x^3 - 25*x^4 - 8*x^5 + 112*x^6 +
(x^3 + 8*x^4 + 16*x^5)*Log[x])*Log[4 - 7*x - Log[x]]^2),x]

[Out]

64/(x + Log[4 - 7*x - Log[x]]) + 192*Defer[Int][1/((-4 + 7*x + Log[x])*(x + Log[4 - 7*x - Log[x]])^2), x] - 4*
Defer[Int][1/(x^3*(-4 + 7*x + Log[x])*(x + Log[4 - 7*x - Log[x]])^2), x] + 4*Defer[Int][1/(x^2*(-4 + 7*x + Log
[x])*(x + Log[4 - 7*x - Log[x]])^2), x] + 20*Defer[Int][1/(x*(-4 + 7*x + Log[x])*(x + Log[4 - 7*x - Log[x]])^2
), x] + 448*Defer[Int][x/((-4 + 7*x + Log[x])*(x + Log[4 - 7*x - Log[x]])^2), x] + 176*Defer[Int][1/((1 + 4*x)
*(-4 + 7*x + Log[x])*(x + Log[4 - 7*x - Log[x]])^2), x] + 64*Defer[Int][Log[x]/((-4 + 7*x + Log[x])*(x + Log[4
- 7*x - Log[x]])^2), x] - 4*Defer[Int][Log[x]/(x^2*(-4 + 7*x + Log[x])*(x + Log[4 - 7*x - Log[x]])^2), x] + 1
6*Defer[Int][Log[x]/(x*(-4 + 7*x + Log[x])*(x + Log[4 - 7*x - Log[x]])^2), x] - 64*Defer[Int][Log[x]/((1 + 4*x
)*(-4 + 7*x + Log[x])*(x + Log[4 - 7*x - Log[x]])^2), x] - 8*Defer[Int][1/(x^3*(x + Log[4 - 7*x - Log[x]])), x
] + 16*Defer[Int][1/(x^2*(x + Log[4 - 7*x - Log[x]])), x] - 256*Defer[Int][1/((1 + 4*x)^2*(x + Log[4 - 7*x - L
og[x]])), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {4 \left (1-x-15 x^2+112 x^3+\left (-8-34 x+84 x^2\right ) \log (4-7 x-\log (x))+\log (x) (x (3+16 x)+2 (1+6 x) \log (4-7 x-\log (x)))\right )}{x^3 (1+4 x)^2 (4-7 x-\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx\\ &=4 \int \frac {1-x-15 x^2+112 x^3+\left (-8-34 x+84 x^2\right ) \log (4-7 x-\log (x))+\log (x) (x (3+16 x)+2 (1+6 x) \log (4-7 x-\log (x)))}{x^3 (1+4 x)^2 (4-7 x-\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx\\ &=4 \int \left (\frac {-1-3 x-7 x^2-x \log (x)}{x^3 (1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}-\frac {2 (1+6 x)}{x^3 (1+4 x)^2 (x+\log (4-7 x-\log (x)))}\right ) \, dx\\ &=4 \int \frac {-1-3 x-7 x^2-x \log (x)}{x^3 (1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx-8 \int \frac {1+6 x}{x^3 (1+4 x)^2 (x+\log (4-7 x-\log (x)))} \, dx\\ &=4 \int \left (\frac {-1-3 x-7 x^2-x \log (x)}{x^3 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}+\frac {4 \left (1+3 x+7 x^2+x \log (x)\right )}{x^2 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}-\frac {16 \left (1+3 x+7 x^2+x \log (x)\right )}{x (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}+\frac {64 \left (1+3 x+7 x^2+x \log (x)\right )}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}\right ) \, dx-8 \int \left (\frac {1}{x^3 (x+\log (4-7 x-\log (x)))}-\frac {2}{x^2 (x+\log (4-7 x-\log (x)))}+\frac {32}{(1+4 x)^2 (x+\log (4-7 x-\log (x)))}\right ) \, dx\\ &=4 \int \frac {-1-3 x-7 x^2-x \log (x)}{x^3 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx-8 \int \frac {1}{x^3 (x+\log (4-7 x-\log (x)))} \, dx+16 \int \frac {1+3 x+7 x^2+x \log (x)}{x^2 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+16 \int \frac {1}{x^2 (x+\log (4-7 x-\log (x)))} \, dx-64 \int \frac {1+3 x+7 x^2+x \log (x)}{x (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+256 \int \frac {1+3 x+7 x^2+x \log (x)}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx-256 \int \frac {1}{(1+4 x)^2 (x+\log (4-7 x-\log (x)))} \, dx\\ &=\frac {64}{x+\log (4-7 x-\log (x))}+4 \int \left (-\frac {1}{x^3 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}-\frac {3}{x^2 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}-\frac {7}{x (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}-\frac {\log (x)}{x^2 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}\right ) \, dx-8 \int \frac {1}{x^3 (x+\log (4-7 x-\log (x)))} \, dx+16 \int \frac {1}{x^2 (x+\log (4-7 x-\log (x)))} \, dx+16 \int \left (\frac {7}{(-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}+\frac {1}{x^2 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}+\frac {3}{x (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}+\frac {\log (x)}{x (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}\right ) \, dx-256 \int \frac {1}{(1+4 x)^2 (x+\log (4-7 x-\log (x)))} \, dx+256 \int \left (\frac {1}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}+\frac {3 x}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}+\frac {7 x^2}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}+\frac {x \log (x)}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}\right ) \, dx\\ &=\frac {64}{x+\log (4-7 x-\log (x))}-4 \int \frac {1}{x^3 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx-4 \int \frac {\log (x)}{x^2 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx-8 \int \frac {1}{x^3 (x+\log (4-7 x-\log (x)))} \, dx-12 \int \frac {1}{x^2 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+16 \int \frac {1}{x^2 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+16 \int \frac {\log (x)}{x (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+16 \int \frac {1}{x^2 (x+\log (4-7 x-\log (x)))} \, dx-28 \int \frac {1}{x (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+48 \int \frac {1}{x (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+112 \int \frac {1}{(-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+256 \int \frac {1}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+256 \int \frac {x \log (x)}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx-256 \int \frac {1}{(1+4 x)^2 (x+\log (4-7 x-\log (x)))} \, dx+768 \int \frac {x}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+1792 \int \frac {x^2}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx\\ &=\frac {64}{x+\log (4-7 x-\log (x))}-4 \int \frac {1}{x^3 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx-4 \int \frac {\log (x)}{x^2 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx-8 \int \frac {1}{x^3 (x+\log (4-7 x-\log (x)))} \, dx-12 \int \frac {1}{x^2 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+16 \int \frac {1}{x^2 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+16 \int \frac {\log (x)}{x (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+16 \int \frac {1}{x^2 (x+\log (4-7 x-\log (x)))} \, dx-28 \int \frac {1}{x (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+48 \int \frac {1}{x (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+112 \int \frac {1}{(-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+256 \int \frac {1}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx-256 \int \frac {1}{(1+4 x)^2 (x+\log (4-7 x-\log (x)))} \, dx+256 \int \left (\frac {\log (x)}{4 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}-\frac {\log (x)}{4 (1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}\right ) \, dx+768 \int \left (\frac {1}{4 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}-\frac {1}{4 (1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}\right ) \, dx+1792 \int \left (-\frac {1}{16 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}+\frac {x}{4 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}+\frac {1}{16 (1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2}\right ) \, dx\\ &=\frac {64}{x+\log (4-7 x-\log (x))}-4 \int \frac {1}{x^3 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx-4 \int \frac {\log (x)}{x^2 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx-8 \int \frac {1}{x^3 (x+\log (4-7 x-\log (x)))} \, dx-12 \int \frac {1}{x^2 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+16 \int \frac {1}{x^2 (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+16 \int \frac {\log (x)}{x (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+16 \int \frac {1}{x^2 (x+\log (4-7 x-\log (x)))} \, dx-28 \int \frac {1}{x (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+48 \int \frac {1}{x (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+64 \int \frac {\log (x)}{(-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx-64 \int \frac {\log (x)}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+112 \int \frac {1}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+192 \int \frac {1}{(-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx-192 \int \frac {1}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx+256 \int \frac {1}{(1+4 x) (-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx-256 \int \frac {1}{(1+4 x)^2 (x+\log (4-7 x-\log (x)))} \, dx+448 \int \frac {x}{(-4+7 x+\log (x)) (x+\log (4-7 x-\log (x)))^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 26, normalized size = 0.93 \begin {gather*} \frac {4}{x^2 (1+4 x) (x+\log (4-7 x-\log (x)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 4*x + 60*x^2 - 448*x^3 + (-12*x - 64*x^2)*Log[x] + (32 + 136*x - 336*x^2 + (-8 - 48*x)*Log[x])
*Log[4 - 7*x - Log[x]])/(-4*x^5 - 25*x^6 - 8*x^7 + 112*x^8 + (x^5 + 8*x^6 + 16*x^7)*Log[x] + (-8*x^4 - 50*x^5
- 16*x^6 + 224*x^7 + (2*x^4 + 16*x^5 + 32*x^6)*Log[x])*Log[4 - 7*x - Log[x]] + (-4*x^3 - 25*x^4 - 8*x^5 + 112*
x^6 + (x^3 + 8*x^4 + 16*x^5)*Log[x])*Log[4 - 7*x - Log[x]]^2),x]

[Out]

4/(x^2*(1 + 4*x)*(x + Log[4 - 7*x - Log[x]]))

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fricas [A]  time = 0.68, size = 33, normalized size = 1.18 \begin {gather*} \frac {4}{4 \, x^{4} + x^{3} + {\left (4 \, x^{3} + x^{2}\right )} \log \left (-7 \, x - \log \relax (x) + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-48*x-8)*log(x)-336*x^2+136*x+32)*log(-log(x)-7*x+4)+(-64*x^2-12*x)*log(x)-448*x^3+60*x^2+4*x-4)/
(((16*x^5+8*x^4+x^3)*log(x)+112*x^6-8*x^5-25*x^4-4*x^3)*log(-log(x)-7*x+4)^2+((32*x^6+16*x^5+2*x^4)*log(x)+224
*x^7-16*x^6-50*x^5-8*x^4)*log(-log(x)-7*x+4)+(16*x^7+8*x^6+x^5)*log(x)+112*x^8-8*x^7-25*x^6-4*x^5),x, algorith
m="fricas")

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-48*x-8)*log(x)-336*x^2+136*x+32)*log(-log(x)-7*x+4)+(-64*x^2-12*x)*log(x)-448*x^3+60*x^2+4*x-4)/
(((16*x^5+8*x^4+x^3)*log(x)+112*x^6-8*x^5-25*x^4-4*x^3)*log(-log(x)-7*x+4)^2+((32*x^6+16*x^5+2*x^4)*log(x)+224
*x^7-16*x^6-50*x^5-8*x^4)*log(-log(x)-7*x+4)+(16*x^7+8*x^6+x^5)*log(x)+112*x^8-8*x^7-25*x^6-4*x^5),x, algorith
m="giac")

[Out]

Timed out

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maple [A]  time = 0.04, size = 27, normalized size = 0.96

method	result	size
risch	\(\frac {4}{\left (4 x +1\right ) x^{2} \left (x +\ln \left (-\ln \relax (x )-7 x +4\right )\right )}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-48*x-8)*ln(x)-336*x^2+136*x+32)*ln(-ln(x)-7*x+4)+(-64*x^2-12*x)*ln(x)-448*x^3+60*x^2+4*x-4)/(((16*x^5+
8*x^4+x^3)*ln(x)+112*x^6-8*x^5-25*x^4-4*x^3)*ln(-ln(x)-7*x+4)^2+((32*x^6+16*x^5+2*x^4)*ln(x)+224*x^7-16*x^6-50
*x^5-8*x^4)*ln(-ln(x)-7*x+4)+(16*x^7+8*x^6+x^5)*ln(x)+112*x^8-8*x^7-25*x^6-4*x^5),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.40, size = 33, normalized size = 1.18 \begin {gather*} \frac {4}{4 \, x^{4} + x^{3} + {\left (4 \, x^{3} + x^{2}\right )} \log \left (-7 \, x - \log \relax (x) + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-48*x-8)*log(x)-336*x^2+136*x+32)*log(-log(x)-7*x+4)+(-64*x^2-12*x)*log(x)-448*x^3+60*x^2+4*x-4)/
(((16*x^5+8*x^4+x^3)*log(x)+112*x^6-8*x^5-25*x^4-4*x^3)*log(-log(x)-7*x+4)^2+((32*x^6+16*x^5+2*x^4)*log(x)+224
*x^7-16*x^6-50*x^5-8*x^4)*log(-log(x)-7*x+4)+(16*x^7+8*x^6+x^5)*log(x)+112*x^8-8*x^7-25*x^6-4*x^5),x, algorith
m="maxima")

[Out]

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

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mupad [B]  time = 2.83, size = 26, normalized size = 0.93 \begin {gather*} \frac {4}{x^2\,\left (4\,x+1\right )\,\left (x+\ln \left (4-\ln \relax (x)-7\,x\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x + log(4 - log(x) - 7*x)*(136*x - log(x)*(48*x + 8) - 336*x^2 + 32) - log(x)*(12*x + 64*x^2) + 60*x^2
- 448*x^3 - 4)/(log(4 - log(x) - 7*x)*(8*x^4 - log(x)*(2*x^4 + 16*x^5 + 32*x^6) + 50*x^5 + 16*x^6 - 224*x^7)
+ log(4 - log(x) - 7*x)^2*(4*x^3 + 25*x^4 + 8*x^5 - 112*x^6 - log(x)*(x^3 + 8*x^4 + 16*x^5)) + 4*x^5 + 25*x^6
+ 8*x^7 - 112*x^8 - log(x)*(x^5 + 8*x^6 + 16*x^7)),x)

[Out]

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

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sympy [A]  time = 0.41, size = 27, normalized size = 0.96 \begin {gather*} \frac {4}{4 x^{4} + x^{3} + \left (4 x^{3} + x^{2}\right ) \log {\left (- 7 x - \log {\relax (x )} + 4 \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-48*x-8)*ln(x)-336*x**2+136*x+32)*ln(-ln(x)-7*x+4)+(-64*x**2-12*x)*ln(x)-448*x**3+60*x**2+4*x-4)/
(((16*x**5+8*x**4+x**3)*ln(x)+112*x**6-8*x**5-25*x**4-4*x**3)*ln(-ln(x)-7*x+4)**2+((32*x**6+16*x**5+2*x**4)*ln
(x)+224*x**7-16*x**6-50*x**5-8*x**4)*ln(-ln(x)-7*x+4)+(16*x**7+8*x**6+x**5)*ln(x)+112*x**8-8*x**7-25*x**6-4*x*
*5),x)

[Out]

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

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