3.60.99 \(\int \frac {e^{-6+2 x} (e^{6-2 x} (-10+x+x^3) \log ^2(2)+4^{2 e^{-6+2 x}} (-2 e^{6-2 x}+4 x \log (4)))}{x^3 \log ^2(2)} \, dx\)

Optimal. Leaf size=33 \[ \frac {x^2+\frac {5-x+\frac {4^{2 e^{-6+2 x}}}{\log ^2(2)}}{x}}{x} \]

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Rubi [F]  time = 0.96, 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 {e^{-6+2 x} \left (e^{6-2 x} \left (-10+x+x^3\right ) \log ^2(2)+4^{2 e^{-6+2 x}} \left (-2 e^{6-2 x}+4 x \log (4)\right )\right )}{x^3 \log ^2(2)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-6 + 2*x)*(E^(6 - 2*x)*(-10 + x + x^3)*Log[2]^2 + 4^(2*E^(-6 + 2*x))*(-2*E^(6 - 2*x) + 4*x*Log[4])))/(
x^3*Log[2]^2),x]

[Out]

5/x^2 - x^(-1) + x - Defer[Int][2^(1 + 4*E^(-6 + 2*x))/x^3, x]/Log[2]^2 + (Log[4]*Defer[Int][(4^(1 + 2*E^(-6 +
2*x))*E^(-6 + 2*x))/x^2, x])/Log[2]^2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {e^{-6+2 x} \left (e^{6-2 x} \left (-10+x+x^3\right ) \log ^2(2)+4^{2 e^{-6+2 x}} \left (-2 e^{6-2 x}+4 x \log (4)\right )\right )}{x^3} \, dx}{\log ^2(2)}\\ &=\frac {\int \left (\frac {-2^{1+4 e^{-6+2 x}}-10 \log ^2(2)+x \log ^2(2)+x^3 \log ^2(2)}{x^3}+\frac {4^{1+2 e^{-6+2 x}} e^{-6+2 x} \log (4)}{x^2}\right ) \, dx}{\log ^2(2)}\\ &=\frac {\int \frac {-2^{1+4 e^{-6+2 x}}-10 \log ^2(2)+x \log ^2(2)+x^3 \log ^2(2)}{x^3} \, dx}{\log ^2(2)}+\frac {\log (4) \int \frac {4^{1+2 e^{-6+2 x}} e^{-6+2 x}}{x^2} \, dx}{\log ^2(2)}\\ &=\frac {\int \left (-\frac {2^{1+4 e^{-6+2 x}}}{x^3}+\frac {\left (-10+x+x^3\right ) \log ^2(2)}{x^3}\right ) \, dx}{\log ^2(2)}+\frac {\log (4) \int \frac {4^{1+2 e^{-6+2 x}} e^{-6+2 x}}{x^2} \, dx}{\log ^2(2)}\\ &=-\frac {\int \frac {2^{1+4 e^{-6+2 x}}}{x^3} \, dx}{\log ^2(2)}+\frac {\log (4) \int \frac {4^{1+2 e^{-6+2 x}} e^{-6+2 x}}{x^2} \, dx}{\log ^2(2)}+\int \frac {-10+x+x^3}{x^3} \, dx\\ &=-\frac {\int \frac {2^{1+4 e^{-6+2 x}}}{x^3} \, dx}{\log ^2(2)}+\frac {\log (4) \int \frac {4^{1+2 e^{-6+2 x}} e^{-6+2 x}}{x^2} \, dx}{\log ^2(2)}+\int \left (1-\frac {10}{x^3}+\frac {1}{x^2}\right ) \, dx\\ &=\frac {5}{x^2}-\frac {1}{x}+x-\frac {\int \frac {2^{1+4 e^{-6+2 x}}}{x^3} \, dx}{\log ^2(2)}+\frac {\log (4) \int \frac {4^{1+2 e^{-6+2 x}} e^{-6+2 x}}{x^2} \, dx}{\log ^2(2)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 35, normalized size = 1.06 \begin {gather*} \frac {5}{x^2}-\frac {1}{x}+x+\frac {2^{-1+4 e^{-6+2 x}} \log (4)}{x^2 \log ^3(2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-6 + 2*x)*(E^(6 - 2*x)*(-10 + x + x^3)*Log[2]^2 + 4^(2*E^(-6 + 2*x))*(-2*E^(6 - 2*x) + 4*x*Log[4
])))/(x^3*Log[2]^2),x]

[Out]

5/x^2 - x^(-1) + x + (2^(-1 + 4*E^(-6 + 2*x))*Log[4])/(x^2*Log[2]^3)

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fricas [A]  time = 0.75, size = 32, normalized size = 0.97 \begin {gather*} \frac {{\left (x^{3} - x + 5\right )} \log \relax (2)^{2} + 2^{4 \, e^{\left (2 \, x - 6\right )}}}{x^{2} \log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(3-x)^2+8*x*log(2))*exp(2*log(2)/exp(3-x)^2)^2+(x^3+x-10)*log(2)^2*exp(3-x)^2)/x^3/log(2)^2/
exp(3-x)^2,x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left ({\left (x^{3} + x - 10\right )} e^{\left (-2 \, x + 6\right )} \log \relax (2)^{2} + 2 \, {\left (4 \, x \log \relax (2) - e^{\left (-2 \, x + 6\right )}\right )} 2^{4 \, e^{\left (2 \, x - 6\right )}}\right )} e^{\left (2 \, x - 6\right )}}{x^{3} \log \relax (2)^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(3-x)^2+8*x*log(2))*exp(2*log(2)/exp(3-x)^2)^2+(x^3+x-10)*log(2)^2*exp(3-x)^2)/x^3/log(2)^2/
exp(3-x)^2,x, algorithm="giac")

[Out]

integrate(((x^3 + x - 10)*e^(-2*x + 6)*log(2)^2 + 2*(4*x*log(2) - e^(-2*x + 6))*2^(4*e^(2*x - 6)))*e^(2*x - 6)
/(x^3*log(2)^2), x)

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maple [A]  time = 0.05, size = 43, normalized size = 1.30

method	result	size
risch	\(x +\frac {-x \ln \relax (2)^{2}+5 \ln \relax (2)^{2}}{\ln \relax (2)^{2} x^{2}}+\frac {4^{2 \,{\mathrm e}^{2 x -6}}}{\ln \relax (2)^{2} x^{2}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*exp(3-x)^2+8*x*ln(2))*exp(2*ln(2)/exp(3-x)^2)^2+(x^3+x-10)*ln(2)^2*exp(3-x)^2)/x^3/ln(2)^2/exp(3-x)^2
,x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.48, size = 44, normalized size = 1.33 \begin {gather*} \frac {x \log \relax (2)^{2} - \frac {\log \relax (2)^{2}}{x} + \frac {5 \, \log \relax (2)^{2}}{x^{2}} + \frac {2^{4 \, e^{\left (2 \, x - 6\right )}}}{x^{2}}}{\log \relax (2)^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(3-x)^2+8*x*log(2))*exp(2*log(2)/exp(3-x)^2)^2+(x^3+x-10)*log(2)^2*exp(3-x)^2)/x^3/log(2)^2/
exp(3-x)^2,x, algorithm="maxima")

[Out]

(x*log(2)^2 - log(2)^2/x + 5*log(2)^2/x^2 + 2^(4*e^(2*x - 6))/x^2)/log(2)^2

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mupad [B]  time = 4.40, size = 34, normalized size = 1.03 \begin {gather*} x+\frac {2^{4\,{\mathrm {e}}^{2\,x-6}}-x\,{\ln \relax (2)}^2+5\,{\ln \relax (2)}^2}{x^2\,{\ln \relax (2)}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(2*x - 6)*(exp(4*exp(2*x - 6)*log(2))*(2*exp(6 - 2*x) - 8*x*log(2)) - exp(6 - 2*x)*log(2)^2*(x + x^3
- 10)))/(x^3*log(2)^2),x)

[Out]

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

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sympy [A]  time = 0.18, size = 29, normalized size = 0.88 \begin {gather*} x + \frac {5 - x}{x^{2}} + \frac {e^{4 e^{2 x - 6} \log {\relax (2 )}}}{x^{2} \log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*exp(3-x)**2+8*x*ln(2))*exp(2*ln(2)/exp(3-x)**2)**2+(x**3+x-10)*ln(2)**2*exp(3-x)**2)/x**3/ln(2)
**2/exp(3-x)**2,x)

[Out]

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

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