\(\int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+(256-192 x+48 x^2-4 x^3) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+(256 x-192 x^2+48 x^3-4 x^4) \log (16)} \, dx\) [7944]

Optimal result

Integrand size = 101, antiderivative size = 24 \[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=\log \left (\frac {e^{2 x}}{(-4+x)^2}-4 \left (x-\frac {x}{\log (16)}\right )\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=\int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx \]

[In]

Int[(-256 + 192*x - 48*x^2 + 4*x^3 + E^(2*x)*(-10 + 2*x)*Log[16] + (256 - 192*x + 48*x^2 - 4*x^3)*Log[16])/(-2
56*x + 192*x^2 - 48*x^3 + 4*x^4 + E^(2*x)*(-4 + x)*Log[16] + (256*x - 192*x^2 + 48*x^3 - 4*x^4)*Log[16]),x]

[Out]

2*x - 2*Log[4 - x] + 192*(1 - Log[16])*Defer[Int][x/(-64*x*(1 - Log[16]) + 32*x^2*(1 - Log[16]) - 4*x^3*(1 - L
og[16]) - E^(2*x)*Log[16]), x] + 8*(1 - Log[16])*Defer[Int][x^3/(-64*x*(1 - Log[16]) + 32*x^2*(1 - Log[16]) -
4*x^3*(1 - Log[16]) - E^(2*x)*Log[16]), x] + 64*(1 - Log[16])*Defer[Int][(64*x*(1 - Log[16]) - 32*x^2*(1 - Log
[16]) + 4*x^3*(1 - Log[16]) + E^(2*x)*Log[16])^(-1), x] + 76*(1 - Log[16])*Defer[Int][x^2/(64*x*(1 - Log[16])
- 32*x^2*(1 - Log[16]) + 4*x^3*(1 - Log[16]) + E^(2*x)*Log[16]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {256-192 x+48 x^2-4 x^3-e^{2 x} (-10+2 x) \log (16)-\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{(4-x) \left (64 x (1-\log (16))-32 x^2 (1-\log (16))+4 x^3 (1-\log (16))+e^{2 x} \log (16)\right )} \, dx \\ & = \int \left (\frac {2 (-5+x)}{-4+x}+\frac {4 \left (16-48 x+19 x^2-2 x^3\right ) (1-\log (16))}{64 x (1-\log (16))-32 x^2 (1-\log (16))+4 x^3 (1-\log (16))+e^{2 x} \log (16)}\right ) \, dx \\ & = 2 \int \frac {-5+x}{-4+x} \, dx+(4 (1-\log (16))) \int \frac {16-48 x+19 x^2-2 x^3}{64 x (1-\log (16))-32 x^2 (1-\log (16))+4 x^3 (1-\log (16))+e^{2 x} \log (16)} \, dx \\ & = 2 \int \left (1+\frac {1}{4-x}\right ) \, dx+(4 (1-\log (16))) \int \left (\frac {48 x}{-64 x (1-\log (16))+32 x^2 (1-\log (16))-4 x^3 (1-\log (16))-e^{2 x} \log (16)}+\frac {2 x^3}{-64 x (1-\log (16))+32 x^2 (1-\log (16))-4 x^3 (1-\log (16))-e^{2 x} \log (16)}+\frac {16}{64 x (1-\log (16))-32 x^2 (1-\log (16))+4 x^3 (1-\log (16))+e^{2 x} \log (16)}+\frac {19 x^2}{64 x (1-\log (16))-32 x^2 (1-\log (16))+4 x^3 (1-\log (16))+e^{2 x} \log (16)}\right ) \, dx \\ & = 2 x-2 \log (4-x)+(8 (1-\log (16))) \int \frac {x^3}{-64 x (1-\log (16))+32 x^2 (1-\log (16))-4 x^3 (1-\log (16))-e^{2 x} \log (16)} \, dx+(64 (1-\log (16))) \int \frac {1}{64 x (1-\log (16))-32 x^2 (1-\log (16))+4 x^3 (1-\log (16))+e^{2 x} \log (16)} \, dx+(76 (1-\log (16))) \int \frac {x^2}{64 x (1-\log (16))-32 x^2 (1-\log (16))+4 x^3 (1-\log (16))+e^{2 x} \log (16)} \, dx+(192 (1-\log (16))) \int \frac {x}{-64 x (1-\log (16))+32 x^2 (1-\log (16))-4 x^3 (1-\log (16))-e^{2 x} \log (16)} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(52\) vs. \(2(24)=48\).

Time = 0.08 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.17 \[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=-2 \log (4-x)+\log \left (-64 x+32 x^2-4 x^3-e^{2 x} \log (16)+64 x \log (16)-32 x^2 \log (16)+4 x^3 \log (16)\right ) \]

[In]

Integrate[(-256 + 192*x - 48*x^2 + 4*x^3 + E^(2*x)*(-10 + 2*x)*Log[16] + (256 - 192*x + 48*x^2 - 4*x^3)*Log[16
])/(-256*x + 192*x^2 - 48*x^3 + 4*x^4 + E^(2*x)*(-4 + x)*Log[16] + (256*x - 192*x^2 + 48*x^3 - 4*x^4)*Log[16])
,x]

[Out]

-2*Log[4 - x] + Log[-64*x + 32*x^2 - 4*x^3 - E^(2*x)*Log[16] + 64*x*Log[16] - 32*x^2*Log[16] + 4*x^3*Log[16]]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs. \(2(21)=42\).

Time = 0.40 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.96

method	result	size
risch	\(-2 \ln \left (x -4\right )+\ln \left ({\mathrm e}^{2 x}-\frac {x \left (4 x^{2} \ln \left (2\right )-32 x \ln \left (2\right )-x^{2}+64 \ln \left (2\right )+8 x -16\right )}{\ln \left (2\right )}\right )\)	\(47\)
norman	\(-2 \ln \left (x -4\right )+\ln \left (4 x^{3} \ln \left (2\right )-32 x^{2} \ln \left (2\right )-\ln \left (2\right ) {\mathrm e}^{2 x}-x^{3}+64 x \ln \left (2\right )+8 x^{2}-16 x \right )\)	\(50\)
parallelrisch	\(\ln \left (\frac {4 x^{3} \ln \left (2\right )-32 x^{2} \ln \left (2\right )-\ln \left (2\right ) {\mathrm e}^{2 x}-x^{3}+64 x \ln \left (2\right )+8 x^{2}-16 x}{4 \ln \left (2\right )-1}\right )-2 \ln \left (x -4\right )\)	\(59\)

[In]

int((4*(2*x-10)*ln(2)*exp(x)^2+4*(-4*x^3+48*x^2-192*x+256)*ln(2)+4*x^3-48*x^2+192*x-256)/(4*(x-4)*ln(2)*exp(x)
^2+4*(-4*x^4+48*x^3-192*x^2+256*x)*ln(2)+4*x^4-48*x^3+192*x^2-256*x),x,method=_RETURNVERBOSE)

[Out]

-2*ln(x-4)+ln(exp(2*x)-x*(4*x^2*ln(2)-32*x*ln(2)-x^2+64*ln(2)+8*x-16)/ln(2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (21) = 42\).

Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.79 \[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=\log \left (x^{3} - 8 \, x^{2} - 4 \, {\left (x^{3} - 8 \, x^{2} + 16 \, x\right )} \log \left (2\right ) + e^{\left (2 \, x\right )} \log \left (2\right ) + 16 \, x\right ) - 2 \, \log \left (x - 4\right ) \]

[In]

integrate((4*(2*x-10)*log(2)*exp(x)^2+4*(-4*x^3+48*x^2-192*x+256)*log(2)+4*x^3-48*x^2+192*x-256)/(4*(x-4)*log(
2)*exp(x)^2+4*(-4*x^4+48*x^3-192*x^2+256*x)*log(2)+4*x^4-48*x^3+192*x^2-256*x),x, algorithm="fricas")

[Out]

log(x^3 - 8*x^2 - 4*(x^3 - 8*x^2 + 16*x)*log(2) + e^(2*x)*log(2) + 16*x) - 2*log(x - 4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (19) = 38\).

Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.12 \[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=- 2 \log {\left (x - 4 \right )} + \log {\left (\frac {- 4 x^{3} \log {\left (2 \right )} + x^{3} - 8 x^{2} + 32 x^{2} \log {\left (2 \right )} - 64 x \log {\left (2 \right )} + 16 x}{\log {\left (2 \right )}} + e^{2 x} \right )} \]

[In]

integrate((4*(2*x-10)*ln(2)*exp(x)**2+4*(-4*x**3+48*x**2-192*x+256)*ln(2)+4*x**3-48*x**2+192*x-256)/(4*(x-4)*l
n(2)*exp(x)**2+4*(-4*x**4+48*x**3-192*x**2+256*x)*ln(2)+4*x**4-48*x**3+192*x**2-256*x),x)

[Out]

-2*log(x - 4) + log((-4*x**3*log(2) + x**3 - 8*x**2 + 32*x**2*log(2) - 64*x*log(2) + 16*x)/log(2) + exp(2*x))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (21) = 42\).

Time = 0.31 (sec) , antiderivative size = 53, normalized size of antiderivative = 2.21 \[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=-2 \, \log \left (x - 4\right ) + \log \left (-\frac {x^{3} {\left (4 \, \log \left (2\right ) - 1\right )} - 8 \, x^{2} {\left (4 \, \log \left (2\right ) - 1\right )} + 16 \, x {\left (4 \, \log \left (2\right ) - 1\right )} - e^{\left (2 \, x\right )} \log \left (2\right )}{\log \left (2\right )}\right ) \]

[In]

integrate((4*(2*x-10)*log(2)*exp(x)^2+4*(-4*x^3+48*x^2-192*x+256)*log(2)+4*x^3-48*x^2+192*x-256)/(4*(x-4)*log(
2)*exp(x)^2+4*(-4*x^4+48*x^3-192*x^2+256*x)*log(2)+4*x^4-48*x^3+192*x^2-256*x),x, algorithm="maxima")

[Out]

-2*log(x - 4) + log(-(x^3*(4*log(2) - 1) - 8*x^2*(4*log(2) - 1) + 16*x*(4*log(2) - 1) - e^(2*x)*log(2))/log(2)
)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (21) = 42\).

Time = 0.29 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.92 \[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=\log \left (-4 \, x^{3} \log \left (2\right ) + x^{3} + 32 \, x^{2} \log \left (2\right ) - 8 \, x^{2} - 64 \, x \log \left (2\right ) + e^{\left (2 \, x\right )} \log \left (2\right ) + 16 \, x\right ) - 2 \, \log \left (x - 4\right ) \]

[In]

integrate((4*(2*x-10)*log(2)*exp(x)^2+4*(-4*x^3+48*x^2-192*x+256)*log(2)+4*x^3-48*x^2+192*x-256)/(4*(x-4)*log(
2)*exp(x)^2+4*(-4*x^4+48*x^3-192*x^2+256*x)*log(2)+4*x^4-48*x^3+192*x^2-256*x),x, algorithm="giac")

[Out]

log(-4*x^3*log(2) + x^3 + 32*x^2*log(2) - 8*x^2 - 64*x*log(2) + e^(2*x)*log(2) + 16*x) - 2*log(x - 4)

Mupad [F(-1)]

Timed out. \[ \int \frac {-256+192 x-48 x^2+4 x^3+e^{2 x} (-10+2 x) \log (16)+\left (256-192 x+48 x^2-4 x^3\right ) \log (16)}{-256 x+192 x^2-48 x^3+4 x^4+e^{2 x} (-4+x) \log (16)+\left (256 x-192 x^2+48 x^3-4 x^4\right ) \log (16)} \, dx=\int \frac {192\,x-4\,\ln \left (2\right )\,\left (4\,x^3-48\,x^2+192\,x-256\right )-48\,x^2+4\,x^3+4\,{\mathrm {e}}^{2\,x}\,\ln \left (2\right )\,\left (2\,x-10\right )-256}{4\,\ln \left (2\right )\,\left (-4\,x^4+48\,x^3-192\,x^2+256\,x\right )-256\,x+192\,x^2-48\,x^3+4\,x^4+4\,{\mathrm {e}}^{2\,x}\,\ln \left (2\right )\,\left (x-4\right )} \,d x \]

[In]

int((192*x - 4*log(2)*(192*x - 48*x^2 + 4*x^3 - 256) - 48*x^2 + 4*x^3 + 4*exp(2*x)*log(2)*(2*x - 10) - 256)/(4
*log(2)*(256*x - 192*x^2 + 48*x^3 - 4*x^4) - 256*x + 192*x^2 - 48*x^3 + 4*x^4 + 4*exp(2*x)*log(2)*(x - 4)),x)

[Out]

int((192*x - 4*log(2)*(192*x - 48*x^2 + 4*x^3 - 256) - 48*x^2 + 4*x^3 + 4*exp(2*x)*log(2)*(2*x - 10) - 256)/(4
*log(2)*(256*x - 192*x^2 + 48*x^3 - 4*x^4) - 256*x + 192*x^2 - 48*x^3 + 4*x^4 + 4*exp(2*x)*log(2)*(x - 4)), x)