3.104.25 \(\int \frac {e^{\frac {(1152+144 e^x) \log ^4(x)}{5 x^4-10 x^3 \log ^2(x)+5 x^2 \log ^4(x)}} ((-4608 x-576 e^x x) \log ^3(x)+(4608 x+e^x (576 x-144 x^2)) \log ^4(x)+(-2304+e^x (-288+144 x)) \log ^6(x))}{-5 x^6+15 x^5 \log ^2(x)-15 x^4 \log ^4(x)+5 x^3 \log ^6(x)} \, dx\)

Optimal. Leaf size=25 \[ e^{\frac {144 \left (8+e^x\right )}{5 \left (-x+\frac {x^2}{\log ^2(x)}\right )^2}} \]

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Rubi [F]  time = 29.51, 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 {\exp \left (\frac {\left (1152+144 e^x\right ) \log ^4(x)}{5 x^4-10 x^3 \log ^2(x)+5 x^2 \log ^4(x)}\right ) \left (\left (-4608 x-576 e^x x\right ) \log ^3(x)+\left (4608 x+e^x \left (576 x-144 x^2\right )\right ) \log ^4(x)+\left (-2304+e^x (-288+144 x)\right ) \log ^6(x)\right )}{-5 x^6+15 x^5 \log ^2(x)-15 x^4 \log ^4(x)+5 x^3 \log ^6(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(((1152 + 144*E^x)*Log[x]^4)/(5*x^4 - 10*x^3*Log[x]^2 + 5*x^2*Log[x]^4))*((-4608*x - 576*E^x*x)*Log[x]^
3 + (4608*x + E^x*(576*x - 144*x^2))*Log[x]^4 + (-2304 + E^x*(-288 + 144*x))*Log[x]^6))/(-5*x^6 + 15*x^5*Log[x
]^2 - 15*x^4*Log[x]^4 + 5*x^3*Log[x]^6),x]

[Out]

(-2304*Defer[Int][E^((144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))/x^3, x])/5 - (288*Defer[Int][E^(x + (1
44*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))/x^3, x])/5 + (144*Defer[Int][E^(x + (144*(8 + E^x)*Log[x]^4)/
(5*x^2*(x - Log[x]^2)^2))/x^2, x])/5 - (2304*Defer[Int][E^((144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))/
(x - Log[x]^2)^3, x])/5 - (288*Defer[Int][E^(x + (144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))/(x - Log[x
]^2)^3, x])/5 + (4608*Defer[Int][(E^((144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))*Log[x])/(x*(x - Log[x]
^2)^3), x])/5 + (576*Defer[Int][(E^(x + (144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))*Log[x])/(x*(x - Log
[x]^2)^3), x])/5 + (144*Defer[Int][E^(x + (144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))/(x - Log[x]^2)^2,
x])/5 + (2304*Defer[Int][E^((144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))/(x*(x - Log[x]^2)^2), x])/5 +
(288*Defer[Int][E^(x + (144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))/(x*(x - Log[x]^2)^2), x])/5 - (4608*
Defer[Int][(E^((144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))*Log[x])/(x^2*(x - Log[x]^2)^2), x])/5 - (576
*Defer[Int][(E^(x + (144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))*Log[x])/(x^2*(x - Log[x]^2)^2), x])/5 +
(2304*Defer[Int][E^((144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))/(x^2*(x - Log[x]^2)), x])/5 + (288*Def
er[Int][E^(x + (144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))/(x^2*(x - Log[x]^2)), x])/5 - (288*Defer[Int
][E^(x + (144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))/(x*(x - Log[x]^2)), x])/5

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {144 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (4 \left (8+e^x\right ) x+\left (-32+e^x (-4+x)\right ) x \log (x)+\left (16-e^x (-2+x)\right ) \log ^3(x)\right )}{5 x^3 \left (x-\log ^2(x)\right )^3} \, dx\\ &=\frac {144}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (4 \left (8+e^x\right ) x+\left (-32+e^x (-4+x)\right ) x \log (x)+\left (16-e^x (-2+x)\right ) \log ^3(x)\right )}{x^3 \left (x-\log ^2(x)\right )^3} \, dx\\ &=\frac {144}{5} \int \left (\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3}-\frac {32 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3}+\frac {16 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3}+\frac {\exp \left (x+\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (4 x-4 x \log (x)+x^2 \log (x)+2 \log ^3(x)-x \log ^3(x)\right )}{x^3 \left (x-\log ^2(x)\right )^3}\right ) \, dx\\ &=\frac {144}{5} \int \frac {\exp \left (x+\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (4 x-4 x \log (x)+x^2 \log (x)+2 \log ^3(x)-x \log ^3(x)\right )}{x^3 \left (x-\log ^2(x)\right )^3} \, dx+\frac {2304}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^6(x)}{x^3 \left (x-\log ^2(x)\right )^3} \, dx+\frac {4608}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x)}{x^2 \left (x-\log ^2(x)\right )^3} \, dx-\frac {4608}{5} \int \frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^4(x)}{x^2 \left (x-\log ^2(x)\right )^3} \, dx\\ &=\frac {144}{5} \int \frac {\exp \left (x+\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log ^3(x) \left (4 x+(-4+x) x \log (x)-(-2+x) \log ^3(x)\right )}{x^3 \left (x-\log ^2(x)\right )^3} \, dx+\frac {2304}{5} \int \left (-\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right )}{x^3}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right )}{\left (x-\log ^2(x)\right )^3}-\frac {3 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right )}{x \left (x-\log ^2(x)\right )^2}+\frac {3 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right )}{x^2 \left (x-\log ^2(x)\right )}\right ) \, dx+\frac {4608}{5} \int \left (\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log (x)}{x \left (x-\log ^2(x)\right )^3}-\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right ) \log (x)}{x^2 \left (x-\log ^2(x)\right )^2}\right ) \, dx-\frac {4608}{5} \int \left (\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right )}{\left (x-\log ^2(x)\right )^3}-\frac {2 \exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right )}{x \left (x-\log ^2(x)\right )^2}+\frac {\exp \left (\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}\right )}{x^2 \left (x-\log ^2(x)\right )}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 28, normalized size = 1.12 \begin {gather*} e^{\frac {144 \left (8+e^x\right ) \log ^4(x)}{5 x^2 \left (x-\log ^2(x)\right )^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(((1152 + 144*E^x)*Log[x]^4)/(5*x^4 - 10*x^3*Log[x]^2 + 5*x^2*Log[x]^4))*((-4608*x - 576*E^x*x)*L
og[x]^3 + (4608*x + E^x*(576*x - 144*x^2))*Log[x]^4 + (-2304 + E^x*(-288 + 144*x))*Log[x]^6))/(-5*x^6 + 15*x^5
*Log[x]^2 - 15*x^4*Log[x]^4 + 5*x^3*Log[x]^6),x]

[Out]

E^((144*(8 + E^x)*Log[x]^4)/(5*x^2*(x - Log[x]^2)^2))

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fricas [A]  time = 0.79, size = 34, normalized size = 1.36 \begin {gather*} e^{\left (\frac {144 \, {\left (e^{x} + 8\right )} \log \relax (x)^{4}}{5 \, {\left (x^{2} \log \relax (x)^{4} - 2 \, x^{3} \log \relax (x)^{2} + x^{4}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((144*x-288)*exp(x)-2304)*log(x)^6+((-144*x^2+576*x)*exp(x)+4608*x)*log(x)^4+(-576*exp(x)*x-4608*x)
*log(x)^3)*exp((144*exp(x)+1152)*log(x)^4/(5*x^2*log(x)^4-10*x^3*log(x)^2+5*x^4))/(5*x^3*log(x)^6-15*x^4*log(x
)^4+15*x^5*log(x)^2-5*x^6),x, algorithm="fricas")

[Out]

e^(144/5*(e^x + 8)*log(x)^4/(x^2*log(x)^4 - 2*x^3*log(x)^2 + x^4))

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giac [B]  time = 0.37, size = 62, normalized size = 2.48 \begin {gather*} e^{\left (\frac {144 \, e^{x} \log \relax (x)^{4}}{5 \, {\left (x^{2} \log \relax (x)^{4} - 2 \, x^{3} \log \relax (x)^{2} + x^{4}\right )}} + \frac {1152 \, \log \relax (x)^{4}}{5 \, {\left (x^{2} \log \relax (x)^{4} - 2 \, x^{3} \log \relax (x)^{2} + x^{4}\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((144*x-288)*exp(x)-2304)*log(x)^6+((-144*x^2+576*x)*exp(x)+4608*x)*log(x)^4+(-576*exp(x)*x-4608*x)
*log(x)^3)*exp((144*exp(x)+1152)*log(x)^4/(5*x^2*log(x)^4-10*x^3*log(x)^2+5*x^4))/(5*x^3*log(x)^6-15*x^4*log(x
)^4+15*x^5*log(x)^2-5*x^6),x, algorithm="giac")

[Out]

e^(144/5*e^x*log(x)^4/(x^2*log(x)^4 - 2*x^3*log(x)^2 + x^4) + 1152/5*log(x)^4/(x^2*log(x)^4 - 2*x^3*log(x)^2 +
x^4))

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maple [A]  time = 0.10, size = 25, normalized size = 1.00

method	result	size
risch	\({\mathrm e}^{\frac {144 \left ({\mathrm e}^{x}+8\right ) \ln \relax (x )^{4}}{5 x^{2} \left (\ln \relax (x )^{2}-x \right )^{2}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((144*x-288)*exp(x)-2304)*ln(x)^6+((-144*x^2+576*x)*exp(x)+4608*x)*ln(x)^4+(-576*exp(x)*x-4608*x)*ln(x)^3
)*exp((144*exp(x)+1152)*ln(x)^4/(5*x^2*ln(x)^4-10*x^3*ln(x)^2+5*x^4))/(5*x^3*ln(x)^6-15*x^4*ln(x)^4+15*x^5*ln(
x)^2-5*x^6),x,method=_RETURNVERBOSE)

[Out]

exp(144/5*(exp(x)+8)*ln(x)^4/x^2/(ln(x)^2-x)^2)

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maxima [B]  time = 4.07, size = 108, normalized size = 4.32 \begin {gather*} e^{\left (\frac {288 \, e^{x}}{5 \, {\left (\log \relax (x)^{4} - x \log \relax (x)^{2}\right )}} + \frac {144 \, e^{x}}{5 \, {\left (\log \relax (x)^{4} - 2 \, x \log \relax (x)^{2} + x^{2}\right )}} + \frac {2304}{5 \, {\left (\log \relax (x)^{4} - x \log \relax (x)^{2}\right )}} + \frac {1152}{5 \, {\left (\log \relax (x)^{4} - 2 \, x \log \relax (x)^{2} + x^{2}\right )}} + \frac {144 \, e^{x}}{5 \, x^{2}} + \frac {1152}{5 \, x^{2}} + \frac {288 \, e^{x}}{5 \, x \log \relax (x)^{2}} + \frac {2304}{5 \, x \log \relax (x)^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((144*x-288)*exp(x)-2304)*log(x)^6+((-144*x^2+576*x)*exp(x)+4608*x)*log(x)^4+(-576*exp(x)*x-4608*x)
*log(x)^3)*exp((144*exp(x)+1152)*log(x)^4/(5*x^2*log(x)^4-10*x^3*log(x)^2+5*x^4))/(5*x^3*log(x)^6-15*x^4*log(x
)^4+15*x^5*log(x)^2-5*x^6),x, algorithm="maxima")

[Out]

e^(288/5*e^x/(log(x)^4 - x*log(x)^2) + 144/5*e^x/(log(x)^4 - 2*x*log(x)^2 + x^2) + 2304/5/(log(x)^4 - x*log(x)
^2) + 1152/5/(log(x)^4 - 2*x*log(x)^2 + x^2) + 144/5*e^x/x^2 + 1152/5/x^2 + 288/5*e^x/(x*log(x)^2) + 2304/5/(x
*log(x)^2))

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mupad [B]  time = 7.10, size = 43, normalized size = 1.72 \begin {gather*} {\mathrm {e}}^{\frac {1152\,{\ln \relax (x)}^4+144\,{\mathrm {e}}^x\,{\ln \relax (x)}^4}{5\,x^4-10\,x^3\,{\ln \relax (x)}^2+5\,x^2\,{\ln \relax (x)}^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((log(x)^4*(144*exp(x) + 1152))/(5*x^2*log(x)^4 - 10*x^3*log(x)^2 + 5*x^4))*(log(x)^6*(exp(x)*(144*x -
288) - 2304) + log(x)^4*(4608*x + exp(x)*(576*x - 144*x^2)) - log(x)^3*(4608*x + 576*x*exp(x))))/(15*x^5*log(
x)^2 - 15*x^4*log(x)^4 + 5*x^3*log(x)^6 - 5*x^6),x)

[Out]

exp((1152*log(x)^4 + 144*exp(x)*log(x)^4)/(5*x^2*log(x)^4 - 10*x^3*log(x)^2 + 5*x^4))

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sympy [A]  time = 1.51, size = 37, normalized size = 1.48 \begin {gather*} e^{\frac {\left (144 e^{x} + 1152\right ) \log {\relax (x )}^{4}}{5 x^{4} - 10 x^{3} \log {\relax (x )}^{2} + 5 x^{2} \log {\relax (x )}^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((144*x-288)*exp(x)-2304)*ln(x)**6+((-144*x**2+576*x)*exp(x)+4608*x)*ln(x)**4+(-576*exp(x)*x-4608*x
)*ln(x)**3)*exp((144*exp(x)+1152)*ln(x)**4/(5*x**2*ln(x)**4-10*x**3*ln(x)**2+5*x**4))/(5*x**3*ln(x)**6-15*x**4
*ln(x)**4+15*x**5*ln(x)**2-5*x**6),x)

[Out]

exp((144*exp(x) + 1152)*log(x)**4/(5*x**4 - 10*x**3*log(x)**2 + 5*x**2*log(x)**4))

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