3.14.54 \(\int \frac {(e^5+x^2+20 x^6) \log (4)+e^{\frac {4 x}{\log (4)}} (16 x^3+4 x^2 \log (4))+e^{\frac {3 x}{\log (4)}} (-48 x^4-32 x^3 \log (4))+e^{\frac {2 x}{\log (4)}} (48 x^5+72 x^4 \log (4))+e^{\frac {x}{\log (4)}} (-16 x^6-64 x^5 \log (4))}{4 e^{\frac {4 x}{\log (4)}} x^3 \log (4)-16 e^{\frac {3 x}{\log (4)}} x^4 \log (4)+24 e^{\frac {2 x}{\log (4)}} x^5 \log (4)-16 e^{\frac {x}{\log (4)}} x^6 \log (4)+(-e^5 x+x^3+4 x^7) \log (4)} \, dx\)

Optimal. Leaf size=28 \[ \log \left (-\frac {e^5}{x}+x+4 x \left (-e^{\frac {x}{\log (4)}}+x\right )^4\right ) \]

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Rubi [F]  time = 19.48, 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 (e^5+x^2+20 x^6\right ) \log (4)+e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )+e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )+e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )+e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )}{4 e^{\frac {4 x}{\log (4)}} x^3 \log (4)-16 e^{\frac {3 x}{\log (4)}} x^4 \log (4)+24 e^{\frac {2 x}{\log (4)}} x^5 \log (4)-16 e^{\frac {x}{\log (4)}} x^6 \log (4)+\left (-e^5 x+x^3+4 x^7\right ) \log (4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((E^5 + x^2 + 20*x^6)*Log[4] + E^((4*x)/Log[4])*(16*x^3 + 4*x^2*Log[4]) + E^((3*x)/Log[4])*(-48*x^4 - 32*x
^3*Log[4]) + E^((2*x)/Log[4])*(48*x^5 + 72*x^4*Log[4]) + E^(x/Log[4])*(-16*x^6 - 64*x^5*Log[4]))/(4*E^((4*x)/L
og[4])*x^3*Log[4] - 16*E^((3*x)/Log[4])*x^4*Log[4] + 24*E^((2*x)/Log[4])*x^5*Log[4] - 16*E^(x/Log[4])*x^6*Log[
4] + (-(E^5*x) + x^3 + 4*x^7)*Log[4]),x]

[Out]

(4*x)/Log[4] + Log[x] - (4*E^5*Defer[Int][(E^5 - x^2 - 4*E^((4*x)/Log[4])*x^2 + 16*E^((3*x)/Log[4])*x^3 - 24*E
^((2*x)/Log[4])*x^4 + 16*E^(x/Log[4])*x^5 - 4*x^6)^(-1), x])/Log[4] - 2*E^5*Defer[Int][1/(x*(E^5 - x^2 - 4*E^(
(4*x)/Log[4])*x^2 + 16*E^((3*x)/Log[4])*x^3 - 24*E^((2*x)/Log[4])*x^4 + 16*E^(x/Log[4])*x^5 - 4*x^6)), x] - (4
*Defer[Int][x^2/(-E^5 + x^2 + 4*E^((4*x)/Log[4])*x^2 - 16*E^((3*x)/Log[4])*x^3 + 24*E^((2*x)/Log[4])*x^4 - 16*
E^(x/Log[4])*x^5 + 4*x^6), x])/Log[4] - 16*Defer[Int][(E^((3*x)/Log[4])*x^2)/(-E^5 + x^2 + 4*E^((4*x)/Log[4])*
x^2 - 16*E^((3*x)/Log[4])*x^3 + 24*E^((2*x)/Log[4])*x^4 - 16*E^(x/Log[4])*x^5 + 4*x^6), x] + 48*Defer[Int][(E^
((2*x)/Log[4])*x^3)/(-E^5 + x^2 + 4*E^((4*x)/Log[4])*x^2 - 16*E^((3*x)/Log[4])*x^3 + 24*E^((2*x)/Log[4])*x^4 -
16*E^(x/Log[4])*x^5 + 4*x^6), x] + (16*Defer[Int][(E^((3*x)/Log[4])*x^3)/(-E^5 + x^2 + 4*E^((4*x)/Log[4])*x^2
- 16*E^((3*x)/Log[4])*x^3 + 24*E^((2*x)/Log[4])*x^4 - 16*E^(x/Log[4])*x^5 + 4*x^6), x])/Log[4] - 48*Defer[Int
][(E^(x/Log[4])*x^4)/(-E^5 + x^2 + 4*E^((4*x)/Log[4])*x^2 - 16*E^((3*x)/Log[4])*x^3 + 24*E^((2*x)/Log[4])*x^4
- 16*E^(x/Log[4])*x^5 + 4*x^6), x] - (48*Defer[Int][(E^((2*x)/Log[4])*x^4)/(-E^5 + x^2 + 4*E^((4*x)/Log[4])*x^
2 - 16*E^((3*x)/Log[4])*x^3 + 24*E^((2*x)/Log[4])*x^4 - 16*E^(x/Log[4])*x^5 + 4*x^6), x])/Log[4] + 16*Defer[In
t][x^5/(-E^5 + x^2 + 4*E^((4*x)/Log[4])*x^2 - 16*E^((3*x)/Log[4])*x^3 + 24*E^((2*x)/Log[4])*x^4 - 16*E^(x/Log[
4])*x^5 + 4*x^6), x] + (48*Defer[Int][(E^(x/Log[4])*x^5)/(-E^5 + x^2 + 4*E^((4*x)/Log[4])*x^2 - 16*E^((3*x)/Lo
g[4])*x^3 + 24*E^((2*x)/Log[4])*x^4 - 16*E^(x/Log[4])*x^5 + 4*x^6), x])/Log[4] - (16*Defer[Int][x^6/(-E^5 + x^
2 + 4*E^((4*x)/Log[4])*x^2 - 16*E^((3*x)/Log[4])*x^3 + 24*E^((2*x)/Log[4])*x^4 - 16*E^(x/Log[4])*x^5 + 4*x^6),
x])/Log[4]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-\left (\left (e^5+x^2+20 x^6\right ) \log (4)\right )-e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )-e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )-e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )-e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )}{x \left (e^5-x^2-4 e^{\frac {4 x}{\log (4)}} x^2+16 e^{\frac {3 x}{\log (4)}} x^3-24 e^{\frac {2 x}{\log (4)}} x^4+16 e^{\frac {x}{\log (4)}} x^5-4 x^6\right ) \log (4)} \, dx\\ &=\frac {\int \frac {-\left (\left (e^5+x^2+20 x^6\right ) \log (4)\right )-e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )-e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )-e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )-e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )}{x \left (e^5-x^2-4 e^{\frac {4 x}{\log (4)}} x^2+16 e^{\frac {3 x}{\log (4)}} x^3-24 e^{\frac {2 x}{\log (4)}} x^4+16 e^{\frac {x}{\log (4)}} x^5-4 x^6\right )} \, dx}{\log (4)}\\ &=\frac {\int \left (\frac {4 x+\log (4)}{x}-\frac {2 \left (-2 e^5 x+2 x^3-8 e^{\frac {3 x}{\log (4)}} x^4+24 e^{\frac {2 x}{\log (4)}} x^5-24 e^{\frac {x}{\log (4)}} x^6+8 x^7-e^5 \log (4)+8 e^{\frac {3 x}{\log (4)}} x^3 \log (4)-24 e^{\frac {2 x}{\log (4)}} x^4 \log (4)+24 e^{\frac {x}{\log (4)}} x^5 \log (4)-8 x^6 \log (4)\right )}{x \left (-e^5+x^2+4 e^{\frac {4 x}{\log (4)}} x^2-16 e^{\frac {3 x}{\log (4)}} x^3+24 e^{\frac {2 x}{\log (4)}} x^4-16 e^{\frac {x}{\log (4)}} x^5+4 x^6\right )}\right ) \, dx}{\log (4)}\\ &=\frac {\int \frac {4 x+\log (4)}{x} \, dx}{\log (4)}-\frac {2 \int \frac {-2 e^5 x+2 x^3-8 e^{\frac {3 x}{\log (4)}} x^4+24 e^{\frac {2 x}{\log (4)}} x^5-24 e^{\frac {x}{\log (4)}} x^6+8 x^7-e^5 \log (4)+8 e^{\frac {3 x}{\log (4)}} x^3 \log (4)-24 e^{\frac {2 x}{\log (4)}} x^4 \log (4)+24 e^{\frac {x}{\log (4)}} x^5 \log (4)-8 x^6 \log (4)}{x \left (-e^5+x^2+4 e^{\frac {4 x}{\log (4)}} x^2-16 e^{\frac {3 x}{\log (4)}} x^3+24 e^{\frac {2 x}{\log (4)}} x^4-16 e^{\frac {x}{\log (4)}} x^5+4 x^6\right )} \, dx}{\log (4)}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [F]  time = 1.81, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e^5+x^2+20 x^6\right ) \log (4)+e^{\frac {4 x}{\log (4)}} \left (16 x^3+4 x^2 \log (4)\right )+e^{\frac {3 x}{\log (4)}} \left (-48 x^4-32 x^3 \log (4)\right )+e^{\frac {2 x}{\log (4)}} \left (48 x^5+72 x^4 \log (4)\right )+e^{\frac {x}{\log (4)}} \left (-16 x^6-64 x^5 \log (4)\right )}{4 e^{\frac {4 x}{\log (4)}} x^3 \log (4)-16 e^{\frac {3 x}{\log (4)}} x^4 \log (4)+24 e^{\frac {2 x}{\log (4)}} x^5 \log (4)-16 e^{\frac {x}{\log (4)}} x^6 \log (4)+\left (-e^5 x+x^3+4 x^7\right ) \log (4)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((E^5 + x^2 + 20*x^6)*Log[4] + E^((4*x)/Log[4])*(16*x^3 + 4*x^2*Log[4]) + E^((3*x)/Log[4])*(-48*x^4
- 32*x^3*Log[4]) + E^((2*x)/Log[4])*(48*x^5 + 72*x^4*Log[4]) + E^(x/Log[4])*(-16*x^6 - 64*x^5*Log[4]))/(4*E^((
4*x)/Log[4])*x^3*Log[4] - 16*E^((3*x)/Log[4])*x^4*Log[4] + 24*E^((2*x)/Log[4])*x^5*Log[4] - 16*E^(x/Log[4])*x^
6*Log[4] + (-(E^5*x) + x^3 + 4*x^7)*Log[4]),x]

[Out]

Integrate[((E^5 + x^2 + 20*x^6)*Log[4] + E^((4*x)/Log[4])*(16*x^3 + 4*x^2*Log[4]) + E^((3*x)/Log[4])*(-48*x^4
- 32*x^3*Log[4]) + E^((2*x)/Log[4])*(48*x^5 + 72*x^4*Log[4]) + E^(x/Log[4])*(-16*x^6 - 64*x^5*Log[4]))/(4*E^((
4*x)/Log[4])*x^3*Log[4] - 16*E^((3*x)/Log[4])*x^4*Log[4] + 24*E^((2*x)/Log[4])*x^5*Log[4] - 16*E^(x/Log[4])*x^
6*Log[4] + (-(E^5*x) + x^3 + 4*x^7)*Log[4]), x]

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fricas [B]  time = 0.79, size = 72, normalized size = 2.57 \begin {gather*} \log \relax (x) + \log \left (\frac {4 \, x^{6} - 16 \, x^{5} e^{\left (\frac {x}{2 \, \log \relax (2)}\right )} + 24 \, x^{4} e^{\frac {x}{\log \relax (2)}} - 16 \, x^{3} e^{\left (\frac {3 \, x}{2 \, \log \relax (2)}\right )} + 4 \, x^{2} e^{\left (\frac {2 \, x}{\log \relax (2)}\right )} + x^{2} - e^{5}}{x^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2*log(2)+16*x^3)*exp(1/2*x/log(2))^4+(-64*x^3*log(2)-48*x^4)*exp(1/2*x/log(2))^3+(144*x^4*log(
2)+48*x^5)*exp(1/2*x/log(2))^2+(-128*x^5*log(2)-16*x^6)*exp(1/2*x/log(2))+2*(exp(5)+20*x^6+x^2)*log(2))/(8*x^3
*log(2)*exp(1/2*x/log(2))^4-32*x^4*log(2)*exp(1/2*x/log(2))^3+48*x^5*log(2)*exp(1/2*x/log(2))^2-32*x^6*log(2)*
exp(1/2*x/log(2))+2*(-x*exp(5)+4*x^7+x^3)*log(2)),x, algorithm="fricas")

[Out]

log(x) + log((4*x^6 - 16*x^5*e^(1/2*x/log(2)) + 24*x^4*e^(x/log(2)) - 16*x^3*e^(3/2*x/log(2)) + 4*x^2*e^(2*x/l
og(2)) + x^2 - e^5)/x^2)

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giac [B]  time = 5.56, size = 70, normalized size = 2.50 \begin {gather*} \log \left (4 \, x^{6} - 16 \, x^{5} e^{\left (\frac {x}{2 \, \log \relax (2)}\right )} + 24 \, x^{4} e^{\frac {x}{\log \relax (2)}} - 16 \, x^{3} e^{\left (\frac {3 \, x}{2 \, \log \relax (2)}\right )} + 4 \, x^{2} e^{\left (\frac {2 \, x}{\log \relax (2)}\right )} + x^{2} - e^{5}\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2*log(2)+16*x^3)*exp(1/2*x/log(2))^4+(-64*x^3*log(2)-48*x^4)*exp(1/2*x/log(2))^3+(144*x^4*log(
2)+48*x^5)*exp(1/2*x/log(2))^2+(-128*x^5*log(2)-16*x^6)*exp(1/2*x/log(2))+2*(exp(5)+20*x^6+x^2)*log(2))/(8*x^3
*log(2)*exp(1/2*x/log(2))^4-32*x^4*log(2)*exp(1/2*x/log(2))^3+48*x^5*log(2)*exp(1/2*x/log(2))^2-32*x^6*log(2)*
exp(1/2*x/log(2))+2*(-x*exp(5)+4*x^7+x^3)*log(2)),x, algorithm="giac")

[Out]

log(4*x^6 - 16*x^5*e^(1/2*x/log(2)) + 24*x^4*e^(x/log(2)) - 16*x^3*e^(3/2*x/log(2)) + 4*x^2*e^(2*x/log(2)) + x
^2 - e^5) - log(x)

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maple [B]  time = 0.17, size = 68, normalized size = 2.43

method	result	size
risch	\(\ln \relax (x )+\ln \left ({\mathrm e}^{\frac {2 x}{\ln \relax (2)}}-4 x \,{\mathrm e}^{\frac {3 x}{2 \ln \relax (2)}}+6 x^{2} {\mathrm e}^{\frac {x}{\ln \relax (2)}}-4 x^{3} {\mathrm e}^{\frac {x}{2 \ln \relax (2)}}-\frac {-4 x^{6}-x^{2}+{\mathrm e}^{5}}{4 x^{2}}\right )\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((8*x^2*ln(2)+16*x^3)*exp(1/2*x/ln(2))^4+(-64*x^3*ln(2)-48*x^4)*exp(1/2*x/ln(2))^3+(144*x^4*ln(2)+48*x^5)*
exp(1/2*x/ln(2))^2+(-128*x^5*ln(2)-16*x^6)*exp(1/2*x/ln(2))+2*(exp(5)+20*x^6+x^2)*ln(2))/(8*x^3*ln(2)*exp(1/2*
x/ln(2))^4-32*x^4*ln(2)*exp(1/2*x/ln(2))^3+48*x^5*ln(2)*exp(1/2*x/ln(2))^2-32*x^6*ln(2)*exp(1/2*x/ln(2))+2*(-x
*exp(5)+4*x^7+x^3)*ln(2)),x,method=_RETURNVERBOSE)

[Out]

ln(x)+ln(exp(2*x/ln(2))-4*x*exp(3/2*x/ln(2))+6*x^2*exp(x/ln(2))-4*x^3*exp(1/2*x/ln(2))-1/4*(-4*x^6-x^2+exp(5))
/x^2)

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maxima [B]  time = 0.71, size = 75, normalized size = 2.68 \begin {gather*} 2 \, \log \relax (x) + \log \left (-\frac {4 \, x^{6} - 16 \, x^{5} e^{\left (\frac {x}{2 \, \log \relax (2)}\right )} + 24 \, x^{4} e^{\frac {x}{\log \relax (2)}} - 16 \, x^{3} e^{\left (\frac {3 \, x}{2 \, \log \relax (2)}\right )} + 4 \, x^{2} e^{\left (\frac {2 \, x}{\log \relax (2)}\right )} + x^{2} - e^{5}}{16 \, x^{3}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x^2*log(2)+16*x^3)*exp(1/2*x/log(2))^4+(-64*x^3*log(2)-48*x^4)*exp(1/2*x/log(2))^3+(144*x^4*log(
2)+48*x^5)*exp(1/2*x/log(2))^2+(-128*x^5*log(2)-16*x^6)*exp(1/2*x/log(2))+2*(exp(5)+20*x^6+x^2)*log(2))/(8*x^3
*log(2)*exp(1/2*x/log(2))^4-32*x^4*log(2)*exp(1/2*x/log(2))^3+48*x^5*log(2)*exp(1/2*x/log(2))^2-32*x^6*log(2)*
exp(1/2*x/log(2))+2*(-x*exp(5)+4*x^7+x^3)*log(2)),x, algorithm="maxima")

[Out]

2*log(x) + log(-1/16*(4*x^6 - 16*x^5*e^(1/2*x/log(2)) + 24*x^4*e^(x/log(2)) - 16*x^3*e^(3/2*x/log(2)) + 4*x^2*
e^(2*x/log(2)) + x^2 - e^5)/x^3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {2\,\ln \relax (2)\,\left (20\,x^6+x^2+{\mathrm {e}}^5\right )+{\mathrm {e}}^{\frac {2\,x}{\ln \relax (2)}}\,\left (16\,x^3+8\,\ln \relax (2)\,x^2\right )-{\mathrm {e}}^{\frac {3\,x}{2\,\ln \relax (2)}}\,\left (48\,x^4+64\,\ln \relax (2)\,x^3\right )-{\mathrm {e}}^{\frac {x}{2\,\ln \relax (2)}}\,\left (16\,x^6+128\,\ln \relax (2)\,x^5\right )+{\mathrm {e}}^{\frac {x}{\ln \relax (2)}}\,\left (48\,x^5+144\,\ln \relax (2)\,x^4\right )}{2\,\ln \relax (2)\,\left (4\,x^7+x^3-{\mathrm {e}}^5\,x\right )+8\,x^3\,{\mathrm {e}}^{\frac {2\,x}{\ln \relax (2)}}\,\ln \relax (2)+48\,x^5\,{\mathrm {e}}^{\frac {x}{\ln \relax (2)}}\,\ln \relax (2)-32\,x^4\,{\mathrm {e}}^{\frac {3\,x}{2\,\ln \relax (2)}}\,\ln \relax (2)-32\,x^6\,{\mathrm {e}}^{\frac {x}{2\,\ln \relax (2)}}\,\ln \relax (2)} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(2)*(exp(5) + x^2 + 20*x^6) + exp((2*x)/log(2))*(8*x^2*log(2) + 16*x^3) - exp((3*x)/(2*log(2)))*(64*
x^3*log(2) + 48*x^4) - exp(x/(2*log(2)))*(128*x^5*log(2) + 16*x^6) + exp(x/log(2))*(144*x^4*log(2) + 48*x^5))/
(2*log(2)*(x^3 - x*exp(5) + 4*x^7) + 8*x^3*exp((2*x)/log(2))*log(2) + 48*x^5*exp(x/log(2))*log(2) - 32*x^4*exp
((3*x)/(2*log(2)))*log(2) - 32*x^6*exp(x/(2*log(2)))*log(2)),x)

[Out]

int((2*log(2)*(exp(5) + x^2 + 20*x^6) + exp((2*x)/log(2))*(8*x^2*log(2) + 16*x^3) - exp((3*x)/(2*log(2)))*(64*
x^3*log(2) + 48*x^4) - exp(x/(2*log(2)))*(128*x^5*log(2) + 16*x^6) + exp(x/log(2))*(144*x^4*log(2) + 48*x^5))/
(2*log(2)*(x^3 - x*exp(5) + 4*x^7) + 8*x^3*exp((2*x)/log(2))*log(2) + 48*x^5*exp(x/log(2))*log(2) - 32*x^4*exp
((3*x)/(2*log(2)))*log(2) - 32*x^6*exp(x/(2*log(2)))*log(2)), x)

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sympy [B]  time = 1.21, size = 68, normalized size = 2.43 \begin {gather*} \log {\relax (x )} + \log {\left (- 4 x^{3} e^{\frac {x}{2 \log {\relax (2 )}}} + 6 x^{2} e^{\frac {x}{\log {\relax (2 )}}} - 4 x e^{\frac {3 x}{2 \log {\relax (2 )}}} + e^{\frac {2 x}{\log {\relax (2 )}}} + \frac {4 x^{6} + x^{2} - e^{5}}{4 x^{2}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((8*x**2*ln(2)+16*x**3)*exp(1/2*x/ln(2))**4+(-64*x**3*ln(2)-48*x**4)*exp(1/2*x/ln(2))**3+(144*x**4*l
n(2)+48*x**5)*exp(1/2*x/ln(2))**2+(-128*x**5*ln(2)-16*x**6)*exp(1/2*x/ln(2))+2*(exp(5)+20*x**6+x**2)*ln(2))/(8
*x**3*ln(2)*exp(1/2*x/ln(2))**4-32*x**4*ln(2)*exp(1/2*x/ln(2))**3+48*x**5*ln(2)*exp(1/2*x/ln(2))**2-32*x**6*ln
(2)*exp(1/2*x/ln(2))+2*(-x*exp(5)+4*x**7+x**3)*ln(2)),x)

[Out]

log(x) + log(-4*x**3*exp(x/(2*log(2))) + 6*x**2*exp(x/log(2)) - 4*x*exp(3*x/(2*log(2))) + exp(2*x/log(2)) + (4
*x**6 + x**2 - exp(5))/(4*x**2))

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