3.99.3 \(\int \frac {-55+220 x-82 x^2+8 x^3+e^{2-x^2} (-50 x+20 x^2-2 x^3)}{(-75-25 x+118 x^2-42 x^3+4 x^4+e^{2-x^2} (25-10 x+x^2)) \log (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x})} \, dx\)

Optimal. Leaf size=33 \[ 2+\log \left (\log \left (-3+e^{2-x^2}-2 x-\frac {x}{5-x}+4 x^2\right )\right ) \]

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Rubi [F]  time = 8.23, 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 {-55+220 x-82 x^2+8 x^3+e^{2-x^2} \left (-50 x+20 x^2-2 x^3\right )}{\left (-75-25 x+118 x^2-42 x^3+4 x^4+e^{2-x^2} \left (25-10 x+x^2\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-55 + 220*x - 82*x^2 + 8*x^3 + E^(2 - x^2)*(-50*x + 20*x^2 - 2*x^3))/((-75 - 25*x + 118*x^2 - 42*x^3 + 4*
x^4 + E^(2 - x^2)*(25 - 10*x + x^2))*Log[(15 + E^(2 - x^2)*(-5 + x) + 8*x - 22*x^2 + 4*x^3)/(-5 + x)]),x]

[Out]

-Defer[Int][1/((-5 + x)*Log[(15 + E^(2 - x^2)*(-5 + x) + 8*x - 22*x^2 + 4*x^3)/(-5 + x)]), x] + 8*Defer[Int][1
/((15 + 8*x - 22*x^2 + 4*x^3)*Log[(15 + E^(2 - x^2)*(-5 + x) + 8*x - 22*x^2 + 4*x^3)/(-5 + x)]), x] - 44*Defer
[Int][x/((15 + 8*x - 22*x^2 + 4*x^3)*Log[(15 + E^(2 - x^2)*(-5 + x) + 8*x - 22*x^2 + 4*x^3)/(-5 + x)]), x] + 1
2*Defer[Int][x^2/((15 + 8*x - 22*x^2 + 4*x^3)*Log[(15 + E^(2 - x^2)*(-5 + x) + 8*x - 22*x^2 + 4*x^3)/(-5 + x)]
), x] - 2*E^2*Defer[Int][1/((E^2*(-5 + x) + E^x^2*(15 + 8*x - 22*x^2 + 4*x^3))*Log[(15 + E^(2 - x^2)*(-5 + x)
+ 8*x - 22*x^2 + 4*x^3)/(-5 + x)]), x] + 10*E^2*Defer[Int][x/((E^2*(-5 + x) + E^x^2*(15 + 8*x - 22*x^2 + 4*x^3
))*Log[(15 + E^(2 - x^2)*(-5 + x) + 8*x - 22*x^2 + 4*x^3)/(-5 + x)]), x] - 2*E^2*Defer[Int][x^2/((E^2*(-5 + x)
+ E^x^2*(15 + 8*x - 22*x^2 + 4*x^3))*Log[(15 + E^(2 - x^2)*(-5 + x) + 8*x - 22*x^2 + 4*x^3)/(-5 + x)]), x] +
85*E^2*Defer[Int][1/((15 + 8*x - 22*x^2 + 4*x^3)*(E^2*(-5 + x) + E^x^2*(15 + 8*x - 22*x^2 + 4*x^3))*Log[(15 +
E^(2 - x^2)*(-5 + x) + 8*x - 22*x^2 + 4*x^3)/(-5 + x)]), x] - 204*E^2*Defer[Int][x/((15 + 8*x - 22*x^2 + 4*x^3
)*(E^2*(-5 + x) + E^x^2*(15 + 8*x - 22*x^2 + 4*x^3))*Log[(15 + E^(2 - x^2)*(-5 + x) + 8*x - 22*x^2 + 4*x^3)/(-
5 + x)]), x] + 38*E^2*Defer[Int][x^2/((15 + 8*x - 22*x^2 + 4*x^3)*(E^2*(-5 + x) + E^x^2*(15 + 8*x - 22*x^2 + 4
*x^3))*Log[(15 + E^(2 - x^2)*(-5 + x) + 8*x - 22*x^2 + 4*x^3)/(-5 + x)]), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-55+220 x-82 x^2+8 x^3}{(-5+x) \left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )}-\frac {e^2 \left (-55+70 x-132 x^2+244 x^3-84 x^4+8 x^5\right )}{\left (15+8 x-22 x^2+4 x^3\right ) \left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )}\right ) \, dx\\ &=-\left (e^2 \int \frac {-55+70 x-132 x^2+244 x^3-84 x^4+8 x^5}{\left (15+8 x-22 x^2+4 x^3\right ) \left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx\right )+\int \frac {-55+220 x-82 x^2+8 x^3}{(-5+x) \left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx\\ &=-\left (e^2 \int \frac {-55+70 x-132 x^2+244 x^3-84 x^4+8 x^5}{\left (15+8 x-22 x^2+4 x^3\right ) \left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx\right )+\int \left (-\frac {1}{(-5+x) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )}+\frac {4 \left (2-11 x+3 x^2\right )}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )}\right ) \, dx\\ &=4 \int \frac {2-11 x+3 x^2}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-e^2 \int \left (\frac {2}{\left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )}-\frac {10 x}{\left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )}+\frac {2 x^2}{\left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )}-\frac {85-204 x+38 x^2}{\left (15+8 x-22 x^2+4 x^3\right ) \left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )}\right ) \, dx-\int \frac {1}{(-5+x) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx\\ &=4 \int \left (\frac {2}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )}-\frac {11 x}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )}+\frac {3 x^2}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )}\right ) \, dx+e^2 \int \frac {85-204 x+38 x^2}{\left (15+8 x-22 x^2+4 x^3\right ) \left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\left (2 e^2\right ) \int \frac {1}{\left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\left (2 e^2\right ) \int \frac {x^2}{\left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+\left (10 e^2\right ) \int \frac {x}{\left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\int \frac {1}{(-5+x) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx\\ &=8 \int \frac {1}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+12 \int \frac {x^2}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-44 \int \frac {x}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+e^2 \int \frac {85-204 x+38 x^2}{\left (15+8 x-22 x^2+4 x^3\right ) \left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\left (2 e^2\right ) \int \frac {1}{\left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\left (2 e^2\right ) \int \frac {x^2}{\left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+\left (10 e^2\right ) \int \frac {x}{\left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\int \frac {1}{(-5+x) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx\\ &=8 \int \frac {1}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+12 \int \frac {x^2}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-44 \int \frac {x}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+e^2 \int \left (\frac {85}{\left (15+8 x-22 x^2+4 x^3\right ) \left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )}-\frac {204 x}{\left (15+8 x-22 x^2+4 x^3\right ) \left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )}+\frac {38 x^2}{\left (15+8 x-22 x^2+4 x^3\right ) \left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )}\right ) \, dx-\left (2 e^2\right ) \int \frac {1}{\left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\left (2 e^2\right ) \int \frac {x^2}{\left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+\left (10 e^2\right ) \int \frac {x}{\left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\int \frac {1}{(-5+x) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx\\ &=8 \int \frac {1}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+12 \int \frac {x^2}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-44 \int \frac {x}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\left (2 e^2\right ) \int \frac {1}{\left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\left (2 e^2\right ) \int \frac {x^2}{\left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+\left (10 e^2\right ) \int \frac {x}{\left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+\left (38 e^2\right ) \int \frac {x^2}{\left (15+8 x-22 x^2+4 x^3\right ) \left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+\left (85 e^2\right ) \int \frac {1}{\left (15+8 x-22 x^2+4 x^3\right ) \left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\left (204 e^2\right ) \int \frac {x}{\left (15+8 x-22 x^2+4 x^3\right ) \left (-5 e^2+15 e^{x^2}+e^2 x+8 e^{x^2} x-22 e^{x^2} x^2+4 e^{x^2} x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\int \frac {1}{(-5+x) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx\\ &=8 \int \frac {1}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+12 \int \frac {x^2}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-44 \int \frac {x}{\left (15+8 x-22 x^2+4 x^3\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\left (2 e^2\right ) \int \frac {1}{\left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\left (2 e^2\right ) \int \frac {x^2}{\left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+\left (10 e^2\right ) \int \frac {x}{\left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+\left (38 e^2\right ) \int \frac {x^2}{\left (15+8 x-22 x^2+4 x^3\right ) \left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx+\left (85 e^2\right ) \int \frac {1}{\left (15+8 x-22 x^2+4 x^3\right ) \left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\left (204 e^2\right ) \int \frac {x}{\left (15+8 x-22 x^2+4 x^3\right ) \left (e^2 (-5+x)+e^{x^2} \left (15+8 x-22 x^2+4 x^3\right )\right ) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx-\int \frac {1}{(-5+x) \log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 36, normalized size = 1.09 \begin {gather*} \log \left (\log \left (\frac {15+e^{2-x^2} (-5+x)+8 x-22 x^2+4 x^3}{-5+x}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-55 + 220*x - 82*x^2 + 8*x^3 + E^(2 - x^2)*(-50*x + 20*x^2 - 2*x^3))/((-75 - 25*x + 118*x^2 - 42*x^
3 + 4*x^4 + E^(2 - x^2)*(25 - 10*x + x^2))*Log[(15 + E^(2 - x^2)*(-5 + x) + 8*x - 22*x^2 + 4*x^3)/(-5 + x)]),x
]

[Out]

Log[Log[(15 + E^(2 - x^2)*(-5 + x) + 8*x - 22*x^2 + 4*x^3)/(-5 + x)]]

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fricas [A]  time = 0.99, size = 35, normalized size = 1.06 \begin {gather*} \log \left (\log \left (\frac {4 \, x^{3} - 22 \, x^{2} + {\left (x - 5\right )} e^{\left (-x^{2} + 2\right )} + 8 \, x + 15}{x - 5}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+20*x^2-50*x)*exp(-x^2+2)+8*x^3-82*x^2+220*x-55)/((x^2-10*x+25)*exp(-x^2+2)+4*x^4-42*x^3+118
*x^2-25*x-75)/log(((x-5)*exp(-x^2+2)+4*x^3-22*x^2+8*x+15)/(x-5)),x, algorithm="fricas")

[Out]

log(log((4*x^3 - 22*x^2 + (x - 5)*e^(-x^2 + 2) + 8*x + 15)/(x - 5)))

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giac [A]  time = 0.42, size = 43, normalized size = 1.30 \begin {gather*} \log \left (\log \left (\frac {4 \, x^{3} - 22 \, x^{2} + x e^{\left (-x^{2} + 2\right )} + 8 \, x - 5 \, e^{\left (-x^{2} + 2\right )} + 15}{x - 5}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+20*x^2-50*x)*exp(-x^2+2)+8*x^3-82*x^2+220*x-55)/((x^2-10*x+25)*exp(-x^2+2)+4*x^4-42*x^3+118
*x^2-25*x-75)/log(((x-5)*exp(-x^2+2)+4*x^3-22*x^2+8*x+15)/(x-5)),x, algorithm="giac")

[Out]

log(log((4*x^3 - 22*x^2 + x*e^(-x^2 + 2) + 8*x - 5*e^(-x^2 + 2) + 15)/(x - 5)))

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maple [A]  time = 0.22, size = 36, normalized size = 1.09

method	result	size
norman	\(\ln \left (\ln \left (\frac {\left (x -5\right ) {\mathrm e}^{-x^{2}+2}+4 x^{3}-22 x^{2}+8 x +15}{x -5}\right )\right )\)	\(36\)
risch	\(\ln \left (\ln \left (x^{3}-\frac {11 x^{2}}{2}+\left (\frac {{\mathrm e}^{-x^{2}+2}}{4}+2\right ) x -\frac {5 \,{\mathrm e}^{-x^{2}+2}}{4}+\frac {15}{4}\right )-\frac {i \left (\pi \,\mathrm {csgn}\left (\frac {i}{x -5}\right ) \mathrm {csgn}\left (i \left (x^{3}-\frac {11 x^{2}}{2}+\left (\frac {{\mathrm e}^{-x^{2}+2}}{4}+2\right ) x -\frac {5 \,{\mathrm e}^{-x^{2}+2}}{4}+\frac {15}{4}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{3}-\frac {11 x^{2}}{2}+\left (\frac {{\mathrm e}^{-x^{2}+2}}{4}+2\right ) x -\frac {5 \,{\mathrm e}^{-x^{2}+2}}{4}+\frac {15}{4}\right )}{x -5}\right )-\pi \,\mathrm {csgn}\left (\frac {i}{x -5}\right ) \mathrm {csgn}\left (\frac {i \left (x^{3}-\frac {11 x^{2}}{2}+\left (\frac {{\mathrm e}^{-x^{2}+2}}{4}+2\right ) x -\frac {5 \,{\mathrm e}^{-x^{2}+2}}{4}+\frac {15}{4}\right )}{x -5}\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (x^{3}-\frac {11 x^{2}}{2}+\left (\frac {{\mathrm e}^{-x^{2}+2}}{4}+2\right ) x -\frac {5 \,{\mathrm e}^{-x^{2}+2}}{4}+\frac {15}{4}\right )\right ) \mathrm {csgn}\left (\frac {i \left (x^{3}-\frac {11 x^{2}}{2}+\left (\frac {{\mathrm e}^{-x^{2}+2}}{4}+2\right ) x -\frac {5 \,{\mathrm e}^{-x^{2}+2}}{4}+\frac {15}{4}\right )}{x -5}\right )^{2}+\pi \mathrm {csgn}\left (\frac {i \left (x^{3}-\frac {11 x^{2}}{2}+\left (\frac {{\mathrm e}^{-x^{2}+2}}{4}+2\right ) x -\frac {5 \,{\mathrm e}^{-x^{2}+2}}{4}+\frac {15}{4}\right )}{x -5}\right )^{3}+4 i \ln \relax (2)-2 i \ln \left (x -5\right )\right )}{2}\right )\)	\(336\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^3+20*x^2-50*x)*exp(-x^2+2)+8*x^3-82*x^2+220*x-55)/((x^2-10*x+25)*exp(-x^2+2)+4*x^4-42*x^3+118*x^2-2
5*x-75)/ln(((x-5)*exp(-x^2+2)+4*x^3-22*x^2+8*x+15)/(x-5)),x,method=_RETURNVERBOSE)

[Out]

ln(ln(((x-5)*exp(-x^2+2)+4*x^3-22*x^2+8*x+15)/(x-5)))

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maxima [A]  time = 0.43, size = 43, normalized size = 1.30 \begin {gather*} \log \left (-x^{2} + \log \left (x e^{2} + {\left (4 \, x^{3} - 22 \, x^{2} + 8 \, x + 15\right )} e^{\left (x^{2}\right )} - 5 \, e^{2}\right ) - \log \left (x - 5\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^3+20*x^2-50*x)*exp(-x^2+2)+8*x^3-82*x^2+220*x-55)/((x^2-10*x+25)*exp(-x^2+2)+4*x^4-42*x^3+118
*x^2-25*x-75)/log(((x-5)*exp(-x^2+2)+4*x^3-22*x^2+8*x+15)/(x-5)),x, algorithm="maxima")

[Out]

log(-x^2 + log(x*e^2 + (4*x^3 - 22*x^2 + 8*x + 15)*e^(x^2) - 5*e^2) - log(x - 5))

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mupad [B]  time = 6.59, size = 35, normalized size = 1.06 \begin {gather*} \ln \left (\ln \left (\frac {8\,x-22\,x^2+4\,x^3+{\mathrm {e}}^2\,{\mathrm {e}}^{-x^2}\,\left (x-5\right )+15}{x-5}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((82*x^2 - 220*x - 8*x^3 + exp(2 - x^2)*(50*x - 20*x^2 + 2*x^3) + 55)/(log((8*x + exp(2 - x^2)*(x - 5) - 22
*x^2 + 4*x^3 + 15)/(x - 5))*(25*x - exp(2 - x^2)*(x^2 - 10*x + 25) - 118*x^2 + 42*x^3 - 4*x^4 + 75)),x)

[Out]

log(log((8*x - 22*x^2 + 4*x^3 + exp(2)*exp(-x^2)*(x - 5) + 15)/(x - 5)))

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sympy [A]  time = 1.28, size = 31, normalized size = 0.94 \begin {gather*} \log {\left (\log {\left (\frac {4 x^{3} - 22 x^{2} + 8 x + \left (x - 5\right ) e^{2 - x^{2}} + 15}{x - 5} \right )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**3+20*x**2-50*x)*exp(-x**2+2)+8*x**3-82*x**2+220*x-55)/((x**2-10*x+25)*exp(-x**2+2)+4*x**4-42
*x**3+118*x**2-25*x-75)/ln(((x-5)*exp(-x**2+2)+4*x**3-22*x**2+8*x+15)/(x-5)),x)

[Out]

log(log((4*x**3 - 22*x**2 + 8*x + (x - 5)*exp(2 - x**2) + 15)/(x - 5)))

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