3.60.6 \(\int \frac {164-36 e^4+32 x+e^x (48+4 x-2 x^2+e^4 (-12+2 x))+(16-4 e^4+4 x) \log (4-e^4+x)}{-256+e^{2 x} (-4+e^4-x)-192 x-48 x^2-4 x^3+e^4 (64+32 x+4 x^2)+e^x (-64-32 x-4 x^2+e^4 (16+4 x))+(-128+e^x (-16+4 e^4-4 x)-64 x-8 x^2+e^4 (32+8 x)) \log (4-e^4+x)+(-16+4 e^4-4 x) \log ^2(4-e^4+x)} \, dx\) [5906]

Optimal. Leaf size=27 \[ \frac {5-x}{4+\frac {e^x}{2}+x+\log \left (4-e^4+x\right )} \]

[Out]

(5-x)/(1/2*exp(x)+ln(4-exp(4)+x)+4+x)

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Rubi [F]
time = 4.57, 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 {164-36 e^4+32 x+e^x \left (48+4 x-2 x^2+e^4 (-12+2 x)\right )+\left (16-4 e^4+4 x\right ) \log \left (4-e^4+x\right )}{-256+e^{2 x} \left (-4+e^4-x\right )-192 x-48 x^2-4 x^3+e^4 \left (64+32 x+4 x^2\right )+e^x \left (-64-32 x-4 x^2+e^4 (16+4 x)\right )+\left (-128+e^x \left (-16+4 e^4-4 x\right )-64 x-8 x^2+e^4 (32+8 x)\right ) \log \left (4-e^4+x\right )+\left (-16+4 e^4-4 x\right ) \log ^2\left (4-e^4+x\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(164 - 36*E^4 + 32*x + E^x*(48 + 4*x - 2*x^2 + E^4*(-12 + 2*x)) + (16 - 4*E^4 + 4*x)*Log[4 - E^4 + x])/(-2
56 + E^(2*x)*(-4 + E^4 - x) - 192*x - 48*x^2 - 4*x^3 + E^4*(64 + 32*x + 4*x^2) + E^x*(-64 - 32*x - 4*x^2 + E^4
*(16 + 4*x)) + (-128 + E^x*(-16 + 4*E^4 - 4*x) - 64*x - 8*x^2 + E^4*(32 + 8*x))*Log[4 - E^4 + x] + (-16 + 4*E^
4 - 4*x)*Log[4 - E^4 + x]^2),x]

[Out]

-4*(11 - 3*E^4)*Defer[Int][(8 + E^x + 2*x + 2*Log[4 - E^4 + x])^(-2), x] - 4*(4 - E^4)*(9 - E^4)*Defer[Int][(8
+ E^x + 2*x + 2*Log[4 - E^4 + x])^(-2), x] + 4*(7 - E^4)*(9 - E^4)*Defer[Int][(8 + E^x + 2*x + 2*Log[4 - E^4
+ x])^(-2), x] - 4*(11 - 3*E^4)*(9 - E^4)*Defer[Int][1/((-4 + E^4 - x)*(8 + E^x + 2*x + 2*Log[4 - E^4 + x])^2)
, x] - 4*(4 - E^4)^2*(9 - E^4)*Defer[Int][1/((-4 + E^4 - x)*(8 + E^x + 2*x + 2*Log[4 - E^4 + x])^2), x] + 4*(4
- E^4)*(7 - E^4)*(9 - E^4)*Defer[Int][1/((-4 + E^4 - x)*(8 + E^x + 2*x + 2*Log[4 - E^4 + x])^2), x] - 4*(7 -
E^4)*Defer[Int][x/(8 + E^x + 2*x + 2*Log[4 - E^4 + x])^2, x] + 4*(9 - E^4)*Defer[Int][x/(8 + E^x + 2*x + 2*Log
[4 - E^4 + x])^2, x] - 4*Defer[Int][x^2/(8 + E^x + 2*x + 2*Log[4 - E^4 + x])^2, x] - 4*(4 - E^4)*Defer[Int][Lo
g[4 - E^4 + x]/(8 + E^x + 2*x + 2*Log[4 - E^4 + x])^2, x] + 4*(9 - E^4)*Defer[Int][Log[4 - E^4 + x]/(8 + E^x +
2*x + 2*Log[4 - E^4 + x])^2, x] - 4*Defer[Int][(x*Log[4 - E^4 + x])/(8 + E^x + 2*x + 2*Log[4 - E^4 + x])^2, x
] - 12*Defer[Int][(8 + E^x + 2*x + 2*Log[4 - E^4 + x])^(-1), x] + 2*Defer[Int][x/(8 + E^x + 2*x + 2*Log[4 - E^
4 + x]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-82 \left (1-\frac {9 e^4}{41}\right )-e^{4+x} (-6+x)-16 x-e^x \left (24+2 x-x^2\right )-\left (8-2 e^4+2 x\right ) \log \left (4-e^4+x\right )\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ &=2 \int \frac {-82 \left (1-\frac {9 e^4}{41}\right )-e^{4+x} (-6+x)-16 x-e^x \left (24+2 x-x^2\right )-\left (8-2 e^4+2 x\right ) \log \left (4-e^4+x\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ &=2 \int \left (\frac {-6+x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )}+\frac {2 (5-x) \left (11 \left (1-\frac {3 e^4}{11}\right )+7 \left (1-\frac {e^4}{7}\right ) x+x^2+4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )+x \log \left (4-e^4+x\right )\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}\right ) \, dx\\ &=2 \int \frac {-6+x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+4 \int \frac {(5-x) \left (11 \left (1-\frac {3 e^4}{11}\right )+7 \left (1-\frac {e^4}{7}\right ) x+x^2+4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )+x \log \left (4-e^4+x\right )\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ &=2 \int \left (-\frac {6}{8+e^x+2 x+2 \log \left (4-e^4+x\right )}+\frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )}\right ) \, dx+4 \int \frac {(5-x) \left (11+7 x+x^2-e^4 (3+x)+\left (4-e^4+x\right ) \log \left (4-e^4+x\right )\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ &=2 \int \frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+4 \int \left (\frac {-11 \left (1-\frac {3 e^4}{11}\right )-7 \left (1-\frac {e^4}{7}\right ) x-x^2-4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )-x \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}+\frac {\left (9-e^4\right ) \left (11 \left (1-\frac {3 e^4}{11}\right )+7 \left (1-\frac {e^4}{7}\right ) x+x^2+4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )+x \log \left (4-e^4+x\right )\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}\right ) \, dx-12 \int \frac {1}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx\\ &=2 \int \frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+4 \int \frac {-11 \left (1-\frac {3 e^4}{11}\right )-7 \left (1-\frac {e^4}{7}\right ) x-x^2-4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )-x \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-12 \int \frac {1}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+\left (4 \left (9-e^4\right )\right ) \int \frac {11 \left (1-\frac {3 e^4}{11}\right )+7 \left (1-\frac {e^4}{7}\right ) x+x^2+4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )+x \log \left (4-e^4+x\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ &=2 \int \frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+4 \int \frac {-11-7 x-x^2+e^4 (3+x)+\left (-4+e^4-x\right ) \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-12 \int \frac {1}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+\left (4 \left (9-e^4\right )\right ) \int \frac {11+7 x+x^2-e^4 (3+x)-\left (-4+e^4-x\right ) \log \left (4-e^4+x\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ &=2 \int \frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+4 \int \left (-\frac {11 \left (1-\frac {3 e^4}{11}\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {7 \left (1-\frac {e^4}{7}\right ) x}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {x^2}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {x \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}\right ) \, dx-12 \int \frac {1}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+\left (4 \left (9-e^4\right )\right ) \int \left (-\frac {11 \left (1-\frac {3 e^4}{11}\right )}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {7 \left (1-\frac {e^4}{7}\right ) x}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {x^2}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {x \log \left (4-e^4+x\right )}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}\right ) \, dx\\ &=2 \int \frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx-4 \int \frac {x^2}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-4 \int \frac {x \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-12 \int \frac {1}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx-\left (4 \left (11-3 e^4\right )\right ) \int \frac {1}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (4-e^4\right )\right ) \int \frac {\log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (7-e^4\right )\right ) \int \frac {x}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (9-e^4\right )\right ) \int \frac {x^2}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (9-e^4\right )\right ) \int \frac {x \log \left (4-e^4+x\right )}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (11-3 e^4\right ) \left (9-e^4\right )\right ) \int \frac {1}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (4-e^4\right ) \left (9-e^4\right )\right ) \int \frac {\log \left (4-e^4+x\right )}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (7-e^4\right ) \left (9-e^4\right )\right ) \int \frac {x}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ &=2 \int \frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx-4 \int \frac {x^2}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-4 \int \frac {x \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-12 \int \frac {1}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx-\left (4 \left (11-3 e^4\right )\right ) \int \frac {1}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (4-e^4\right )\right ) \int \frac {\log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (7-e^4\right )\right ) \int \frac {x}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (9-e^4\right )\right ) \int \left (\frac {4 \left (1-\frac {e^4}{4}\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}+\frac {\left (-4+e^4\right )^2}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {x}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}\right ) \, dx-\left (4 \left (9-e^4\right )\right ) \int \left (-\frac {\log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}+\frac {\left (-4+e^4\right ) \log \left (4-e^4+x\right )}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}\right ) \, dx-\left (4 \left (11-3 e^4\right ) \left (9-e^4\right )\right ) \int \frac {1}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (4-e^4\right ) \left (9-e^4\right )\right ) \int \frac {\log \left (4-e^4+x\right )}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (7-e^4\right ) \left (9-e^4\right )\right ) \int \left (-\frac {1}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}+\frac {-4+e^4}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}\right ) \, dx\\ &=2 \int \frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx-4 \int \frac {x^2}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-4 \int \frac {x \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-12 \int \frac {1}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx-\left (4 \left (11-3 e^4\right )\right ) \int \frac {1}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (4-e^4\right )\right ) \int \frac {\log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (7-e^4\right )\right ) \int \frac {x}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx+\left (4 \left (9-e^4\right )\right ) \int \frac {x}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx+\left (4 \left (9-e^4\right )\right ) \int \frac {\log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (11-3 e^4\right ) \left (9-e^4\right )\right ) \int \frac {1}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (4-e^4\right ) \left (9-e^4\right )\right ) \int \frac {1}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (4-e^4\right )^2 \left (9-e^4\right )\right ) \int \frac {1}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx+\left (4 \left (7-e^4\right ) \left (9-e^4\right )\right ) \int \frac {1}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx+\left (4 \left (4-e^4\right ) \left (7-e^4\right ) \left (9-e^4\right )\right ) \int \frac {1}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.05, size = 28, normalized size = 1.04 \begin {gather*} \frac {2 (5-x)}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(164 - 36*E^4 + 32*x + E^x*(48 + 4*x - 2*x^2 + E^4*(-12 + 2*x)) + (16 - 4*E^4 + 4*x)*Log[4 - E^4 + x
])/(-256 + E^(2*x)*(-4 + E^4 - x) - 192*x - 48*x^2 - 4*x^3 + E^4*(64 + 32*x + 4*x^2) + E^x*(-64 - 32*x - 4*x^2
+ E^4*(16 + 4*x)) + (-128 + E^x*(-16 + 4*E^4 - 4*x) - 64*x - 8*x^2 + E^4*(32 + 8*x))*Log[4 - E^4 + x] + (-16
+ 4*E^4 - 4*x)*Log[4 - E^4 + x]^2),x]

[Out]

(2*(5 - x))/(8 + E^x + 2*x + 2*Log[4 - E^4 + x])

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Maple [A]
time = 0.21, size = 25, normalized size = 0.93

method	result	size
risch	\(-\frac {2 \left (x -5\right )}{{\mathrm e}^{x}+2 \ln \left (4-{\mathrm e}^{4}+x \right )+2 x +8}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*exp(4)+4*x+16)*ln(4-exp(4)+x)+((2*x-12)*exp(4)-2*x^2+4*x+48)*exp(x)-36*exp(4)+32*x+164)/((4*exp(4)-16
-4*x)*ln(4-exp(4)+x)^2+((4*exp(4)-16-4*x)*exp(x)+(8*x+32)*exp(4)-8*x^2-64*x-128)*ln(4-exp(4)+x)+(exp(4)-x-4)*e
xp(x)^2+((4*x+16)*exp(4)-4*x^2-32*x-64)*exp(x)+(4*x^2+32*x+64)*exp(4)-4*x^3-48*x^2-192*x-256),x,method=_RETURN
VERBOSE)

[Out]

-2*(x-5)/(exp(x)+2*ln(4-exp(4)+x)+2*x+8)

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Maxima [A]
time = 0.49, size = 24, normalized size = 0.89 \begin {gather*} -\frac {2 \, {\left (x - 5\right )}}{2 \, x + e^{x} + 2 \, \log \left (x - e^{4} + 4\right ) + 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(4)+4*x+16)*log(4-exp(4)+x)+((2*x-12)*exp(4)-2*x^2+4*x+48)*exp(x)-36*exp(4)+32*x+164)/((4*ex
p(4)-16-4*x)*log(4-exp(4)+x)^2+((4*exp(4)-16-4*x)*exp(x)+(8*x+32)*exp(4)-8*x^2-64*x-128)*log(4-exp(4)+x)+(exp(
4)-x-4)*exp(x)^2+((4*x+16)*exp(4)-4*x^2-32*x-64)*exp(x)+(4*x^2+32*x+64)*exp(4)-4*x^3-48*x^2-192*x-256),x, algo
rithm="maxima")

[Out]

-2*(x - 5)/(2*x + e^x + 2*log(x - e^4 + 4) + 8)

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Fricas [A]
time = 0.35, size = 24, normalized size = 0.89 \begin {gather*} -\frac {2 \, {\left (x - 5\right )}}{2 \, x + e^{x} + 2 \, \log \left (x - e^{4} + 4\right ) + 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(4)+4*x+16)*log(4-exp(4)+x)+((2*x-12)*exp(4)-2*x^2+4*x+48)*exp(x)-36*exp(4)+32*x+164)/((4*ex
p(4)-16-4*x)*log(4-exp(4)+x)^2+((4*exp(4)-16-4*x)*exp(x)+(8*x+32)*exp(4)-8*x^2-64*x-128)*log(4-exp(4)+x)+(exp(
4)-x-4)*exp(x)^2+((4*x+16)*exp(4)-4*x^2-32*x-64)*exp(x)+(4*x^2+32*x+64)*exp(4)-4*x^3-48*x^2-192*x-256),x, algo
rithm="fricas")

[Out]

-2*(x - 5)/(2*x + e^x + 2*log(x - e^4 + 4) + 8)

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Sympy [A]
time = 0.14, size = 22, normalized size = 0.81 \begin {gather*} \frac {10 - 2 x}{2 x + e^{x} + 2 \log {\left (x - e^{4} + 4 \right )} + 8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(4)+4*x+16)*ln(4-exp(4)+x)+((2*x-12)*exp(4)-2*x**2+4*x+48)*exp(x)-36*exp(4)+32*x+164)/((4*ex
p(4)-16-4*x)*ln(4-exp(4)+x)**2+((4*exp(4)-16-4*x)*exp(x)+(8*x+32)*exp(4)-8*x**2-64*x-128)*ln(4-exp(4)+x)+(exp(
4)-x-4)*exp(x)**2+((4*x+16)*exp(4)-4*x**2-32*x-64)*exp(x)+(4*x**2+32*x+64)*exp(4)-4*x**3-48*x**2-192*x-256),x)

[Out]

(10 - 2*x)/(2*x + exp(x) + 2*log(x - exp(4) + 4) + 8)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 8653 vs. \(2 (24) = 48\).
time = 0.53, size = 8653, normalized size = 320.48 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*exp(4)+4*x+16)*log(4-exp(4)+x)+((2*x-12)*exp(4)-2*x^2+4*x+48)*exp(x)-36*exp(4)+32*x+164)/((4*ex
p(4)-16-4*x)*log(4-exp(4)+x)^2+((4*exp(4)-16-4*x)*exp(x)+(8*x+32)*exp(4)-8*x^2-64*x-128)*log(4-exp(4)+x)+(exp(
4)-x-4)*exp(x)^2+((4*x+16)*exp(4)-4*x^2-32*x-64)*exp(x)+(4*x^2+32*x+64)*exp(4)-4*x^3-48*x^2-192*x-256),x, algo
rithm="giac")

[Out]

-2*(8*(x - e^4 + 4)^8*e^4 + 2*(x - e^4 + 4)^8*e^(2*x + 4) + 8*(x - e^4 + 4)^8*e^(x + 4) + 24*(x - e^4 + 4)^7*e
^4*log(x - e^4 + 4) + 6*(x - e^4 + 4)^7*e^(2*x + 4)*log(x - e^4 + 4) + 24*(x - e^4 + 4)^7*e^(x + 4)*log(x - e^
4 + 4) + 24*(x - e^4 + 4)^6*e^4*log(x - e^4 + 4)^2 + 6*(x - e^4 + 4)^6*e^(2*x + 4)*log(x - e^4 + 4)^2 + 24*(x
- e^4 + 4)^6*e^(x + 4)*log(x - e^4 + 4)^2 + 8*(x - e^4 + 4)^5*e^4*log(x - e^4 + 4)^3 + 2*(x - e^4 + 4)^5*e^(2*
x + 4)*log(x - e^4 + 4)^3 + 8*(x - e^4 + 4)^5*e^(x + 4)*log(x - e^4 + 4)^3 + 32*(x - e^4 + 4)^7*e^8 - 72*(x -
e^4 + 4)^7*e^4 + (x - e^4 + 4)^7*e^(3*x + 4) + 8*(x - e^4 + 4)^7*e^(2*x + 8) - 18*(x - e^4 + 4)^7*e^(2*x + 4)
+ 32*(x - e^4 + 4)^7*e^(x + 8) - 76*(x - e^4 + 4)^7*e^(x + 4) + 72*(x - e^4 + 4)^6*e^8*log(x - e^4 + 4) - 200*
(x - e^4 + 4)^6*e^4*log(x - e^4 + 4) + 2*(x - e^4 + 4)^6*e^(3*x + 4)*log(x - e^4 + 4) + 18*(x - e^4 + 4)^6*e^(
2*x + 8)*log(x - e^4 + 4) - 54*(x - e^4 + 4)^6*e^(2*x + 4)*log(x - e^4 + 4) + 72*(x - e^4 + 4)^6*e^(x + 8)*log
(x - e^4 + 4) - 216*(x - e^4 + 4)^6*e^(x + 4)*log(x - e^4 + 4) + 48*(x - e^4 + 4)^5*e^8*log(x - e^4 + 4)^2 - 1
84*(x - e^4 + 4)^5*e^4*log(x - e^4 + 4)^2 + (x - e^4 + 4)^5*e^(3*x + 4)*log(x - e^4 + 4)^2 + 12*(x - e^4 + 4)^
5*e^(2*x + 8)*log(x - e^4 + 4)^2 - 54*(x - e^4 + 4)^5*e^(2*x + 4)*log(x - e^4 + 4)^2 + 48*(x - e^4 + 4)^5*e^(x
+ 8)*log(x - e^4 + 4)^2 - 204*(x - e^4 + 4)^5*e^(x + 4)*log(x - e^4 + 4)^2 + 8*(x - e^4 + 4)^4*e^8*log(x - e^
4 + 4)^3 - 56*(x - e^4 + 4)^4*e^4*log(x - e^4 + 4)^3 + 2*(x - e^4 + 4)^4*e^(2*x + 8)*log(x - e^4 + 4)^3 - 18*(
x - e^4 + 4)^4*e^(2*x + 4)*log(x - e^4 + 4)^3 + 8*(x - e^4 + 4)^4*e^(x + 8)*log(x - e^4 + 4)^3 - 64*(x - e^4 +
4)^4*e^(x + 4)*log(x - e^4 + 4)^3 + 48*(x - e^4 + 4)^6*e^12 - 200*(x - e^4 + 4)^6*e^8 - 32*(x - e^4 + 4)^6*e^
4 + 3*(x - e^4 + 4)^6*e^(3*x + 8) - 11*(x - e^4 + 4)^6*e^(3*x + 4) + 12*(x - e^4 + 4)^6*e^(2*x + 12) - 54*(x -
e^4 + 4)^6*e^(2*x + 8) - 6*(x - e^4 + 4)^6*e^(2*x + 4) + 48*(x - e^4 + 4)^6*e^(x + 12) - 220*(x - e^4 + 4)^6*
e^(x + 8) + 12*(x - e^4 + 4)^6*e^(x + 4) + 72*(x - e^4 + 4)^5*e^12*log(x - e^4 + 4) - 352*(x - e^4 + 4)^5*e^8*
log(x - e^4 + 4) - 208*(x - e^4 + 4)^5*e^4*log(x - e^4 + 4) + 4*(x - e^4 + 4)^5*e^(3*x + 8)*log(x - e^4 + 4) -
20*(x - e^4 + 4)^5*e^(3*x + 4)*log(x - e^4 + 4) + 18*(x - e^4 + 4)^5*e^(2*x + 12)*log(x - e^4 + 4) - 108*(x -
e^4 + 4)^5*e^(2*x + 8)*log(x - e^4 + 4) - 6*(x - e^4 + 4)^5*e^(2*x + 4)*log(x - e^4 + 4) + 72*(x - e^4 + 4)^5
*e^(x + 12)*log(x - e^4 + 4) - 408*(x - e^4 + 4)^5*e^(x + 8)*log(x - e^4 + 4) - 48*(x - e^4 + 4)^5*e^(x + 4)*l
og(x - e^4 + 4) + 24*(x - e^4 + 4)^4*e^12*log(x - e^4 + 4)^2 - 136*(x - e^4 + 4)^4*e^8*log(x - e^4 + 4)^2 - 31
2*(x - e^4 + 4)^4*e^4*log(x - e^4 + 4)^2 + (x - e^4 + 4)^4*e^(3*x + 8)*log(x - e^4 + 4)^2 - 9*(x - e^4 + 4)^4*
e^(3*x + 4)*log(x - e^4 + 4)^2 + 6*(x - e^4 + 4)^4*e^(2*x + 12)*log(x - e^4 + 4)^2 - 54*(x - e^4 + 4)^4*e^(2*x
+ 8)*log(x - e^4 + 4)^2 + 24*(x - e^4 + 4)^4*e^(x + 12)*log(x - e^4 + 4)^2 - 180*(x - e^4 + 4)^4*e^(x + 8)*lo
g(x - e^4 + 4)^2 - 132*(x - e^4 + 4)^4*e^(x + 4)*log(x - e^4 + 4)^2 + 16*(x - e^4 + 4)^3*e^8*log(x - e^4 + 4)^
3 - 136*(x - e^4 + 4)^3*e^4*log(x - e^4 + 4)^3 + 8*(x - e^4 + 4)^3*e^(x + 8)*log(x - e^4 + 4)^3 - 72*(x - e^4
+ 4)^3*e^(x + 4)*log(x - e^4 + 4)^3 + 32*(x - e^4 + 4)^5*e^16 - 168*(x - e^4 + 4)^5*e^12 - 240*(x - e^4 + 4)^5
*e^8 + 272*(x - e^4 + 4)^5*e^4 + 3*(x - e^4 + 4)^5*e^(3*x + 12) - 22*(x - e^4 + 4)^5*e^(3*x + 8) + 17*(x - e^4
+ 4)^5*e^(3*x + 4) + 8*(x - e^4 + 4)^5*e^(2*x + 16) - 54*(x - e^4 + 4)^5*e^(2*x + 12) - 12*(x - e^4 + 4)^5*e^
(2*x + 8) + 46*(x - e^4 + 4)^5*e^(2*x + 4) + 32*(x - e^4 + 4)^5*e^(x + 16) - 204*(x - e^4 + 4)^5*e^(x + 12) -
72*(x - e^4 + 4)^5*e^(x + 8) + 208*(x - e^4 + 4)^5*e^(x + 4) + 24*(x - e^4 + 4)^4*e^16*log(x - e^4 + 4) - 104*
(x - e^4 + 4)^4*e^12*log(x - e^4 + 4) - 688*(x - e^4 + 4)^4*e^8*log(x - e^4 + 4) + 512*(x - e^4 + 4)^4*e^4*log
(x - e^4 + 4) + 2*(x - e^4 + 4)^4*e^(3*x + 12)*log(x - e^4 + 4) - 20*(x - e^4 + 4)^4*e^(3*x + 8)*log(x - e^4 +
4) + 16*(x - e^4 + 4)^4*e^(3*x + 4)*log(x - e^4 + 4) + 6*(x - e^4 + 4)^4*e^(2*x + 16)*log(x - e^4 + 4) - 54*(
x - e^4 + 4)^4*e^(2*x + 12)*log(x - e^4 + 4) - 6*(x - e^4 + 4)^4*e^(2*x + 8)*log(x - e^4 + 4) + 42*(x - e^4 +
4)^4*e^(2*x + 4)*log(x - e^4 + 4) + 24*(x - e^4 + 4)^4*e^(x + 16)*log(x - e^4 + 4) - 168*(x - e^4 + 4)^4*e^(x
+ 12)*log(x - e^4 + 4) - 312*(x - e^4 + 4)^4*e^(x + 8)*log(x - e^4 + 4) + 408*(x - e^4 + 4)^4*e^(x + 4)*log(x
- e^4 + 4) + 48*(x - e^4 + 4)^3*e^12*log(x - e^4 + 4)^2 - 432*(x - e^4 + 4)^3*e^8*log(x - e^4 + 4)^2 + 168*(x
- e^4 + 4)^3*e^4*log(x - e^4 + 4)^2 + 24*(x - e^4 + 4)^3*e^(x + 12)*log(x - e^4 + 4)^2 - 240*(x - e^4 + 4)^3*e
^(x + 8)*log(x - e^4 + 4)^2 + 204*(x - e^4 + 4)^3*e^(x + 4)*log(x - e^4 + 4)^2 + 8*(x - e^4 + 4)^2*e^8*log(x -
e^4 + 4)^3 - 72*(x - e^4 + 4)^2*e^4*log(x - e^4 + 4)^3 + 8*(x - e^4 + 4)^4*e^20 - 24*(x - e^4 + 4)^4*e^16 - 3
76*(x - e^4 + 4)^4*e^12 + 496*(x - e^4 + 4)^4*e^8 + 176*(x - e^4 + 4)^4*e^4 + (x - e^4 + 4)^4*e^(3*x + 16) - 1
1*(x - e^4 + 4)^4*e^(3*x + 12) + 15*(x - e^4 + ...

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {32\,x-36\,{\mathrm {e}}^4+\ln \left (x-{\mathrm {e}}^4+4\right )\,\left (4\,x-4\,{\mathrm {e}}^4+16\right )+{\mathrm {e}}^x\,\left (4\,x-2\,x^2+{\mathrm {e}}^4\,\left (2\,x-12\right )+48\right )+164}{192\,x+\ln \left (x-{\mathrm {e}}^4+4\right )\,\left (64\,x+{\mathrm {e}}^x\,\left (4\,x-4\,{\mathrm {e}}^4+16\right )+8\,x^2-{\mathrm {e}}^4\,\left (8\,x+32\right )+128\right )-{\mathrm {e}}^4\,\left (4\,x^2+32\,x+64\right )+{\mathrm {e}}^{2\,x}\,\left (x-{\mathrm {e}}^4+4\right )+{\mathrm {e}}^x\,\left (32\,x+4\,x^2-{\mathrm {e}}^4\,\left (4\,x+16\right )+64\right )+{\ln \left (x-{\mathrm {e}}^4+4\right )}^2\,\left (4\,x-4\,{\mathrm {e}}^4+16\right )+48\,x^2+4\,x^3+256} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(32*x - 36*exp(4) + log(x - exp(4) + 4)*(4*x - 4*exp(4) + 16) + exp(x)*(4*x - 2*x^2 + exp(4)*(2*x - 12) +
48) + 164)/(192*x + log(x - exp(4) + 4)*(64*x + exp(x)*(4*x - 4*exp(4) + 16) + 8*x^2 - exp(4)*(8*x + 32) + 12
8) - exp(4)*(32*x + 4*x^2 + 64) + exp(2*x)*(x - exp(4) + 4) + exp(x)*(32*x + 4*x^2 - exp(4)*(4*x + 16) + 64) +
log(x - exp(4) + 4)^2*(4*x - 4*exp(4) + 16) + 48*x^2 + 4*x^3 + 256),x)

[Out]

int(-(32*x - 36*exp(4) + log(x - exp(4) + 4)*(4*x - 4*exp(4) + 16) + exp(x)*(4*x - 2*x^2 + exp(4)*(2*x - 12) +
48) + 164)/(192*x + log(x - exp(4) + 4)*(64*x + exp(x)*(4*x - 4*exp(4) + 16) + 8*x^2 - exp(4)*(8*x + 32) + 12
8) - exp(4)*(32*x + 4*x^2 + 64) + exp(2*x)*(x - exp(4) + 4) + exp(x)*(32*x + 4*x^2 - exp(4)*(4*x + 16) + 64) +
log(x - exp(4) + 4)^2*(4*x - 4*exp(4) + 16) + 48*x^2 + 4*x^3 + 256), x)

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