3.55.39 \(\int \frac {-108+144 x-36 x^2+(54-144 x+54 x^2) \log (x^2)+e^{-5+e^x} (-18+18 x+(9-18 x+e^x (9 x-9 x^2)) \log (x^2))}{\log ^2(x^2)} \, dx\)

Optimal. Leaf size=26 \[ \frac {3 (3-3 x) \left (6+e^{-5+e^x}-2 x\right ) x}{\log \left (x^2\right )} \]

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Rubi [F]  time = 1.94, 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 {-108+144 x-36 x^2+\left (54-144 x+54 x^2\right ) \log \left (x^2\right )+e^{-5+e^x} \left (-18+18 x+\left (9-18 x+e^x \left (9 x-9 x^2\right )\right ) \log \left (x^2\right )\right )}{\log ^2\left (x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-108 + 144*x - 36*x^2 + (54 - 144*x + 54*x^2)*Log[x^2] + E^(-5 + E^x)*(-18 + 18*x + (9 - 18*x + E^x*(9*x
- 9*x^2))*Log[x^2]))/Log[x^2]^2,x]

[Out]

(54*x)/Log[x^2] - (72*x^2)/Log[x^2] + (18*x^3)/Log[x^2] - (18*Defer[Int][E^E^x/Log[x^2]^2, x])/E^5 + (18*Defer
[Int][(E^E^x*x)/Log[x^2]^2, x])/E^5 + (9*Defer[Int][E^E^x/Log[x^2], x])/E^5 - (18*Defer[Int][(E^E^x*x)/Log[x^2
], x])/E^5 + 9*Defer[Int][(E^(-5 + E^x + x)*x)/Log[x^2], x] - 9*Defer[Int][(E^(-5 + E^x + x)*x^2)/Log[x^2], x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {9 e^{-5+e^x+x} (-1+x) x}{\log \left (x^2\right )}+\frac {9 \left (-12 e^5-2 e^{e^x}+16 e^5 x+2 e^{e^x} x-4 e^5 x^2+6 e^5 \log \left (x^2\right )+e^{e^x} \log \left (x^2\right )-16 e^5 x \log \left (x^2\right )-2 e^{e^x} x \log \left (x^2\right )+6 e^5 x^2 \log \left (x^2\right )\right )}{e^5 \log ^2\left (x^2\right )}\right ) \, dx\\ &=-\left (9 \int \frac {e^{-5+e^x+x} (-1+x) x}{\log \left (x^2\right )} \, dx\right )+\frac {9 \int \frac {-12 e^5-2 e^{e^x}+16 e^5 x+2 e^{e^x} x-4 e^5 x^2+6 e^5 \log \left (x^2\right )+e^{e^x} \log \left (x^2\right )-16 e^5 x \log \left (x^2\right )-2 e^{e^x} x \log \left (x^2\right )+6 e^5 x^2 \log \left (x^2\right )}{\log ^2\left (x^2\right )} \, dx}{e^5}\\ &=-\left (9 \int \left (-\frac {e^{-5+e^x+x} x}{\log \left (x^2\right )}+\frac {e^{-5+e^x+x} x^2}{\log \left (x^2\right )}\right ) \, dx\right )+\frac {9 \int \frac {2 \left (e^{e^x}-2 e^5 (-3+x)\right ) (-1+x)+\left (e^{e^x} (1-2 x)+2 e^5 \left (3-8 x+3 x^2\right )\right ) \log \left (x^2\right )}{\log ^2\left (x^2\right )} \, dx}{e^5}\\ &=9 \int \frac {e^{-5+e^x+x} x}{\log \left (x^2\right )} \, dx-9 \int \frac {e^{-5+e^x+x} x^2}{\log \left (x^2\right )} \, dx+\frac {9 \int \left (-\frac {e^{e^x} \left (2-2 x-\log \left (x^2\right )+2 x \log \left (x^2\right )\right )}{\log ^2\left (x^2\right )}+\frac {2 e^5 \left (-6+8 x-2 x^2+3 \log \left (x^2\right )-8 x \log \left (x^2\right )+3 x^2 \log \left (x^2\right )\right )}{\log ^2\left (x^2\right )}\right ) \, dx}{e^5}\\ &=9 \int \frac {e^{-5+e^x+x} x}{\log \left (x^2\right )} \, dx-9 \int \frac {e^{-5+e^x+x} x^2}{\log \left (x^2\right )} \, dx+18 \int \frac {-6+8 x-2 x^2+3 \log \left (x^2\right )-8 x \log \left (x^2\right )+3 x^2 \log \left (x^2\right )}{\log ^2\left (x^2\right )} \, dx-\frac {9 \int \frac {e^{e^x} \left (2-2 x-\log \left (x^2\right )+2 x \log \left (x^2\right )\right )}{\log ^2\left (x^2\right )} \, dx}{e^5}\\ &=9 \int \frac {e^{-5+e^x+x} x}{\log \left (x^2\right )} \, dx-9 \int \frac {e^{-5+e^x+x} x^2}{\log \left (x^2\right )} \, dx+18 \int \left (-\frac {2 \left (3-4 x+x^2\right )}{\log ^2\left (x^2\right )}+\frac {3-8 x+3 x^2}{\log \left (x^2\right )}\right ) \, dx-\frac {9 \int \left (-\frac {2 e^{e^x} (-1+x)}{\log ^2\left (x^2\right )}+\frac {e^{e^x} (-1+2 x)}{\log \left (x^2\right )}\right ) \, dx}{e^5}\\ &=9 \int \frac {e^{-5+e^x+x} x}{\log \left (x^2\right )} \, dx-9 \int \frac {e^{-5+e^x+x} x^2}{\log \left (x^2\right )} \, dx+18 \int \frac {3-8 x+3 x^2}{\log \left (x^2\right )} \, dx-36 \int \frac {3-4 x+x^2}{\log ^2\left (x^2\right )} \, dx-\frac {9 \int \frac {e^{e^x} (-1+2 x)}{\log \left (x^2\right )} \, dx}{e^5}+\frac {18 \int \frac {e^{e^x} (-1+x)}{\log ^2\left (x^2\right )} \, dx}{e^5}\\ &=9 \int \frac {e^{-5+e^x+x} x}{\log \left (x^2\right )} \, dx-9 \int \frac {e^{-5+e^x+x} x^2}{\log \left (x^2\right )} \, dx+18 \int \left (\frac {3}{\log \left (x^2\right )}-\frac {8 x}{\log \left (x^2\right )}+\frac {3 x^2}{\log \left (x^2\right )}\right ) \, dx-36 \int \left (\frac {3}{\log ^2\left (x^2\right )}-\frac {4 x}{\log ^2\left (x^2\right )}+\frac {x^2}{\log ^2\left (x^2\right )}\right ) \, dx-\frac {9 \int \left (-\frac {e^{e^x}}{\log \left (x^2\right )}+\frac {2 e^{e^x} x}{\log \left (x^2\right )}\right ) \, dx}{e^5}+\frac {18 \int \left (-\frac {e^{e^x}}{\log ^2\left (x^2\right )}+\frac {e^{e^x} x}{\log ^2\left (x^2\right )}\right ) \, dx}{e^5}\\ &=9 \int \frac {e^{-5+e^x+x} x}{\log \left (x^2\right )} \, dx-9 \int \frac {e^{-5+e^x+x} x^2}{\log \left (x^2\right )} \, dx-36 \int \frac {x^2}{\log ^2\left (x^2\right )} \, dx+54 \int \frac {1}{\log \left (x^2\right )} \, dx+54 \int \frac {x^2}{\log \left (x^2\right )} \, dx-108 \int \frac {1}{\log ^2\left (x^2\right )} \, dx+144 \int \frac {x}{\log ^2\left (x^2\right )} \, dx-144 \int \frac {x}{\log \left (x^2\right )} \, dx+\frac {9 \int \frac {e^{e^x}}{\log \left (x^2\right )} \, dx}{e^5}-\frac {18 \int \frac {e^{e^x}}{\log ^2\left (x^2\right )} \, dx}{e^5}+\frac {18 \int \frac {e^{e^x} x}{\log ^2\left (x^2\right )} \, dx}{e^5}-\frac {18 \int \frac {e^{e^x} x}{\log \left (x^2\right )} \, dx}{e^5}\\ &=\frac {54 x}{\log \left (x^2\right )}-\frac {72 x^2}{\log \left (x^2\right )}+\frac {18 x^3}{\log \left (x^2\right )}+9 \int \frac {e^{-5+e^x+x} x}{\log \left (x^2\right )} \, dx-9 \int \frac {e^{-5+e^x+x} x^2}{\log \left (x^2\right )} \, dx-54 \int \frac {1}{\log \left (x^2\right )} \, dx-54 \int \frac {x^2}{\log \left (x^2\right )} \, dx-72 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,x^2\right )+144 \int \frac {x}{\log \left (x^2\right )} \, dx+\frac {9 \int \frac {e^{e^x}}{\log \left (x^2\right )} \, dx}{e^5}-\frac {18 \int \frac {e^{e^x}}{\log ^2\left (x^2\right )} \, dx}{e^5}+\frac {18 \int \frac {e^{e^x} x}{\log ^2\left (x^2\right )} \, dx}{e^5}-\frac {18 \int \frac {e^{e^x} x}{\log \left (x^2\right )} \, dx}{e^5}+\frac {\left (27 x^3\right ) \operatorname {Subst}\left (\int \frac {e^{3 x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{\left (x^2\right )^{3/2}}+\frac {(27 x) \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2}}\\ &=\frac {27 x \text {Ei}\left (\frac {\log \left (x^2\right )}{2}\right )}{\sqrt {x^2}}+\frac {27 x^3 \text {Ei}\left (\frac {3 \log \left (x^2\right )}{2}\right )}{\left (x^2\right )^{3/2}}+\frac {54 x}{\log \left (x^2\right )}-\frac {72 x^2}{\log \left (x^2\right )}+\frac {18 x^3}{\log \left (x^2\right )}-72 \text {li}\left (x^2\right )+9 \int \frac {e^{-5+e^x+x} x}{\log \left (x^2\right )} \, dx-9 \int \frac {e^{-5+e^x+x} x^2}{\log \left (x^2\right )} \, dx+72 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,x^2\right )+\frac {9 \int \frac {e^{e^x}}{\log \left (x^2\right )} \, dx}{e^5}-\frac {18 \int \frac {e^{e^x}}{\log ^2\left (x^2\right )} \, dx}{e^5}+\frac {18 \int \frac {e^{e^x} x}{\log ^2\left (x^2\right )} \, dx}{e^5}-\frac {18 \int \frac {e^{e^x} x}{\log \left (x^2\right )} \, dx}{e^5}-\frac {\left (27 x^3\right ) \operatorname {Subst}\left (\int \frac {e^{3 x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{\left (x^2\right )^{3/2}}-\frac {(27 x) \operatorname {Subst}\left (\int \frac {e^{x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2}}\\ &=\frac {54 x}{\log \left (x^2\right )}-\frac {72 x^2}{\log \left (x^2\right )}+\frac {18 x^3}{\log \left (x^2\right )}+9 \int \frac {e^{-5+e^x+x} x}{\log \left (x^2\right )} \, dx-9 \int \frac {e^{-5+e^x+x} x^2}{\log \left (x^2\right )} \, dx+\frac {9 \int \frac {e^{e^x}}{\log \left (x^2\right )} \, dx}{e^5}-\frac {18 \int \frac {e^{e^x}}{\log ^2\left (x^2\right )} \, dx}{e^5}+\frac {18 \int \frac {e^{e^x} x}{\log ^2\left (x^2\right )} \, dx}{e^5}-\frac {18 \int \frac {e^{e^x} x}{\log \left (x^2\right )} \, dx}{e^5}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.61, size = 31, normalized size = 1.19 \begin {gather*} \frac {9 \left (-e^{e^x}+2 e^5 (-3+x)\right ) (-1+x) x}{e^5 \log \left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-108 + 144*x - 36*x^2 + (54 - 144*x + 54*x^2)*Log[x^2] + E^(-5 + E^x)*(-18 + 18*x + (9 - 18*x + E^x
*(9*x - 9*x^2))*Log[x^2]))/Log[x^2]^2,x]

[Out]

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

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fricas [A]  time = 1.20, size = 36, normalized size = 1.38 \begin {gather*} \frac {9 \, {\left (2 \, x^{3} - 8 \, x^{2} - {\left (x^{2} - x\right )} e^{\left (e^{x} - 5\right )} + 6 \, x\right )}}{\log \left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-9*x^2+9*x)*exp(x)-18*x+9)*log(x^2)+18*x-18)*exp(exp(x)-5)+(54*x^2-144*x+54)*log(x^2)-36*x^2+144
*x-108)/log(x^2)^2,x, algorithm="fricas")

[Out]

9*(2*x^3 - 8*x^2 - (x^2 - x)*e^(e^x - 5) + 6*x)/log(x^2)

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giac [A]  time = 0.23, size = 43, normalized size = 1.65 \begin {gather*} \frac {9 \, {\left (2 \, x^{3} e^{5} - 8 \, x^{2} e^{5} - x^{2} e^{\left (e^{x}\right )} + 6 \, x e^{5} + x e^{\left (e^{x}\right )}\right )} e^{\left (-5\right )}}{\log \left (x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-9*x^2+9*x)*exp(x)-18*x+9)*log(x^2)+18*x-18)*exp(exp(x)-5)+(54*x^2-144*x+54)*log(x^2)-36*x^2+144
*x-108)/log(x^2)^2,x, algorithm="giac")

[Out]

9*(2*x^3*e^5 - 8*x^2*e^5 - x^2*e^(e^x) + 6*x*e^5 + x*e^(e^x))*e^(-5)/log(x^2)

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maple [F]  time = 0.16, size = 0, normalized size = 0.00 \[\int \frac {\left (\left (\left (-9 x^{2}+9 x \right ) {\mathrm e}^{x}-18 x +9\right ) \ln \left (x^{2}\right )+18 x -18\right ) {\mathrm e}^{{\mathrm e}^{x}-5}+\left (54 x^{2}-144 x +54\right ) \ln \left (x^{2}\right )-36 x^{2}+144 x -108}{\ln \left (x^{2}\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((-9*x^2+9*x)*exp(x)-18*x+9)*ln(x^2)+18*x-18)*exp(exp(x)-5)+(54*x^2-144*x+54)*ln(x^2)-36*x^2+144*x-108)/
ln(x^2)^2,x)

[Out]

int(((((-9*x^2+9*x)*exp(x)-18*x+9)*ln(x^2)+18*x-18)*exp(exp(x)-5)+(54*x^2-144*x+54)*ln(x^2)-36*x^2+144*x-108)/
ln(x^2)^2,x)

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maxima [A]  time = 0.39, size = 40, normalized size = 1.54 \begin {gather*} \frac {9 \, {\left (2 \, x^{3} e^{5} - 8 \, x^{2} e^{5} + 6 \, x e^{5} - {\left (x^{2} - x\right )} e^{\left (e^{x}\right )}\right )} e^{\left (-5\right )}}{2 \, \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-9*x^2+9*x)*exp(x)-18*x+9)*log(x^2)+18*x-18)*exp(exp(x)-5)+(54*x^2-144*x+54)*log(x^2)-36*x^2+144
*x-108)/log(x^2)^2,x, algorithm="maxima")

[Out]

9/2*(2*x^3*e^5 - 8*x^2*e^5 + 6*x*e^5 - (x^2 - x)*e^(e^x))*e^(-5)/log(x)

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mupad [B]  time = 3.49, size = 71, normalized size = 2.73 \begin {gather*} 27\,x+\frac {18\,x\,\left (x^2-4\,x+3\right )-9\,x\,\ln \left (x^2\right )\,\left (3\,x^2-8\,x+3\right )}{\ln \left (x^2\right )}-72\,x^2+27\,x^3+\frac {{\mathrm {e}}^{{\mathrm {e}}^x-5}\,\left (9\,x-9\,x^2\right )}{\ln \left (x^2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((144*x + log(x^2)*(54*x^2 - 144*x + 54) + exp(exp(x) - 5)*(18*x + log(x^2)*(exp(x)*(9*x - 9*x^2) - 18*x +
9) - 18) - 36*x^2 - 108)/log(x^2)^2,x)

[Out]

27*x + (18*x*(x^2 - 4*x + 3) - 9*x*log(x^2)*(3*x^2 - 8*x + 3))/log(x^2) - 72*x^2 + 27*x^3 + (exp(exp(x) - 5)*(
9*x - 9*x^2))/log(x^2)

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sympy [A]  time = 0.32, size = 37, normalized size = 1.42 \begin {gather*} \frac {\left (- 9 x^{2} + 9 x\right ) e^{e^{x} - 5}}{\log {\left (x^{2} \right )}} + \frac {18 x^{3} - 72 x^{2} + 54 x}{\log {\left (x^{2} \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((-9*x**2+9*x)*exp(x)-18*x+9)*ln(x**2)+18*x-18)*exp(exp(x)-5)+(54*x**2-144*x+54)*ln(x**2)-36*x**2+
144*x-108)/ln(x**2)**2,x)

[Out]

(-9*x**2 + 9*x)*exp(exp(x) - 5)/log(x**2) + (18*x**3 - 72*x**2 + 54*x)/log(x**2)

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