3.8.99 \(\int \frac {16 x-4 e^x x+e^{2 x} x+(-112-128 x-16 x^2+e^{2 x} (-6-7 x-x^2)+e^x (52+60 x+8 x^2)) \log (\frac {-28-4 x+e^x (6+x)}{-4+e^x})}{(112 x^3+16 x^4+e^x (-52 x^3-8 x^4)+e^{2 x} (6 x^3+x^4)) \log (\frac {-28-4 x+e^x (6+x)}{-4+e^x})+(224 x^2+32 x^3+e^x (-104 x^2-16 x^3)+e^{2 x} (12 x^2+2 x^3)) \log (\frac {-28-4 x+e^x (6+x)}{-4+e^x}) \log (\frac {x}{\log (\frac {-28-4 x+e^x (6+x)}{-4+e^x})})+(112 x+16 x^2+e^x (-52 x-8 x^2)+e^{2 x} (6 x+x^2)) \log (\frac {-28-4 x+e^x (6+x)}{-4+e^x}) \log ^2(\frac {x}{\log (\frac {-28-4 x+e^x (6+x)}{-4+e^x})})} \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{x+\log \left (\frac {x}{\log \left (6+\frac {4}{4-e^x}+x\right )}\right )} \]

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Rubi [F]  time = 7.42, 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 {16 x-4 e^x x+e^{2 x} x+\left (-112-128 x-16 x^2+e^{2 x} \left (-6-7 x-x^2\right )+e^x \left (52+60 x+8 x^2\right )\right ) \log \left (\frac {-28-4 x+e^x (6+x)}{-4+e^x}\right )}{\left (112 x^3+16 x^4+e^x \left (-52 x^3-8 x^4\right )+e^{2 x} \left (6 x^3+x^4\right )\right ) \log \left (\frac {-28-4 x+e^x (6+x)}{-4+e^x}\right )+\left (224 x^2+32 x^3+e^x \left (-104 x^2-16 x^3\right )+e^{2 x} \left (12 x^2+2 x^3\right )\right ) \log \left (\frac {-28-4 x+e^x (6+x)}{-4+e^x}\right ) \log \left (\frac {x}{\log \left (\frac {-28-4 x+e^x (6+x)}{-4+e^x}\right )}\right )+\left (112 x+16 x^2+e^x \left (-52 x-8 x^2\right )+e^{2 x} \left (6 x+x^2\right )\right ) \log \left (\frac {-28-4 x+e^x (6+x)}{-4+e^x}\right ) \log ^2\left (\frac {x}{\log \left (\frac {-28-4 x+e^x (6+x)}{-4+e^x}\right )}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(16*x - 4*E^x*x + E^(2*x)*x + (-112 - 128*x - 16*x^2 + E^(2*x)*(-6 - 7*x - x^2) + E^x*(52 + 60*x + 8*x^2))
*Log[(-28 - 4*x + E^x*(6 + x))/(-4 + E^x)])/((112*x^3 + 16*x^4 + E^x*(-52*x^3 - 8*x^4) + E^(2*x)*(6*x^3 + x^4)
)*Log[(-28 - 4*x + E^x*(6 + x))/(-4 + E^x)] + (224*x^2 + 32*x^3 + E^x*(-104*x^2 - 16*x^3) + E^(2*x)*(12*x^2 +
2*x^3))*Log[(-28 - 4*x + E^x*(6 + x))/(-4 + E^x)]*Log[x/Log[(-28 - 4*x + E^x*(6 + x))/(-4 + E^x)]] + (112*x +
16*x^2 + E^x*(-52*x - 8*x^2) + E^(2*x)*(6*x + x^2))*Log[(-28 - 4*x + E^x*(6 + x))/(-4 + E^x)]*Log[x/Log[(-28 -
4*x + E^x*(6 + x))/(-4 + E^x)]]^2),x]

[Out]

-Defer[Int][(x + Log[x/Log[(E^x*(6 + x) - 4*(7 + x))/(-4 + E^x)]])^(-2), x] - Defer[Int][1/(x*(x + Log[x/Log[(
E^x*(6 + x) - 4*(7 + x))/(-4 + E^x)]])^2), x] - 4*Defer[Int][1/((-4 + E^x)*Log[(E^x*(6 + x) - 4*(7 + x))/(-4 +
E^x)]*(x + Log[x/Log[(E^x*(6 + x) - 4*(7 + x))/(-4 + E^x)]])^2), x] + Defer[Int][1/((6 + x)*Log[(E^x*(6 + x)
- 4*(7 + x))/(-4 + E^x)]*(x + Log[x/Log[(E^x*(6 + x) - 4*(7 + x))/(-4 + E^x)]])^2), x] + 28*Defer[Int][1/((-28
+ 6*E^x - 4*x + E^x*x)*Log[(E^x*(6 + x) - 4*(7 + x))/(-4 + E^x)]*(x + Log[x/Log[(E^x*(6 + x) - 4*(7 + x))/(-4
+ E^x)]])^2), x] + 4*Defer[Int][x/((-28 + 6*E^x - 4*x + E^x*x)*Log[(E^x*(6 + x) - 4*(7 + x))/(-4 + E^x)]*(x +
Log[x/Log[(E^x*(6 + x) - 4*(7 + x))/(-4 + E^x)]])^2), x] + 4*Defer[Int][1/((6 + x)*(-28 + 6*E^x - 4*x + E^x*x
)*Log[(E^x*(6 + x) - 4*(7 + x))/(-4 + E^x)]*(x + Log[x/Log[(E^x*(6 + x) - 4*(7 + x))/(-4 + E^x)]])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\frac {1+x}{x}+\frac {16-4 e^x+e^{2 x}}{\left (-4+e^x\right ) \left (e^x (6+x)-4 (7+x)\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}}{\left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx\\ &=\int \left (-\frac {4}{\left (-4+e^x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}+\frac {4 \left (43+13 x+x^2\right )}{(6+x) \left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}+\frac {x-6 \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )-7 x \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )-x^2 \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}{x (6+x) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {1}{\left (-4+e^x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx\right )+4 \int \frac {43+13 x+x^2}{(6+x) \left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+\int \frac {x-6 \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )-7 x \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )-x^2 \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}{x (6+x) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx\\ &=-\left (4 \int \frac {1}{\left (-4+e^x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx\right )+4 \int \left (\frac {7}{\left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}+\frac {x}{\left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}+\frac {1}{(6+x) \left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}\right ) \, dx+\int \frac {x-\left (6+7 x+x^2\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}{x (6+x) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx\\ &=-\left (4 \int \frac {1}{\left (-4+e^x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx\right )+4 \int \frac {x}{\left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+4 \int \frac {1}{(6+x) \left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+28 \int \frac {1}{\left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+\int \left (\frac {x-6 \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )-7 x \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )-x^2 \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}{6 x \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}+\frac {-x+6 \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )+7 x \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )+x^2 \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}{6 (6+x) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}\right ) \, dx\\ &=\frac {1}{6} \int \frac {x-6 \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )-7 x \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )-x^2 \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}{x \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+\frac {1}{6} \int \frac {-x+6 \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )+7 x \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )+x^2 \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}{(6+x) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx-4 \int \frac {1}{\left (-4+e^x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+4 \int \frac {x}{\left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+4 \int \frac {1}{(6+x) \left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+28 \int \frac {1}{\left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx\\ &=\frac {1}{6} \int \frac {x-\left (6+7 x+x^2\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}{x \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+\frac {1}{6} \int \frac {-x+\left (6+7 x+x^2\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}{(6+x) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx-4 \int \frac {1}{\left (-4+e^x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+4 \int \frac {x}{\left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+4 \int \frac {1}{(6+x) \left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+28 \int \frac {1}{\left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx\\ &=\frac {1}{6} \int \left (-\frac {7}{\left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}-\frac {6}{x \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}-\frac {x}{\left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}+\frac {1}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}\right ) \, dx+\frac {1}{6} \int \left (\frac {6}{(6+x) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}+\frac {7 x}{(6+x) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}+\frac {x^2}{(6+x) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}-\frac {x}{(6+x) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2}\right ) \, dx-4 \int \frac {1}{\left (-4+e^x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+4 \int \frac {x}{\left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+4 \int \frac {1}{(6+x) \left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+28 \int \frac {1}{\left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx\\ &=-\left (\frac {1}{6} \int \frac {x}{\left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx\right )+\frac {1}{6} \int \frac {x^2}{(6+x) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+\frac {1}{6} \int \frac {1}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx-\frac {1}{6} \int \frac {x}{(6+x) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx-\frac {7}{6} \int \frac {1}{\left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+\frac {7}{6} \int \frac {x}{(6+x) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx-4 \int \frac {1}{\left (-4+e^x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+4 \int \frac {x}{\left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+4 \int \frac {1}{(6+x) \left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+28 \int \frac {1}{\left (-28+6 e^x-4 x+e^x x\right ) \log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right ) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx-\int \frac {1}{x \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx+\int \frac {1}{(6+x) \left (x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )\right )^2} \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.32, size = 31, normalized size = 1.29 \begin {gather*} \frac {1}{x+\log \left (\frac {x}{\log \left (\frac {e^x (6+x)-4 (7+x)}{-4+e^x}\right )}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(16*x - 4*E^x*x + E^(2*x)*x + (-112 - 128*x - 16*x^2 + E^(2*x)*(-6 - 7*x - x^2) + E^x*(52 + 60*x + 8
*x^2))*Log[(-28 - 4*x + E^x*(6 + x))/(-4 + E^x)])/((112*x^3 + 16*x^4 + E^x*(-52*x^3 - 8*x^4) + E^(2*x)*(6*x^3
+ x^4))*Log[(-28 - 4*x + E^x*(6 + x))/(-4 + E^x)] + (224*x^2 + 32*x^3 + E^x*(-104*x^2 - 16*x^3) + E^(2*x)*(12*
x^2 + 2*x^3))*Log[(-28 - 4*x + E^x*(6 + x))/(-4 + E^x)]*Log[x/Log[(-28 - 4*x + E^x*(6 + x))/(-4 + E^x)]] + (11
2*x + 16*x^2 + E^x*(-52*x - 8*x^2) + E^(2*x)*(6*x + x^2))*Log[(-28 - 4*x + E^x*(6 + x))/(-4 + E^x)]*Log[x/Log[
(-28 - 4*x + E^x*(6 + x))/(-4 + E^x)]]^2),x]

[Out]

(x + Log[x/Log[(E^x*(6 + x) - 4*(7 + x))/(-4 + E^x)]])^(-1)

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fricas [A]  time = 0.99, size = 28, normalized size = 1.17 \begin {gather*} \frac {1}{x + \log \left (\frac {x}{\log \left (\frac {{\left (x + 6\right )} e^{x} - 4 \, x - 28}{e^{x} - 4}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-7*x-6)*exp(x)^2+(8*x^2+60*x+52)*exp(x)-16*x^2-128*x-112)*log(((x+6)*exp(x)-4*x-28)/(exp(x)-4
))+x*exp(x)^2-4*exp(x)*x+16*x)/(((x^2+6*x)*exp(x)^2+(-8*x^2-52*x)*exp(x)+16*x^2+112*x)*log(((x+6)*exp(x)-4*x-2
8)/(exp(x)-4))*log(x/log(((x+6)*exp(x)-4*x-28)/(exp(x)-4)))^2+((2*x^3+12*x^2)*exp(x)^2+(-16*x^3-104*x^2)*exp(x
)+32*x^3+224*x^2)*log(((x+6)*exp(x)-4*x-28)/(exp(x)-4))*log(x/log(((x+6)*exp(x)-4*x-28)/(exp(x)-4)))+((x^4+6*x
^3)*exp(x)^2+(-8*x^4-52*x^3)*exp(x)+16*x^4+112*x^3)*log(((x+6)*exp(x)-4*x-28)/(exp(x)-4))),x, algorithm="frica
s")

[Out]

1/(x + log(x/log(((x + 6)*e^x - 4*x - 28)/(e^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((((-x^2-7*x-6)*exp(x)^2+(8*x^2+60*x+52)*exp(x)-16*x^2-128*x-112)*log(((x+6)*exp(x)-4*x-28)/(exp(x)-4
))+x*exp(x)^2-4*exp(x)*x+16*x)/(((x^2+6*x)*exp(x)^2+(-8*x^2-52*x)*exp(x)+16*x^2+112*x)*log(((x+6)*exp(x)-4*x-2
8)/(exp(x)-4))*log(x/log(((x+6)*exp(x)-4*x-28)/(exp(x)-4)))^2+((2*x^3+12*x^2)*exp(x)^2+(-16*x^3-104*x^2)*exp(x
)+32*x^3+224*x^2)*log(((x+6)*exp(x)-4*x-28)/(exp(x)-4))*log(x/log(((x+6)*exp(x)-4*x-28)/(exp(x)-4)))+((x^4+6*x
^3)*exp(x)^2+(-8*x^4-52*x^3)*exp(x)+16*x^4+112*x^3)*log(((x+6)*exp(x)-4*x-28)/(exp(x)-4))),x, algorithm="giac"
)

[Out]

Timed out

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maple [C]  time = 2.66, size = 2495, normalized size = 103.96

method	result	size
risch	\(\text {Expression too large to display}\)	\(2495\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2-7*x-6)*exp(x)^2+(8*x^2+60*x+52)*exp(x)-16*x^2-128*x-112)*ln(((x+6)*exp(x)-4*x-28)/(exp(x)-4))+x*ex
p(x)^2-4*exp(x)*x+16*x)/(((x^2+6*x)*exp(x)^2+(-8*x^2-52*x)*exp(x)+16*x^2+112*x)*ln(((x+6)*exp(x)-4*x-28)/(exp(
x)-4))*ln(x/ln(((x+6)*exp(x)-4*x-28)/(exp(x)-4)))^2+((2*x^3+12*x^2)*exp(x)^2+(-16*x^3-104*x^2)*exp(x)+32*x^3+2
24*x^2)*ln(((x+6)*exp(x)-4*x-28)/(exp(x)-4))*ln(x/ln(((x+6)*exp(x)-4*x-28)/(exp(x)-4)))+((x^4+6*x^3)*exp(x)^2+
(-8*x^4-52*x^3)*exp(x)+16*x^4+112*x^3)*ln(((x+6)*exp(x)-4*x-28)/(exp(x)-4))),x,method=_RETURNVERBOSE)

[Out]

2*I/(Pi*csgn(I*x)*csgn(I/(-Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp
(x)-28)/(exp(x)-4))+Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2+Pi*csgn(I*(x*(exp(x)
-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2-Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)
-4))^3+2*I*ln(exp(x)-4)-2*I*ln(x*(exp(x)-4)+6*exp(x)-28)))*csgn(I*x/(-Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-
4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))+Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*ex
p(x)-28)/(exp(x)-4))^2+Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2-P
i*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^3+2*I*ln(exp(x)-4)-2*I*ln(x*(exp(x)-4)+6*exp(x)-28)))-Pi*csgn(
I*x)*csgn(I*x/(-Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(ex
p(x)-4))+Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2+Pi*csgn(I*(x*(exp(x)-4)+6*exp(x
)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2-Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^3+2*I*
ln(exp(x)-4)-2*I*ln(x*(exp(x)-4)+6*exp(x)-28)))^2+Pi*csgn(I*x/(-Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*e
xp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))+Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-2
8)/(exp(x)-4))^2+Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2-Pi*csgn
(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^3+2*I*ln(exp(x)-4)-2*I*ln(x*(exp(x)-4)+6*exp(x)-28)))*csgn(x/(-Pi*cs
gn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))+Pi*csgn(I/(e
xp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2+Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp
(x)-4)+6*exp(x)-28)/(exp(x)-4))^2-Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^3+2*I*ln(exp(x)-4)-2*I*ln(x
*(exp(x)-4)+6*exp(x)-28)))+Pi*csgn(x/(-Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp
(x)-4)+6*exp(x)-28)/(exp(x)-4))+Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2+Pi*csgn(
I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2-Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)
-28)/(exp(x)-4))^3+2*I*ln(exp(x)-4)-2*I*ln(x*(exp(x)-4)+6*exp(x)-28)))^2+Pi*csgn(I/(-Pi*csgn(I/(exp(x)-4))*csg
n(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))+Pi*csgn(I/(exp(x)-4))*csgn(I*(x*
(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2+Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/
(exp(x)-4))^2-Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^3+2*I*ln(exp(x)-4)-2*I*ln(x*(exp(x)-4)+6*exp(x)
-28)))*csgn(I*x/(-Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(
exp(x)-4))+Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2+Pi*csgn(I*(x*(exp(x)-4)+6*exp
(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2-Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^3+2*
I*ln(exp(x)-4)-2*I*ln(x*(exp(x)-4)+6*exp(x)-28)))^2-Pi*csgn(I*x/(-Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6
*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))+Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)
-28)/(exp(x)-4))^2+Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2-Pi*cs
gn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^3+2*I*ln(exp(x)-4)-2*I*ln(x*(exp(x)-4)+6*exp(x)-28)))^3+Pi*csgn(I*
x/(-Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))+Pi*
csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2+Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(
I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2-Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^3+2*I*ln(exp(x)-4)
-2*I*ln(x*(exp(x)-4)+6*exp(x)-28)))*csgn(x/(-Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(
x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))+Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2+Pi
*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2-Pi*csgn(I*(x*(exp(x)-4)+6*
exp(x)-28)/(exp(x)-4))^3+2*I*ln(exp(x)-4)-2*I*ln(x*(exp(x)-4)+6*exp(x)-28)))^2+Pi*csgn(x/(-Pi*csgn(I/(exp(x)-4
))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))+Pi*csgn(I/(exp(x)-4))*csgn
(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2+Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x
)-28)/(exp(x)-4))^2-Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^3+2*I*ln(exp(x)-4)-2*I*ln(x*(exp(x)-4)+6*
exp(x)-28)))^3-Pi+2*I*ln(2)+2*I*x+2*I*ln(x)-2*I*ln(Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*cs
gn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))-Pi*csgn(I/(exp(x)-4))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4)
)^2-Pi*csgn(I*(x*(exp(x)-4)+6*exp(x)-28))*csgn(I*(x*(exp(x)-4)+6*exp(x)-28)/(exp(x)-4))^2+Pi*csgn(I*(x*(exp(x)
-4)+6*exp(x)-28)/(exp(x)-4))^3-2*I*ln(exp(x)-4)+2*I*ln(x*(exp(x)-4)+6*exp(x)-28)))

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maxima [A]  time = 16.77, size = 29, normalized size = 1.21 \begin {gather*} \frac {1}{x + \log \relax (x) - \log \left (\log \left ({\left (x + 6\right )} e^{x} - 4 \, x - 28\right ) - \log \left (e^{x} - 4\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-7*x-6)*exp(x)^2+(8*x^2+60*x+52)*exp(x)-16*x^2-128*x-112)*log(((x+6)*exp(x)-4*x-28)/(exp(x)-4
))+x*exp(x)^2-4*exp(x)*x+16*x)/(((x^2+6*x)*exp(x)^2+(-8*x^2-52*x)*exp(x)+16*x^2+112*x)*log(((x+6)*exp(x)-4*x-2
8)/(exp(x)-4))*log(x/log(((x+6)*exp(x)-4*x-28)/(exp(x)-4)))^2+((2*x^3+12*x^2)*exp(x)^2+(-16*x^3-104*x^2)*exp(x
)+32*x^3+224*x^2)*log(((x+6)*exp(x)-4*x-28)/(exp(x)-4))*log(x/log(((x+6)*exp(x)-4*x-28)/(exp(x)-4)))+((x^4+6*x
^3)*exp(x)^2+(-8*x^4-52*x^3)*exp(x)+16*x^4+112*x^3)*log(((x+6)*exp(x)-4*x-28)/(exp(x)-4))),x, algorithm="maxim
a")

[Out]

1/(x + log(x) - log(log((x + 6)*e^x - 4*x - 28) - log(e^x - 4)))

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mupad [B]  time = 2.71, size = 30, normalized size = 1.25 \begin {gather*} \frac {1}{x+\ln \left (\frac {x}{\ln \left (-\frac {4\,x-{\mathrm {e}}^x\,\left (x+6\right )+28}{{\mathrm {e}}^x-4}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x + x*exp(2*x) - 4*x*exp(x) - log(-(4*x - exp(x)*(x + 6) + 28)/(exp(x) - 4))*(128*x + exp(2*x)*(7*x +
x^2 + 6) - exp(x)*(60*x + 8*x^2 + 52) + 16*x^2 + 112))/(log(-(4*x - exp(x)*(x + 6) + 28)/(exp(x) - 4))*(exp(2*
x)*(6*x^3 + x^4) - exp(x)*(52*x^3 + 8*x^4) + 112*x^3 + 16*x^4) + log(x/log(-(4*x - exp(x)*(x + 6) + 28)/(exp(x
) - 4)))^2*log(-(4*x - exp(x)*(x + 6) + 28)/(exp(x) - 4))*(112*x + exp(2*x)*(6*x + x^2) - exp(x)*(52*x + 8*x^2
) + 16*x^2) + log(x/log(-(4*x - exp(x)*(x + 6) + 28)/(exp(x) - 4)))*log(-(4*x - exp(x)*(x + 6) + 28)/(exp(x) -
4))*(exp(2*x)*(12*x^2 + 2*x^3) - exp(x)*(104*x^2 + 16*x^3) + 224*x^2 + 32*x^3)),x)

[Out]

1/(x + log(x/log(-(4*x - exp(x)*(x + 6) + 28)/(exp(x) - 4))))

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sympy [A]  time = 8.02, size = 24, normalized size = 1.00 \begin {gather*} \frac {1}{x + \log {\left (\frac {x}{\log {\left (\frac {- 4 x + \left (x + 6\right ) e^{x} - 28}{e^{x} - 4} \right )}} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2-7*x-6)*exp(x)**2+(8*x**2+60*x+52)*exp(x)-16*x**2-128*x-112)*ln(((x+6)*exp(x)-4*x-28)/(exp(x
)-4))+x*exp(x)**2-4*exp(x)*x+16*x)/(((x**2+6*x)*exp(x)**2+(-8*x**2-52*x)*exp(x)+16*x**2+112*x)*ln(((x+6)*exp(x
)-4*x-28)/(exp(x)-4))*ln(x/ln(((x+6)*exp(x)-4*x-28)/(exp(x)-4)))**2+((2*x**3+12*x**2)*exp(x)**2+(-16*x**3-104*
x**2)*exp(x)+32*x**3+224*x**2)*ln(((x+6)*exp(x)-4*x-28)/(exp(x)-4))*ln(x/ln(((x+6)*exp(x)-4*x-28)/(exp(x)-4)))
+((x**4+6*x**3)*exp(x)**2+(-8*x**4-52*x**3)*exp(x)+16*x**4+112*x**3)*ln(((x+6)*exp(x)-4*x-28)/(exp(x)-4))),x)

[Out]

1/(x + log(x/log((-4*x + (x + 6)*exp(x) - 28)/(exp(x) - 4))))

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