3.94.97 \(\int \frac {-16 e^{\frac {16}{\log (x)}}+(-3365 x+149462 x^2-2478498 x^3+19518724 x^4) \log ^2(x)+e^{\frac {12}{\log (x)}} (96-2256 x+188 x \log ^2(x))+e^{\frac {8}{\log (x)}} (-272+9008 x-106032 x^2+(-1126 x+26508 x^2) \log ^2(x))+e^{\frac {4}{\log (x)}} (288-12752 x+211312 x^2-1661168 x^3+(3188 x-105656 x^2+1245876 x^3) \log ^2(x))}{x \log ^2(x)} \, dx\)

Optimal. Leaf size=22 \[ x+\left (5+x+\left (-2+e^{\frac {4}{\log (x)}}+47 x\right )^2\right )^2 \]

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Rubi [B]  time = 1.74, antiderivative size = 87, normalized size of antiderivative = 3.95, number of steps used = 7, number of rules used = 3, integrand size = 140, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {6742, 2209, 2288} \begin {gather*} 4879681 x^4-826166 x^3+74731 x^2+2 \left (6627 x^2-563 x+17\right ) e^{\frac {8}{\log (x)}}-4 \left (-103823 x^3+13207 x^2-797 x+18\right ) e^{\frac {4}{\log (x)}}-3365 x+e^{\frac {16}{\log (x)}}-4 (2-47 x) e^{\frac {12}{\log (x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-16*E^(16/Log[x]) + (-3365*x + 149462*x^2 - 2478498*x^3 + 19518724*x^4)*Log[x]^2 + E^(12/Log[x])*(96 - 22
56*x + 188*x*Log[x]^2) + E^(8/Log[x])*(-272 + 9008*x - 106032*x^2 + (-1126*x + 26508*x^2)*Log[x]^2) + E^(4/Log
[x])*(288 - 12752*x + 211312*x^2 - 1661168*x^3 + (3188*x - 105656*x^2 + 1245876*x^3)*Log[x]^2))/(x*Log[x]^2),x
]

[Out]

E^(16/Log[x]) - 4*E^(12/Log[x])*(2 - 47*x) - 3365*x + 74731*x^2 - 826166*x^3 + 4879681*x^4 + 2*E^(8/Log[x])*(1
7 - 563*x + 6627*x^2) - 4*E^(4/Log[x])*(18 - 797*x + 13207*x^2 - 103823*x^3)

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-3365+149462 x-2478498 x^2+19518724 x^3-\frac {16 e^{\frac {16}{\log (x)}}}{x \log ^2(x)}+\frac {4 e^{\frac {12}{\log (x)}} \left (24-564 x+47 x \log ^2(x)\right )}{x \log ^2(x)}+\frac {2 e^{\frac {8}{\log (x)}} \left (-136+4504 x-53016 x^2-563 x \log ^2(x)+13254 x^2 \log ^2(x)\right )}{x \log ^2(x)}+\frac {4 e^{\frac {4}{\log (x)}} \left (72-3188 x+52828 x^2-415292 x^3+797 x \log ^2(x)-26414 x^2 \log ^2(x)+311469 x^3 \log ^2(x)\right )}{x \log ^2(x)}\right ) \, dx\\ &=-3365 x+74731 x^2-826166 x^3+4879681 x^4+2 \int \frac {e^{\frac {8}{\log (x)}} \left (-136+4504 x-53016 x^2-563 x \log ^2(x)+13254 x^2 \log ^2(x)\right )}{x \log ^2(x)} \, dx+4 \int \frac {e^{\frac {12}{\log (x)}} \left (24-564 x+47 x \log ^2(x)\right )}{x \log ^2(x)} \, dx+4 \int \frac {e^{\frac {4}{\log (x)}} \left (72-3188 x+52828 x^2-415292 x^3+797 x \log ^2(x)-26414 x^2 \log ^2(x)+311469 x^3 \log ^2(x)\right )}{x \log ^2(x)} \, dx-16 \int \frac {e^{\frac {16}{\log (x)}}}{x \log ^2(x)} \, dx\\ &=-4 e^{\frac {12}{\log (x)}} (2-47 x)-3365 x+74731 x^2-826166 x^3+4879681 x^4+2 e^{\frac {8}{\log (x)}} \left (17-563 x+6627 x^2\right )-4 e^{\frac {4}{\log (x)}} \left (18-797 x+13207 x^2-103823 x^3\right )-16 \operatorname {Subst}\left (\int \frac {e^{16/x}}{x^2} \, dx,x,\log (x)\right )\\ &=e^{\frac {16}{\log (x)}}-4 e^{\frac {12}{\log (x)}} (2-47 x)-3365 x+74731 x^2-826166 x^3+4879681 x^4+2 e^{\frac {8}{\log (x)}} \left (17-563 x+6627 x^2\right )-4 e^{\frac {4}{\log (x)}} \left (18-797 x+13207 x^2-103823 x^3\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.36, size = 87, normalized size = 3.95 \begin {gather*} e^{\frac {16}{\log (x)}}-e^{\frac {12}{\log (x)}} (8-188 x)-3365 x+74731 x^2-826166 x^3+4879681 x^4-e^{\frac {8}{\log (x)}} (-34-2 x (-563+6627 x))-e^{\frac {4}{\log (x)}} \left (72-4 x \left (797-13207 x+103823 x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16*E^(16/Log[x]) + (-3365*x + 149462*x^2 - 2478498*x^3 + 19518724*x^4)*Log[x]^2 + E^(12/Log[x])*(9
6 - 2256*x + 188*x*Log[x]^2) + E^(8/Log[x])*(-272 + 9008*x - 106032*x^2 + (-1126*x + 26508*x^2)*Log[x]^2) + E^
(4/Log[x])*(288 - 12752*x + 211312*x^2 - 1661168*x^3 + (3188*x - 105656*x^2 + 1245876*x^3)*Log[x]^2))/(x*Log[x
]^2),x]

[Out]

E^(16/Log[x]) - E^(12/Log[x])*(8 - 188*x) - 3365*x + 74731*x^2 - 826166*x^3 + 4879681*x^4 - E^(8/Log[x])*(-34
- 2*x*(-563 + 6627*x)) - E^(4/Log[x])*(72 - 4*x*(797 - 13207*x + 103823*x^2))

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fricas [B]  time = 0.53, size = 83, normalized size = 3.77 \begin {gather*} 4879681 \, x^{4} - 826166 \, x^{3} + 74731 \, x^{2} + 4 \, {\left (47 \, x - 2\right )} e^{\frac {12}{\log \relax (x)}} + 2 \, {\left (6627 \, x^{2} - 563 \, x + 17\right )} e^{\frac {8}{\log \relax (x)}} + 4 \, {\left (103823 \, x^{3} - 13207 \, x^{2} + 797 \, x - 18\right )} e^{\frac {4}{\log \relax (x)}} - 3365 \, x + e^{\frac {16}{\log \relax (x)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(4/log(x))^4+(188*x*log(x)^2-2256*x+96)*exp(4/log(x))^3+((26508*x^2-1126*x)*log(x)^2-106032*
x^2+9008*x-272)*exp(4/log(x))^2+((1245876*x^3-105656*x^2+3188*x)*log(x)^2-1661168*x^3+211312*x^2-12752*x+288)*
exp(4/log(x))+(19518724*x^4-2478498*x^3+149462*x^2-3365*x)*log(x)^2)/x/log(x)^2,x, algorithm="fricas")

[Out]

4879681*x^4 - 826166*x^3 + 74731*x^2 + 4*(47*x - 2)*e^(12/log(x)) + 2*(6627*x^2 - 563*x + 17)*e^(8/log(x)) + 4
*(103823*x^3 - 13207*x^2 + 797*x - 18)*e^(4/log(x)) - 3365*x + e^(16/log(x))

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giac [B]  time = 0.15, size = 119, normalized size = 5.41 \begin {gather*} 4879681 \, x^{4} + 415292 \, x^{3} e^{\frac {4}{\log \relax (x)}} - 826166 \, x^{3} + 13254 \, x^{2} e^{\frac {8}{\log \relax (x)}} - 52828 \, x^{2} e^{\frac {4}{\log \relax (x)}} + 74731 \, x^{2} + 188 \, x e^{\frac {12}{\log \relax (x)}} - 1126 \, x e^{\frac {8}{\log \relax (x)}} + 3188 \, x e^{\frac {4}{\log \relax (x)}} - 3365 \, x + e^{\frac {16}{\log \relax (x)}} - 8 \, e^{\frac {12}{\log \relax (x)}} + 34 \, e^{\frac {8}{\log \relax (x)}} - 72 \, e^{\frac {4}{\log \relax (x)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(4/log(x))^4+(188*x*log(x)^2-2256*x+96)*exp(4/log(x))^3+((26508*x^2-1126*x)*log(x)^2-106032*
x^2+9008*x-272)*exp(4/log(x))^2+((1245876*x^3-105656*x^2+3188*x)*log(x)^2-1661168*x^3+211312*x^2-12752*x+288)*
exp(4/log(x))+(19518724*x^4-2478498*x^3+149462*x^2-3365*x)*log(x)^2)/x/log(x)^2,x, algorithm="giac")

[Out]

4879681*x^4 + 415292*x^3*e^(4/log(x)) - 826166*x^3 + 13254*x^2*e^(8/log(x)) - 52828*x^2*e^(4/log(x)) + 74731*x
^2 + 188*x*e^(12/log(x)) - 1126*x*e^(8/log(x)) + 3188*x*e^(4/log(x)) - 3365*x + e^(16/log(x)) - 8*e^(12/log(x)
) + 34*e^(8/log(x)) - 72*e^(4/log(x))

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maple [B]  time = 0.10, size = 81, normalized size = 3.68

method	result	size
risch	\(4879681 x^{4}+{\mathrm e}^{\frac {16}{\ln \relax (x )}}-826166 x^{3}+74731 x^{2}-3365 x +\left (-8+188 x \right ) {\mathrm e}^{\frac {12}{\ln \relax (x )}}+\left (13254 x^{2}-1126 x +34\right ) {\mathrm e}^{\frac {8}{\ln \relax (x )}}+\left (415292 x^{3}-52828 x^{2}+3188 x -72\right ) {\mathrm e}^{\frac {4}{\ln \relax (x )}}\)	\(81\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-16*exp(4/ln(x))^4+(188*x*ln(x)^2-2256*x+96)*exp(4/ln(x))^3+((26508*x^2-1126*x)*ln(x)^2-106032*x^2+9008*x
-272)*exp(4/ln(x))^2+((1245876*x^3-105656*x^2+3188*x)*ln(x)^2-1661168*x^3+211312*x^2-12752*x+288)*exp(4/ln(x))
+(19518724*x^4-2478498*x^3+149462*x^2-3365*x)*ln(x)^2)/x/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

4879681*x^4+exp(16/ln(x))-826166*x^3+74731*x^2-3365*x+(-8+188*x)*exp(12/ln(x))+(13254*x^2-1126*x+34)*exp(8/ln(
x))+(415292*x^3-52828*x^2+3188*x-72)*exp(4/ln(x))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 4879681 \, x^{4} - 826166 \, x^{3} + 74731 \, x^{2} - 3365 \, x + e^{\frac {16}{\log \relax (x)}} + \int \frac {4 \, {\left (47 \, x \log \relax (x)^{2} - 564 \, x + 24\right )} e^{\frac {12}{\log \relax (x)}}}{x \log \relax (x)^{2}}\,{d x} + \int \frac {2 \, {\left ({\left (13254 \, x^{2} - 563 \, x\right )} \log \relax (x)^{2} - 53016 \, x^{2} + 4504 \, x - 136\right )} e^{\frac {8}{\log \relax (x)}}}{x \log \relax (x)^{2}}\,{d x} + \int -\frac {4 \, {\left (415292 \, x^{3} - {\left (311469 \, x^{3} - 26414 \, x^{2} + 797 \, x\right )} \log \relax (x)^{2} - 52828 \, x^{2} + 3188 \, x - 72\right )} e^{\frac {4}{\log \relax (x)}}}{x \log \relax (x)^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(4/log(x))^4+(188*x*log(x)^2-2256*x+96)*exp(4/log(x))^3+((26508*x^2-1126*x)*log(x)^2-106032*
x^2+9008*x-272)*exp(4/log(x))^2+((1245876*x^3-105656*x^2+3188*x)*log(x)^2-1661168*x^3+211312*x^2-12752*x+288)*
exp(4/log(x))+(19518724*x^4-2478498*x^3+149462*x^2-3365*x)*log(x)^2)/x/log(x)^2,x, algorithm="maxima")

[Out]

4879681*x^4 - 826166*x^3 + 74731*x^2 - 3365*x + e^(16/log(x)) + integrate(4*(47*x*log(x)^2 - 564*x + 24)*e^(12
/log(x))/(x*log(x)^2), x) + integrate(2*((13254*x^2 - 563*x)*log(x)^2 - 53016*x^2 + 4504*x - 136)*e^(8/log(x))
/(x*log(x)^2), x) + integrate(-4*(415292*x^3 - (311469*x^3 - 26414*x^2 + 797*x)*log(x)^2 - 52828*x^2 + 3188*x
- 72)*e^(4/log(x))/(x*log(x)^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} -\int \frac {16\,{\mathrm {e}}^{\frac {16}{\ln \relax (x)}}-{\mathrm {e}}^{\frac {4}{\ln \relax (x)}}\,\left ({\ln \relax (x)}^2\,\left (1245876\,x^3-105656\,x^2+3188\,x\right )-12752\,x+211312\,x^2-1661168\,x^3+288\right )+{\ln \relax (x)}^2\,\left (-19518724\,x^4+2478498\,x^3-149462\,x^2+3365\,x\right )-{\mathrm {e}}^{\frac {12}{\ln \relax (x)}}\,\left (188\,x\,{\ln \relax (x)}^2-2256\,x+96\right )+{\mathrm {e}}^{\frac {8}{\ln \relax (x)}}\,\left ({\ln \relax (x)}^2\,\left (1126\,x-26508\,x^2\right )-9008\,x+106032\,x^2+272\right )}{x\,{\ln \relax (x)}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*exp(16/log(x)) - exp(4/log(x))*(log(x)^2*(3188*x - 105656*x^2 + 1245876*x^3) - 12752*x + 211312*x^2 -
1661168*x^3 + 288) + log(x)^2*(3365*x - 149462*x^2 + 2478498*x^3 - 19518724*x^4) - exp(12/log(x))*(188*x*log(
x)^2 - 2256*x + 96) + exp(8/log(x))*(log(x)^2*(1126*x - 26508*x^2) - 9008*x + 106032*x^2 + 272))/(x*log(x)^2),
x)

[Out]

-int((16*exp(16/log(x)) - exp(4/log(x))*(log(x)^2*(3188*x - 105656*x^2 + 1245876*x^3) - 12752*x + 211312*x^2 -
1661168*x^3 + 288) + log(x)^2*(3365*x - 149462*x^2 + 2478498*x^3 - 19518724*x^4) - exp(12/log(x))*(188*x*log(
x)^2 - 2256*x + 96) + exp(8/log(x))*(log(x)^2*(1126*x - 26508*x^2) - 9008*x + 106032*x^2 + 272))/(x*log(x)^2),
x)

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sympy [B]  time = 5.59, size = 75, normalized size = 3.41 \begin {gather*} 4879681 x^{4} - 826166 x^{3} + 74731 x^{2} - 3365 x + \left (188 x - 8\right ) e^{\frac {12}{\log {\relax (x )}}} + \left (13254 x^{2} - 1126 x + 34\right ) e^{\frac {8}{\log {\relax (x )}}} + \left (415292 x^{3} - 52828 x^{2} + 3188 x - 72\right ) e^{\frac {4}{\log {\relax (x )}}} + e^{\frac {16}{\log {\relax (x )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-16*exp(4/ln(x))**4+(188*x*ln(x)**2-2256*x+96)*exp(4/ln(x))**3+((26508*x**2-1126*x)*ln(x)**2-106032
*x**2+9008*x-272)*exp(4/ln(x))**2+((1245876*x**3-105656*x**2+3188*x)*ln(x)**2-1661168*x**3+211312*x**2-12752*x
+288)*exp(4/ln(x))+(19518724*x**4-2478498*x**3+149462*x**2-3365*x)*ln(x)**2)/x/ln(x)**2,x)

[Out]

4879681*x**4 - 826166*x**3 + 74731*x**2 - 3365*x + (188*x - 8)*exp(12/log(x)) + (13254*x**2 - 1126*x + 34)*exp
(8/log(x)) + (415292*x**3 - 52828*x**2 + 3188*x - 72)*exp(4/log(x)) + exp(16/log(x))

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