3.25.29 \(\int \frac {-56+20 x-22 x^2+9 x^3-x^4+(28-10 x+x^2) \log (2)+(-16-8 x^2+2 x^3+8 \log (2)) \log (x)+(-40+18 x-2 x^2+(20-9 x+x^2) \log (2)+(-16+4 x+(8-2 x) \log (2)) \log (x)) \log (\frac {4-x}{5 x-x^2+2 x \log (x)})}{40 x^2-18 x^3+2 x^4+(-20 x^2+9 x^3-x^4) \log (2)+(16 x^2-4 x^3+(-8 x^2+2 x^3) \log (2)) \log (x)} \, dx\)

Optimal. Leaf size=38 \[ 1-\frac {x}{2-\log (2)}+\frac {\log \left (\frac {-4+x}{-x+x (x-2 (2+\log (x)))}\right )}{x} \]

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Rubi [F]  time = 5.20, 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 {-56+20 x-22 x^2+9 x^3-x^4+\left (28-10 x+x^2\right ) \log (2)+\left (-16-8 x^2+2 x^3+8 \log (2)\right ) \log (x)+\left (-40+18 x-2 x^2+\left (20-9 x+x^2\right ) \log (2)+(-16+4 x+(8-2 x) \log (2)) \log (x)\right ) \log \left (\frac {4-x}{5 x-x^2+2 x \log (x)}\right )}{40 x^2-18 x^3+2 x^4+\left (-20 x^2+9 x^3-x^4\right ) \log (2)+\left (16 x^2-4 x^3+\left (-8 x^2+2 x^3\right ) \log (2)\right ) \log (x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-56 + 20*x - 22*x^2 + 9*x^3 - x^4 + (28 - 10*x + x^2)*Log[2] + (-16 - 8*x^2 + 2*x^3 + 8*Log[2])*Log[x] +
(-40 + 18*x - 2*x^2 + (20 - 9*x + x^2)*Log[2] + (-16 + 4*x + (8 - 2*x)*Log[2])*Log[x])*Log[(4 - x)/(5*x - x^2
+ 2*x*Log[x])])/(40*x^2 - 18*x^3 + 2*x^4 + (-20*x^2 + 9*x^3 - x^4)*Log[2] + (16*x^2 - 4*x^3 + (-8*x^2 + 2*x^3)
*Log[2])*Log[x]),x]

[Out]

-x^(-1) - x/(2 - Log[2]) + (8 - Log[16])/(4*x*(2 - Log[2])) - Log[4 - x]/4 + ((8 - Log[16])*Log[4 - x])/(16*(2
- Log[2])) + Log[x]/4 - ((8 - Log[16])*Log[x])/(16*(2 - Log[2])) + Log[(4 - x)/(x*(5 - x + 2*Log[x]))]/x - De
fer[Int][1/((-4 + x)*(-5 + x - 2*Log[x])), x]/(2*(2 - Log[2])) + (Log[2]*Defer[Int][1/((-4 + x)*(-5 + x - 2*Lo
g[x])), x])/(4*(2 - Log[2])) + ((8 - Log[16])*Defer[Int][1/((-4 + x)*(-5 + x - 2*Log[x])), x])/(16*(2 - Log[2]
)) - 2*Defer[Int][1/(x^2*(-5 + x - 2*Log[x])), x] + (14*Defer[Int][1/(x^2*(-5 + x - 2*Log[x])), x])/(2 - Log[2
]) - (7*Log[2]*Defer[Int][1/(x^2*(-5 + x - 2*Log[x])), x])/(2 - Log[2]) - (5*(8 - Log[16])*Defer[Int][1/(x^2*(
-5 + x - 2*Log[x])), x])/(4*(2 - Log[2])) + Defer[Int][1/(x*(-5 + x - 2*Log[x])), x] - (3*Defer[Int][1/(x*(-5
+ x - 2*Log[x])), x])/(2*(2 - Log[2])) + (3*Log[2]*Defer[Int][1/(x*(-5 + x - 2*Log[x])), x])/(4*(2 - Log[2]))
- ((8 - Log[16])*Defer[Int][1/(x*(-5 + x - 2*Log[x])), x])/(16*(2 - Log[2]))

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-56+20 x-22 x^2+9 x^3-x^4+\left (28-10 x+x^2\right ) \log (2)+\left (-16-8 x^2+2 x^3+8 \log (2)\right ) \log (x)+\left (-40+18 x-2 x^2+\left (20-9 x+x^2\right ) \log (2)+(-16+4 x+(8-2 x) \log (2)) \log (x)\right ) \log \left (\frac {4-x}{5 x-x^2+2 x \log (x)}\right )}{(4-x) x^2 (2-\log (2)) (5-x+2 \log (x))} \, dx\\ &=\frac {\int \frac {-56+20 x-22 x^2+9 x^3-x^4+\left (28-10 x+x^2\right ) \log (2)+\left (-16-8 x^2+2 x^3+8 \log (2)\right ) \log (x)+\left (-40+18 x-2 x^2+\left (20-9 x+x^2\right ) \log (2)+(-16+4 x+(8-2 x) \log (2)) \log (x)\right ) \log \left (\frac {4-x}{5 x-x^2+2 x \log (x)}\right )}{(4-x) x^2 (5-x+2 \log (x))} \, dx}{2-\log (2)}\\ &=\frac {\int \frac {-56+20 x-22 x^2+9 x^3-x^4+\left (28-10 x+x^2\right ) \log (2)+2 \left (-8-4 x^2+x^3+\log (16)\right ) \log (x)+(-4+x) (-2+\log (2)) (-5+x-2 \log (x)) \log \left (\frac {-4+x}{x (-5+x-2 \log (x))}\right )}{(4-x) x^2 (5-x+2 \log (x))} \, dx}{2-\log (2)}\\ &=\frac {\int \left (-\frac {22}{(-4+x) (-5+x-2 \log (x))}-\frac {56}{(-4+x) x^2 (-5+x-2 \log (x))}+\frac {20}{(-4+x) x (-5+x-2 \log (x))}+\frac {9 x}{(-4+x) (-5+x-2 \log (x))}-\frac {x^2}{(-4+x) (-5+x-2 \log (x))}+\frac {\left (28-10 x+x^2\right ) \log (2)}{(-4+x) x^2 (-5+x-2 \log (x))}+\frac {2 \left (-8-4 x^2+x^3+\log (16)\right ) \log (x)}{(-4+x) x^2 (-5+x-2 \log (x))}+\frac {(-2+\log (2)) \log \left (\frac {-4+x}{x (-5+x-2 \log (x))}\right )}{x^2}\right ) \, dx}{2-\log (2)}\\ &=-\frac {\int \frac {x^2}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {2 \int \frac {\left (-8-4 x^2+x^3+\log (16)\right ) \log (x)}{(-4+x) x^2 (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {9 \int \frac {x}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {20 \int \frac {1}{(-4+x) x (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {22 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {56 \int \frac {1}{(-4+x) x^2 (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {\log (2) \int \frac {28-10 x+x^2}{(-4+x) x^2 (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\int \frac {\log \left (\frac {-4+x}{x (-5+x-2 \log (x))}\right )}{x^2} \, dx\\ &=\frac {\log \left (\frac {4-x}{x (5-x+2 \log (x))}\right )}{x}-\frac {\int \left (\frac {4}{-5+x-2 \log (x)}+\frac {16}{(-4+x) (-5+x-2 \log (x))}+\frac {x}{-5+x-2 \log (x)}\right ) \, dx}{2-\log (2)}+\frac {2 \int \left (\frac {8+4 x^2-x^3-\log (16)}{2 (-4+x) x^2}+\frac {(-5+x) \left (-8-4 x^2+x^3+\log (16)\right )}{2 (-4+x) x^2 (-5+x-2 \log (x))}\right ) \, dx}{2-\log (2)}+\frac {9 \int \left (\frac {1}{-5+x-2 \log (x)}+\frac {4}{(-4+x) (-5+x-2 \log (x))}\right ) \, dx}{2-\log (2)}+\frac {20 \int \left (\frac {1}{4 (-4+x) (-5+x-2 \log (x))}-\frac {1}{4 x (-5+x-2 \log (x))}\right ) \, dx}{2-\log (2)}-\frac {22 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {56 \int \left (\frac {1}{16 (-4+x) (-5+x-2 \log (x))}-\frac {1}{4 x^2 (-5+x-2 \log (x))}-\frac {1}{16 x (-5+x-2 \log (x))}\right ) \, dx}{2-\log (2)}+\frac {\log (2) \int \left (\frac {1}{4 (-4+x) (-5+x-2 \log (x))}-\frac {7}{x^2 (-5+x-2 \log (x))}+\frac {3}{4 x (-5+x-2 \log (x))}\right ) \, dx}{2-\log (2)}+\int \frac {28-10 x+x^2+8 \log (x)}{(-4+x) x^2 (-5+x-2 \log (x))} \, dx\\ &=\frac {\log \left (\frac {4-x}{x (5-x+2 \log (x))}\right )}{x}+\frac {\int \frac {8+4 x^2-x^3-\log (16)}{(-4+x) x^2} \, dx}{2-\log (2)}-\frac {\int \frac {x}{-5+x-2 \log (x)} \, dx}{2-\log (2)}+\frac {\int \frac {(-5+x) \left (-8-4 x^2+x^3+\log (16)\right )}{(-4+x) x^2 (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {7 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2 (2-\log (2))}+\frac {7 \int \frac {1}{x (-5+x-2 \log (x))} \, dx}{2 (2-\log (2))}-\frac {4 \int \frac {1}{-5+x-2 \log (x)} \, dx}{2-\log (2)}+\frac {5 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {5 \int \frac {1}{x (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {9 \int \frac {1}{-5+x-2 \log (x)} \, dx}{2-\log (2)}+\frac {14 \int \frac {1}{x^2 (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {16 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {22 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {36 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {\log (2) \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{4 (2-\log (2))}+\frac {(3 \log (2)) \int \frac {1}{x (-5+x-2 \log (x))} \, dx}{4 (2-\log (2))}-\frac {(7 \log (2)) \int \frac {1}{x^2 (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\int \left (-\frac {4}{(-4+x) x^2}+\frac {-2+x}{x^2 (-5+x-2 \log (x))}\right ) \, dx\\ &=\frac {\log \left (\frac {4-x}{x (5-x+2 \log (x))}\right )}{x}-4 \int \frac {1}{(-4+x) x^2} \, dx+\frac {\int \left (-1+\frac {8-\log (16)}{16 (-4+x)}+\frac {-8+\log (16)}{4 x^2}+\frac {-8+\log (16)}{16 x}\right ) \, dx}{2-\log (2)}+\frac {\int \left (-\frac {5}{-5+x-2 \log (x)}+\frac {x}{-5+x-2 \log (x)}+\frac {8-\log (16)}{16 (-4+x) (-5+x-2 \log (x))}+\frac {5 (-8+\log (16))}{4 x^2 (-5+x-2 \log (x))}+\frac {-8+\log (16)}{16 x (-5+x-2 \log (x))}\right ) \, dx}{2-\log (2)}-\frac {\int \frac {x}{-5+x-2 \log (x)} \, dx}{2-\log (2)}-\frac {7 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2 (2-\log (2))}+\frac {7 \int \frac {1}{x (-5+x-2 \log (x))} \, dx}{2 (2-\log (2))}-\frac {4 \int \frac {1}{-5+x-2 \log (x)} \, dx}{2-\log (2)}+\frac {5 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {5 \int \frac {1}{x (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {9 \int \frac {1}{-5+x-2 \log (x)} \, dx}{2-\log (2)}+\frac {14 \int \frac {1}{x^2 (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {16 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {22 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {36 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {\log (2) \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{4 (2-\log (2))}+\frac {(3 \log (2)) \int \frac {1}{x (-5+x-2 \log (x))} \, dx}{4 (2-\log (2))}-\frac {(7 \log (2)) \int \frac {1}{x^2 (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\int \frac {-2+x}{x^2 (-5+x-2 \log (x))} \, dx\\ &=-\frac {x}{2-\log (2)}+\frac {8-\log (16)}{4 x (2-\log (2))}+\frac {(8-\log (16)) \log (4-x)}{16 (2-\log (2))}-\frac {(8-\log (16)) \log (x)}{16 (2-\log (2))}+\frac {\log \left (\frac {4-x}{x (5-x+2 \log (x))}\right )}{x}-4 \int \left (\frac {1}{16 (-4+x)}-\frac {1}{4 x^2}-\frac {1}{16 x}\right ) \, dx-\frac {7 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2 (2-\log (2))}+\frac {7 \int \frac {1}{x (-5+x-2 \log (x))} \, dx}{2 (2-\log (2))}-\frac {4 \int \frac {1}{-5+x-2 \log (x)} \, dx}{2-\log (2)}-\frac {5 \int \frac {1}{-5+x-2 \log (x)} \, dx}{2-\log (2)}+\frac {5 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {5 \int \frac {1}{x (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {9 \int \frac {1}{-5+x-2 \log (x)} \, dx}{2-\log (2)}+\frac {14 \int \frac {1}{x^2 (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {16 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {22 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {36 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {\log (2) \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{4 (2-\log (2))}+\frac {(3 \log (2)) \int \frac {1}{x (-5+x-2 \log (x))} \, dx}{4 (2-\log (2))}-\frac {(7 \log (2)) \int \frac {1}{x^2 (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {(8-\log (16)) \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{16 (2-\log (2))}-\frac {(8-\log (16)) \int \frac {1}{x (-5+x-2 \log (x))} \, dx}{16 (2-\log (2))}+\frac {(5 (-8+\log (16))) \int \frac {1}{x^2 (-5+x-2 \log (x))} \, dx}{4 (2-\log (2))}+\int \left (-\frac {2}{x^2 (-5+x-2 \log (x))}+\frac {1}{x (-5+x-2 \log (x))}\right ) \, dx\\ &=-\frac {1}{x}-\frac {x}{2-\log (2)}+\frac {8-\log (16)}{4 x (2-\log (2))}-\frac {1}{4} \log (4-x)+\frac {(8-\log (16)) \log (4-x)}{16 (2-\log (2))}+\frac {\log (x)}{4}-\frac {(8-\log (16)) \log (x)}{16 (2-\log (2))}+\frac {\log \left (\frac {4-x}{x (5-x+2 \log (x))}\right )}{x}-2 \int \frac {1}{x^2 (-5+x-2 \log (x))} \, dx-\frac {7 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2 (2-\log (2))}+\frac {7 \int \frac {1}{x (-5+x-2 \log (x))} \, dx}{2 (2-\log (2))}-\frac {4 \int \frac {1}{-5+x-2 \log (x)} \, dx}{2-\log (2)}-\frac {5 \int \frac {1}{-5+x-2 \log (x)} \, dx}{2-\log (2)}+\frac {5 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {5 \int \frac {1}{x (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {9 \int \frac {1}{-5+x-2 \log (x)} \, dx}{2-\log (2)}+\frac {14 \int \frac {1}{x^2 (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {16 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}-\frac {22 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {36 \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {\log (2) \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{4 (2-\log (2))}+\frac {(3 \log (2)) \int \frac {1}{x (-5+x-2 \log (x))} \, dx}{4 (2-\log (2))}-\frac {(7 \log (2)) \int \frac {1}{x^2 (-5+x-2 \log (x))} \, dx}{2-\log (2)}+\frac {(8-\log (16)) \int \frac {1}{(-4+x) (-5+x-2 \log (x))} \, dx}{16 (2-\log (2))}-\frac {(8-\log (16)) \int \frac {1}{x (-5+x-2 \log (x))} \, dx}{16 (2-\log (2))}+\frac {(5 (-8+\log (16))) \int \frac {1}{x^2 (-5+x-2 \log (x))} \, dx}{4 (2-\log (2))}+\int \frac {1}{x (-5+x-2 \log (x))} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.15, size = 34, normalized size = 0.89 \begin {gather*} \frac {x+\frac {(-2+\log (2)) \log \left (\frac {-4+x}{x (-5+x-2 \log (x))}\right )}{x}}{-2+\log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-56 + 20*x - 22*x^2 + 9*x^3 - x^4 + (28 - 10*x + x^2)*Log[2] + (-16 - 8*x^2 + 2*x^3 + 8*Log[2])*Log
[x] + (-40 + 18*x - 2*x^2 + (20 - 9*x + x^2)*Log[2] + (-16 + 4*x + (8 - 2*x)*Log[2])*Log[x])*Log[(4 - x)/(5*x
- x^2 + 2*x*Log[x])])/(40*x^2 - 18*x^3 + 2*x^4 + (-20*x^2 + 9*x^3 - x^4)*Log[2] + (16*x^2 - 4*x^3 + (-8*x^2 +
2*x^3)*Log[2])*Log[x]),x]

[Out]

(x + ((-2 + Log[2])*Log[(-4 + x)/(x*(-5 + x - 2*Log[x]))])/x)/(-2 + Log[2])

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fricas [A]  time = 0.89, size = 39, normalized size = 1.03 \begin {gather*} \frac {x^{2} + {\left (\log \relax (2) - 2\right )} \log \left (\frac {x - 4}{x^{2} - 2 \, x \log \relax (x) - 5 \, x}\right )}{x \log \relax (2) - 2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-2*x+8)*log(2)+4*x-16)*log(x)+(x^2-9*x+20)*log(2)-2*x^2+18*x-40)*log((-x+4)/(2*x*log(x)-x^2+5*x)
)+(8*log(2)+2*x^3-8*x^2-16)*log(x)+(x^2-10*x+28)*log(2)-x^4+9*x^3-22*x^2+20*x-56)/(((2*x^3-8*x^2)*log(2)-4*x^3
+16*x^2)*log(x)+(-x^4+9*x^3-20*x^2)*log(2)+2*x^4-18*x^3+40*x^2),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.46, size = 37, normalized size = 0.97 \begin {gather*} \frac {x}{\log \relax (2) - 2} - \frac {\log \left (x - 2 \, \log \relax (x) - 5\right )}{x} + \frac {\log \left (x - 4\right )}{x} - \frac {\log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-2*x+8)*log(2)+4*x-16)*log(x)+(x^2-9*x+20)*log(2)-2*x^2+18*x-40)*log((-x+4)/(2*x*log(x)-x^2+5*x)
)+(8*log(2)+2*x^3-8*x^2-16)*log(x)+(x^2-10*x+28)*log(2)-x^4+9*x^3-22*x^2+20*x-56)/(((2*x^3-8*x^2)*log(2)-4*x^3
+16*x^2)*log(x)+(-x^4+9*x^3-20*x^2)*log(2)+2*x^4-18*x^3+40*x^2),x, algorithm="giac")

[Out]

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

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maple [C]  time = 0.32, size = 660, normalized size = 17.37

method	result	size
risch	\(-\frac {\ln \left (-2 \ln \relax (x )+x -5\right )}{x}+\frac {-i \pi \ln \relax (2) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{2 \ln \relax (x )-x +5}\right ) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{x \left (2 \ln \relax (x )-x +5\right )}\right )^{2}-2 i \pi \,\mathrm {csgn}\left (i \left (x -4\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{2 \ln \relax (x )-x +5}\right )^{2}+2 i \pi \,\mathrm {csgn}\left (\frac {i}{2 \ln \relax (x )-x +5}\right ) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{2 \ln \relax (x )-x +5}\right )^{2}-i \pi \ln \relax (2) \mathrm {csgn}\left (\frac {i}{2 \ln \relax (x )-x +5}\right ) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{2 \ln \relax (x )-x +5}\right )^{2}+2 i \pi \,\mathrm {csgn}\left (\frac {i \left (x -4\right )}{2 \ln \relax (x )-x +5}\right ) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{x \left (2 \ln \relax (x )-x +5\right )}\right )^{2}+2 i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{2 \ln \relax (x )-x +5}\right ) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{x \left (2 \ln \relax (x )-x +5\right )}\right )+i \pi \ln \relax (2) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{2 \ln \relax (x )-x +5}\right )^{3}-i \pi \ln \relax (2) \mathrm {csgn}\left (i \left (x -4\right )\right ) \mathrm {csgn}\left (\frac {i}{2 \ln \relax (x )-x +5}\right ) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{2 \ln \relax (x )-x +5}\right )+i \pi \ln \relax (2) \mathrm {csgn}\left (i \left (x -4\right )\right ) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{2 \ln \relax (x )-x +5}\right )^{2}+2 i \pi \,\mathrm {csgn}\left (i \left (x -4\right )\right ) \mathrm {csgn}\left (\frac {i}{2 \ln \relax (x )-x +5}\right ) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{2 \ln \relax (x )-x +5}\right )+i \pi \ln \relax (2) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{x \left (2 \ln \relax (x )-x +5\right )}\right )^{2}-2 i \pi \,\mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{x \left (2 \ln \relax (x )-x +5\right )}\right )^{2}-i \pi \ln \relax (2) \mathrm {csgn}\left (\frac {i}{x}\right ) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{2 \ln \relax (x )-x +5}\right ) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{x \left (2 \ln \relax (x )-x +5\right )}\right )-2 i \pi \mathrm {csgn}\left (\frac {i \left (x -4\right )}{2 \ln \relax (x )-x +5}\right )^{3}+i \pi \ln \relax (2) \mathrm {csgn}\left (\frac {i \left (x -4\right )}{x \left (2 \ln \relax (x )-x +5\right )}\right )^{3}-2 i \pi \mathrm {csgn}\left (\frac {i \left (x -4\right )}{x \left (2 \ln \relax (x )-x +5\right )}\right )^{3}-2 \ln \relax (2) \ln \relax (x )+2 \ln \relax (2) \ln \left (x -4\right )+2 x^{2}+4 \ln \relax (x )-4 \ln \left (x -4\right )}{2 \left (\ln \relax (2)-2\right ) x}\)	\(660\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((-2*x+8)*ln(2)+4*x-16)*ln(x)+(x^2-9*x+20)*ln(2)-2*x^2+18*x-40)*ln((-x+4)/(2*x*ln(x)-x^2+5*x))+(8*ln(2)+
2*x^3-8*x^2-16)*ln(x)+(x^2-10*x+28)*ln(2)-x^4+9*x^3-22*x^2+20*x-56)/(((2*x^3-8*x^2)*ln(2)-4*x^3+16*x^2)*ln(x)+
(-x^4+9*x^3-20*x^2)*ln(2)+2*x^4-18*x^3+40*x^2),x,method=_RETURNVERBOSE)

[Out]

-1/x*ln(-2*ln(x)+x-5)+1/2*(-I*Pi*ln(2)*csgn(I*(x-4)/(2*ln(x)-x+5))*csgn(I*(x-4)/x/(2*ln(x)-x+5))^2-2*I*Pi*csgn
(I*(x-4))*csgn(I*(x-4)/(2*ln(x)-x+5))^2+2*I*Pi*csgn(I/(2*ln(x)-x+5))*csgn(I*(x-4)/(2*ln(x)-x+5))^2-I*Pi*ln(2)*
csgn(I/(2*ln(x)-x+5))*csgn(I*(x-4)/(2*ln(x)-x+5))^2+2*I*Pi*csgn(I*(x-4)/(2*ln(x)-x+5))*csgn(I*(x-4)/x/(2*ln(x)
-x+5))^2+2*I*Pi*csgn(I/x)*csgn(I*(x-4)/(2*ln(x)-x+5))*csgn(I*(x-4)/x/(2*ln(x)-x+5))+I*Pi*ln(2)*csgn(I*(x-4)/(2
*ln(x)-x+5))^3-I*Pi*ln(2)*csgn(I*(x-4))*csgn(I/(2*ln(x)-x+5))*csgn(I*(x-4)/(2*ln(x)-x+5))+I*Pi*ln(2)*csgn(I*(x
-4))*csgn(I*(x-4)/(2*ln(x)-x+5))^2+2*I*Pi*csgn(I*(x-4))*csgn(I/(2*ln(x)-x+5))*csgn(I*(x-4)/(2*ln(x)-x+5))+I*Pi
*ln(2)*csgn(I/x)*csgn(I*(x-4)/x/(2*ln(x)-x+5))^2-2*I*Pi*csgn(I/x)*csgn(I*(x-4)/x/(2*ln(x)-x+5))^2-I*Pi*ln(2)*c
sgn(I/x)*csgn(I*(x-4)/(2*ln(x)-x+5))*csgn(I*(x-4)/x/(2*ln(x)-x+5))-2*I*Pi*csgn(I*(x-4)/(2*ln(x)-x+5))^3+I*Pi*l
n(2)*csgn(I*(x-4)/x/(2*ln(x)-x+5))^3-2*I*Pi*csgn(I*(x-4)/x/(2*ln(x)-x+5))^3-2*ln(2)*ln(x)+2*ln(2)*ln(x-4)+2*x^
2+4*ln(x)-4*ln(x-4))/(ln(2)-2)/x

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maxima [A]  time = 0.66, size = 45, normalized size = 1.18 \begin {gather*} \frac {x^{2} - {\left (\log \relax (2) - 2\right )} \log \left (x - 2 \, \log \relax (x) - 5\right ) + {\left (\log \relax (2) - 2\right )} \log \left (x - 4\right ) - {\left (\log \relax (2) - 2\right )} \log \relax (x)}{x {\left (\log \relax (2) - 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-2*x+8)*log(2)+4*x-16)*log(x)+(x^2-9*x+20)*log(2)-2*x^2+18*x-40)*log((-x+4)/(2*x*log(x)-x^2+5*x)
)+(8*log(2)+2*x^3-8*x^2-16)*log(x)+(x^2-10*x+28)*log(2)-x^4+9*x^3-22*x^2+20*x-56)/(((2*x^3-8*x^2)*log(2)-4*x^3
+16*x^2)*log(x)+(-x^4+9*x^3-20*x^2)*log(2)+2*x^4-18*x^3+40*x^2),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 2.13, size = 35, normalized size = 0.92 \begin {gather*} \frac {x}{\ln \relax (2)-2}+\frac {\ln \left (-\frac {x-4}{5\,x+2\,x\,\ln \relax (x)-x^2}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(20*x - log(-(x - 4)/(5*x + 2*x*log(x) - x^2))*(log(x)*(log(2)*(2*x - 8) - 4*x + 16) - 18*x + 2*x^2 - log
(2)*(x^2 - 9*x + 20) + 40) - 22*x^2 + 9*x^3 - x^4 + log(2)*(x^2 - 10*x + 28) + log(x)*(8*log(2) - 8*x^2 + 2*x^
3 - 16) - 56)/(log(2)*(20*x^2 - 9*x^3 + x^4) + log(x)*(log(2)*(8*x^2 - 2*x^3) - 16*x^2 + 4*x^3) - 40*x^2 + 18*
x^3 - 2*x^4),x)

[Out]

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

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sympy [A]  time = 0.64, size = 26, normalized size = 0.68 \begin {gather*} \frac {x}{-2 + \log {\relax (2 )}} + \frac {\log {\left (\frac {4 - x}{- x^{2} + 2 x \log {\relax (x )} + 5 x} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-2*x+8)*ln(2)+4*x-16)*ln(x)+(x**2-9*x+20)*ln(2)-2*x**2+18*x-40)*ln((-x+4)/(2*x*ln(x)-x**2+5*x))+
(8*ln(2)+2*x**3-8*x**2-16)*ln(x)+(x**2-10*x+28)*ln(2)-x**4+9*x**3-22*x**2+20*x-56)/(((2*x**3-8*x**2)*ln(2)-4*x
**3+16*x**2)*ln(x)+(-x**4+9*x**3-20*x**2)*ln(2)+2*x**4-18*x**3+40*x**2),x)

[Out]

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

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