3.102.57 \(\int \frac {(-20-20 x^2-5 x^4) \log (3)+e^{-4 e^x+4 x+4 x^2} (e^x (-32 x-16 x^3) \log (3)+(8+32 x+60 x^2+16 x^3+32 x^4) \log (3))}{16+16 x^2+4 x^4} \, dx\)

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

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Rubi [F]  time = 6.83, 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 {\left (-20-20 x^2-5 x^4\right ) \log (3)+e^{-4 e^x+4 x+4 x^2} \left (e^x \left (-32 x-16 x^3\right ) \log (3)+\left (8+32 x+60 x^2+16 x^3+32 x^4\right ) \log (3)\right )}{16+16 x^2+4 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-20 - 20*x^2 - 5*x^4)*Log[3] + E^(-4*E^x + 4*x + 4*x^2)*(E^x*(-32*x - 16*x^3)*Log[3] + (8 + 32*x + 60*x^
2 + 16*x^3 + 32*x^4)*Log[3]))/(16 + 16*x^2 + 4*x^4),x]

[Out]

(-5*x*Log[3])/4 + 8*Log[3]*Defer[Int][E^(-4*(E^x - x - x^2)), x] - (Log[3]*Defer[Int][1/(E^(4*(E^x - x - x^2))
*(I*Sqrt[2] - x)^2), x])/2 - 2*Log[3]*Defer[Int][1/(E^(4*(E^x - x - x^2))*(I*Sqrt[2] - x)), x] - (4*I)*Sqrt[2]
*Log[3]*Defer[Int][1/(E^(4*(E^x - x - x^2))*(I*Sqrt[2] - x)), x] + 2*Log[3]*Defer[Int][E^(x - 4*(E^x - x - x^2
))/(I*Sqrt[2] - x), x] - (Log[3]*Defer[Int][1/(E^(4*(E^x - x - x^2))*(I*Sqrt[2] + x)^2), x])/2 + 2*Log[3]*Defe
r[Int][1/(E^(4*(E^x - x - x^2))*(I*Sqrt[2] + x)), x] - (4*I)*Sqrt[2]*Log[3]*Defer[Int][1/(E^(4*(E^x - x - x^2)
)*(I*Sqrt[2] + x)), x] - 2*Log[3]*Defer[Int][E^(x - 4*(E^x - x - x^2))/(I*Sqrt[2] + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=4 \int \frac {\left (-20-20 x^2-5 x^4\right ) \log (3)+e^{-4 e^x+4 x+4 x^2} \left (e^x \left (-32 x-16 x^3\right ) \log (3)+\left (8+32 x+60 x^2+16 x^3+32 x^4\right ) \log (3)\right )}{\left (8+4 x^2\right )^2} \, dx\\ &=4 \int \left (-\frac {5 \log (3)}{16}+\frac {e^{-4 \left (e^x-x-x^2\right )} \left (2+8 x-8 e^x x+15 x^2+4 x^3-4 e^x x^3+8 x^4\right ) \log (3)}{4 \left (2+x^2\right )^2}\right ) \, dx\\ &=-\frac {5}{4} x \log (3)+\log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )} \left (2+8 x-8 e^x x+15 x^2+4 x^3-4 e^x x^3+8 x^4\right )}{\left (2+x^2\right )^2} \, dx\\ &=-\frac {5}{4} x \log (3)+\log (3) \int \left (\frac {2 e^{-4 \left (e^x-x-x^2\right )}}{\left (2+x^2\right )^2}+\frac {8 e^{-4 \left (e^x-x-x^2\right )} x}{\left (2+x^2\right )^2}+\frac {15 e^{-4 \left (e^x-x-x^2\right )} x^2}{\left (2+x^2\right )^2}+\frac {4 e^{-4 \left (e^x-x-x^2\right )} x^3}{\left (2+x^2\right )^2}+\frac {8 e^{-4 \left (e^x-x-x^2\right )} x^4}{\left (2+x^2\right )^2}-\frac {4 e^{x-4 \left (e^x-x-x^2\right )} x}{2+x^2}\right ) \, dx\\ &=-\frac {5}{4} x \log (3)+(2 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (2+x^2\right )^2} \, dx+(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )} x^3}{\left (2+x^2\right )^2} \, dx-(4 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )} x}{2+x^2} \, dx+(8 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )} x}{\left (2+x^2\right )^2} \, dx+(8 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )} x^4}{\left (2+x^2\right )^2} \, dx+(15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )} x^2}{\left (2+x^2\right )^2} \, dx\\ &=-\frac {5}{4} x \log (3)+(2 \log (3)) \int \left (-\frac {e^{-4 \left (e^x-x-x^2\right )}}{8 \left (i \sqrt {2}-x\right )^2}-\frac {e^{-4 \left (e^x-x-x^2\right )}}{8 \left (i \sqrt {2}+x\right )^2}-\frac {e^{-4 \left (e^x-x-x^2\right )}}{4 \left (-2-x^2\right )}\right ) \, dx-(4 \log (3)) \int \left (-\frac {e^{x-4 \left (e^x-x-x^2\right )}}{2 \left (i \sqrt {2}-x\right )}+\frac {e^{x-4 \left (e^x-x-x^2\right )}}{2 \left (i \sqrt {2}+x\right )}\right ) \, dx+(4 \log (3)) \int \left (-\frac {2 e^{-4 \left (e^x-x-x^2\right )} x}{\left (2+x^2\right )^2}+\frac {e^{-4 \left (e^x-x-x^2\right )} x}{2+x^2}\right ) \, dx+(8 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )} x}{\left (2+x^2\right )^2} \, dx+(8 \log (3)) \int \left (e^{-4 \left (e^x-x-x^2\right )}+\frac {4 e^{-4 \left (e^x-x-x^2\right )}}{\left (2+x^2\right )^2}-\frac {4 e^{-4 \left (e^x-x-x^2\right )}}{2+x^2}\right ) \, dx+(15 \log (3)) \int \left (-\frac {2 e^{-4 \left (e^x-x-x^2\right )}}{\left (2+x^2\right )^2}+\frac {e^{-4 \left (e^x-x-x^2\right )}}{2+x^2}\right ) \, dx\\ &=-\frac {5}{4} x \log (3)-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-\frac {1}{2} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{-2-x^2} \, dx+(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx-(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx+(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )} x}{2+x^2} \, dx+(8 \log (3)) \int e^{-4 \left (e^x-x-x^2\right )} \, dx+(15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{2+x^2} \, dx-(30 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (2+x^2\right )^2} \, dx+(32 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (2+x^2\right )^2} \, dx-(32 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{2+x^2} \, dx\\ &=-\frac {5}{4} x \log (3)-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-\frac {1}{2} \log (3) \int \left (-\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}-x\right )}-\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}+x\right )}\right ) \, dx+(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx-(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx+(4 \log (3)) \int \left (-\frac {e^{-4 \left (e^x-x-x^2\right )}}{2 \left (i \sqrt {2}-x\right )}+\frac {e^{-4 \left (e^x-x-x^2\right )}}{2 \left (i \sqrt {2}+x\right )}\right ) \, dx+(8 \log (3)) \int e^{-4 \left (e^x-x-x^2\right )} \, dx+(15 \log (3)) \int \left (\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}-x\right )}+\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}+x\right )}\right ) \, dx-(30 \log (3)) \int \left (-\frac {e^{-4 \left (e^x-x-x^2\right )}}{8 \left (i \sqrt {2}-x\right )^2}-\frac {e^{-4 \left (e^x-x-x^2\right )}}{8 \left (i \sqrt {2}+x\right )^2}-\frac {e^{-4 \left (e^x-x-x^2\right )}}{4 \left (-2-x^2\right )}\right ) \, dx-(32 \log (3)) \int \left (\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}-x\right )}+\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}+x\right )}\right ) \, dx+(32 \log (3)) \int \left (-\frac {e^{-4 \left (e^x-x-x^2\right )}}{8 \left (i \sqrt {2}-x\right )^2}-\frac {e^{-4 \left (e^x-x-x^2\right )}}{8 \left (i \sqrt {2}+x\right )^2}-\frac {e^{-4 \left (e^x-x-x^2\right )}}{4 \left (-2-x^2\right )}\right ) \, dx\\ &=-\frac {5}{4} x \log (3)-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-(2 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx+(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx+(2 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx-(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx+\frac {1}{4} (15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx+\frac {1}{4} (15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx+\frac {1}{2} (15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{-2-x^2} \, dx+(8 \log (3)) \int e^{-4 \left (e^x-x-x^2\right )} \, dx-(8 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{-2-x^2} \, dx+\frac {(i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx}{4 \sqrt {2}}+\frac {(i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx}{4 \sqrt {2}}+\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx}{2 \sqrt {2}}+\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx}{2 \sqrt {2}}-\left (8 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx-\left (8 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx\\ &=-\frac {5}{4} x \log (3)-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-(2 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx+(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx+(2 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx-(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx+\frac {1}{4} (15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx+\frac {1}{4} (15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx+\frac {1}{2} (15 \log (3)) \int \left (-\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}-x\right )}-\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}+x\right )}\right ) \, dx+(8 \log (3)) \int e^{-4 \left (e^x-x-x^2\right )} \, dx-(8 \log (3)) \int \left (-\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}-x\right )}-\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}+x\right )}\right ) \, dx+\frac {(i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx}{4 \sqrt {2}}+\frac {(i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx}{4 \sqrt {2}}+\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx}{2 \sqrt {2}}+\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx}{2 \sqrt {2}}-\left (8 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx-\left (8 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx\\ &=-\frac {5}{4} x \log (3)-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-(2 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx+(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx+(2 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx-(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx+\frac {1}{4} (15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx+\frac {1}{4} (15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx+(8 \log (3)) \int e^{-4 \left (e^x-x-x^2\right )} \, dx+\frac {(i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx}{4 \sqrt {2}}+\frac {(i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx}{4 \sqrt {2}}-\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx}{4 \sqrt {2}}-\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx}{4 \sqrt {2}}+\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx}{2 \sqrt {2}}+\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx}{2 \sqrt {2}}+\left (2 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx+\left (2 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx-\left (8 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx-\left (8 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.36, size = 32, normalized size = 0.97 \begin {gather*} \frac {1}{4} x \left (-5+\frac {4 e^{-4 e^x+4 x (1+x)}}{2+x^2}\right ) \log (3) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-20 - 20*x^2 - 5*x^4)*Log[3] + E^(-4*E^x + 4*x + 4*x^2)*(E^x*(-32*x - 16*x^3)*Log[3] + (8 + 32*x +
60*x^2 + 16*x^3 + 32*x^4)*Log[3]))/(16 + 16*x^2 + 4*x^4),x]

[Out]

(x*(-5 + (4*E^(-4*E^x + 4*x*(1 + x)))/(2 + x^2))*Log[3])/4

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fricas [A]  time = 1.12, size = 40, normalized size = 1.21 \begin {gather*} \frac {4 \, x e^{\left (4 \, x^{2} + 4 \, x - 4 \, e^{x}\right )} \log \relax (3) - 5 \, {\left (x^{3} + 2 \, x\right )} \log \relax (3)}{4 \, {\left (x^{2} + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x^3-32*x)*log(3)*exp(x)+(32*x^4+16*x^3+60*x^2+32*x+8)*log(3))*exp(-4*exp(x)+4*x^2+4*x)+(-5*x^
4-20*x^2-20)*log(3))/(4*x^4+16*x^2+16),x, algorithm="fricas")

[Out]

1/4*(4*x*e^(4*x^2 + 4*x - 4*e^x)*log(3) - 5*(x^3 + 2*x)*log(3))/(x^2 + 2)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {4 \, {\left (4 \, {\left (x^{3} + 2 \, x\right )} e^{x} \log \relax (3) - {\left (8 \, x^{4} + 4 \, x^{3} + 15 \, x^{2} + 8 \, x + 2\right )} \log \relax (3)\right )} e^{\left (4 \, x^{2} + 4 \, x - 4 \, e^{x}\right )} + 5 \, {\left (x^{4} + 4 \, x^{2} + 4\right )} \log \relax (3)}{4 \, {\left (x^{4} + 4 \, x^{2} + 4\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x^3-32*x)*log(3)*exp(x)+(32*x^4+16*x^3+60*x^2+32*x+8)*log(3))*exp(-4*exp(x)+4*x^2+4*x)+(-5*x^
4-20*x^2-20)*log(3))/(4*x^4+16*x^2+16),x, algorithm="giac")

[Out]

integrate(-1/4*(4*(4*(x^3 + 2*x)*e^x*log(3) - (8*x^4 + 4*x^3 + 15*x^2 + 8*x + 2)*log(3))*e^(4*x^2 + 4*x - 4*e^
x) + 5*(x^4 + 4*x^2 + 4)*log(3))/(x^4 + 4*x^2 + 4), x)

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maple [A]  time = 0.30, size = 32, normalized size = 0.97

method	result	size
risch	\(-\frac {5 x \ln \relax (3)}{4}+\frac {\ln \relax (3) x \,{\mathrm e}^{-4 \,{\mathrm e}^{x}+4 x^{2}+4 x}}{x^{2}+2}\)	\(32\)
norman	\(\frac {x \ln \relax (3) {\mathrm e}^{-4 \,{\mathrm e}^{x}+4 x^{2}+4 x}-\frac {5 x \ln \relax (3)}{2}-\frac {5 x^{3} \ln \relax (3)}{4}}{x^{2}+2}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-16*x^3-32*x)*ln(3)*exp(x)+(32*x^4+16*x^3+60*x^2+32*x+8)*ln(3))*exp(-4*exp(x)+4*x^2+4*x)+(-5*x^4-20*x^2
-20)*ln(3))/(4*x^4+16*x^2+16),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.51, size = 108, normalized size = 3.27 \begin {gather*} \frac {5}{8} \, {\left (3 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 2 \, x - \frac {2 \, x}{x^{2} + 2}\right )} \log \relax (3) - \frac {5}{8} \, {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {2 \, x}{x^{2} + 2}\right )} \log \relax (3) - \frac {5}{4} \, {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {2 \, x}{x^{2} + 2}\right )} \log \relax (3) + \frac {x e^{\left (4 \, x^{2} + 4 \, x - 4 \, e^{x}\right )} \log \relax (3)}{x^{2} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x^3-32*x)*log(3)*exp(x)+(32*x^4+16*x^3+60*x^2+32*x+8)*log(3))*exp(-4*exp(x)+4*x^2+4*x)+(-5*x^
4-20*x^2-20)*log(3))/(4*x^4+16*x^2+16),x, algorithm="maxima")

[Out]

5/8*(3*sqrt(2)*arctan(1/2*sqrt(2)*x) - 2*x - 2*x/(x^2 + 2))*log(3) - 5/8*(sqrt(2)*arctan(1/2*sqrt(2)*x) + 2*x/
(x^2 + 2))*log(3) - 5/4*(sqrt(2)*arctan(1/2*sqrt(2)*x) - 2*x/(x^2 + 2))*log(3) + x*e^(4*x^2 + 4*x - 4*e^x)*log
(3)/(x^2 + 2)

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mupad [B]  time = 0.52, size = 32, normalized size = 0.97 \begin {gather*} \frac {x\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{4\,x^2}\,{\mathrm {e}}^{-4\,{\mathrm {e}}^x}\,\ln \relax (3)}{x^2+2}-\frac {5\,x\,\ln \relax (3)}{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(4*x - 4*exp(x) + 4*x^2)*(log(3)*(32*x + 60*x^2 + 16*x^3 + 32*x^4 + 8) - exp(x)*log(3)*(32*x + 16*x^3)
) - log(3)*(20*x^2 + 5*x^4 + 20))/(16*x^2 + 4*x^4 + 16),x)

[Out]

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

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sympy [A]  time = 0.25, size = 32, normalized size = 0.97 \begin {gather*} - \frac {5 x \log {\relax (3 )}}{4} + \frac {x e^{4 x^{2} + 4 x - 4 e^{x}} \log {\relax (3 )}}{x^{2} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-16*x**3-32*x)*ln(3)*exp(x)+(32*x**4+16*x**3+60*x**2+32*x+8)*ln(3))*exp(-4*exp(x)+4*x**2+4*x)+(-5
*x**4-20*x**2-20)*ln(3))/(4*x**4+16*x**2+16),x)

[Out]

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

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