3.44.25 \(\int \frac {64 x^2+27 e^{\frac {3 (-1+2 x)}{x}} x^4-81 e^{\frac {2 (-1+2 x)}{x}} x^5-27 x^7+128 x^2 \log (2)+64 x^2 \log ^2(2)+e^{\frac {-1+2 x}{x}} (-32-32 x+81 x^6+(-64-64 x) \log (2)+(-32-32 x) \log ^2(2))}{9 e^{\frac {3 (-1+2 x)}{x}} x^4-27 e^{\frac {2 (-1+2 x)}{x}} x^5+27 e^{\frac {-1+2 x}{x}} x^6-9 x^7} \, dx\)

Optimal. Leaf size=36 \[ 3 x+\frac {(4+4 \log (2))^2}{9 \left (e^{\frac {-1+2 x}{x}}-x\right )^2 x^2} \]

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Rubi [F]  time = 2.72, 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 {64 x^2+27 e^{\frac {3 (-1+2 x)}{x}} x^4-81 e^{\frac {2 (-1+2 x)}{x}} x^5-27 x^7+128 x^2 \log (2)+64 x^2 \log ^2(2)+e^{\frac {-1+2 x}{x}} \left (-32-32 x+81 x^6+(-64-64 x) \log (2)+(-32-32 x) \log ^2(2)\right )}{9 e^{\frac {3 (-1+2 x)}{x}} x^4-27 e^{\frac {2 (-1+2 x)}{x}} x^5+27 e^{\frac {-1+2 x}{x}} x^6-9 x^7} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(64*x^2 + 27*E^((3*(-1 + 2*x))/x)*x^4 - 81*E^((2*(-1 + 2*x))/x)*x^5 - 27*x^7 + 128*x^2*Log[2] + 64*x^2*Log
[2]^2 + E^((-1 + 2*x)/x)*(-32 - 32*x + 81*x^6 + (-64 - 64*x)*Log[2] + (-32 - 32*x)*Log[2]^2))/(9*E^((3*(-1 + 2
*x))/x)*x^4 - 27*E^((2*(-1 + 2*x))/x)*x^5 + 27*E^((-1 + 2*x)/x)*x^6 - 9*x^7),x]

[Out]

3*x + (16*(1 + Log[2])^2)/(9*x^4) + (32*E^6*(1 + Log[2])^2*Defer[Int][1/(x^6*(-E^2 + E^x^(-1)*x)^3), x])/9 - (
32*E^6*(1 + Log[2])^2*Defer[Int][1/(x^5*(-E^2 + E^x^(-1)*x)^3), x])/9 + (64*E^4*(1 + Log[2])^2*Defer[Int][1/(x
^6*(-E^2 + E^x^(-1)*x)^2), x])/9 - (128*E^4*(1 + Log[2])^2*Defer[Int][1/(x^5*(-E^2 + E^x^(-1)*x)^2), x])/9 + (
32*E^2*(1 + Log[2])^2*Defer[Int][1/(x^6*(-E^2 + E^x^(-1)*x)), x])/9 - (160*E^2*(1 + Log[2])^2*Defer[Int][1/(x^
5*(-E^2 + E^x^(-1)*x)), x])/9

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {27 e^{\frac {3 (-1+2 x)}{x}} x^4-81 e^{\frac {2 (-1+2 x)}{x}} x^5-27 x^7+64 x^2 \log ^2(2)+x^2 (64+128 \log (2))+e^{\frac {-1+2 x}{x}} \left (-32-32 x+81 x^6+(-64-64 x) \log (2)+(-32-32 x) \log ^2(2)\right )}{9 e^{\frac {3 (-1+2 x)}{x}} x^4-27 e^{\frac {2 (-1+2 x)}{x}} x^5+27 e^{\frac {-1+2 x}{x}} x^6-9 x^7} \, dx\\ &=\int \frac {27 e^{\frac {3 (-1+2 x)}{x}} x^4-81 e^{\frac {2 (-1+2 x)}{x}} x^5-27 x^7+x^2 \left (64+128 \log (2)+64 \log ^2(2)\right )+e^{\frac {-1+2 x}{x}} \left (-32-32 x+81 x^6+(-64-64 x) \log (2)+(-32-32 x) \log ^2(2)\right )}{9 e^{\frac {3 (-1+2 x)}{x}} x^4-27 e^{\frac {2 (-1+2 x)}{x}} x^5+27 e^{\frac {-1+2 x}{x}} x^6-9 x^7} \, dx\\ &=\int \frac {27 e^6 x^4-81 e^{4+\frac {1}{x}} x^5-e^{3/x} x^2 \left (27 x^5-64 (1+\log (2))^2\right )-e^{2+\frac {2}{x}} \left (-81 x^6+32 (1+\log (2))^2+32 x (1+\log (2))^2\right )}{9 x^4 \left (e^2-e^{\frac {1}{x}} x\right )^3} \, dx\\ &=\frac {1}{9} \int \frac {27 e^6 x^4-81 e^{4+\frac {1}{x}} x^5-e^{3/x} x^2 \left (27 x^5-64 (1+\log (2))^2\right )-e^{2+\frac {2}{x}} \left (-81 x^6+32 (1+\log (2))^2+32 x (1+\log (2))^2\right )}{x^4 \left (e^2-e^{\frac {1}{x}} x\right )^3} \, dx\\ &=\frac {1}{9} \int \left (\frac {32 e^6 (-1+x) (1+\log (2))^2}{x^6 \left (e^2-e^{\frac {1}{x}} x\right )^3}-\frac {64 e^4 (-1+2 x) (1+\log (2))^2}{x^6 \left (e^2-e^{\frac {1}{x}} x\right )^2}+\frac {32 e^2 (-1+5 x) (1+\log (2))^2}{x^6 \left (e^2-e^{\frac {1}{x}} x\right )}+\frac {-64+27 x^5-128 \log (2)-64 \log ^2(2)}{x^5}\right ) \, dx\\ &=\frac {1}{9} \int \frac {-64+27 x^5-128 \log (2)-64 \log ^2(2)}{x^5} \, dx+\frac {1}{9} \left (32 e^2 (1+\log (2))^2\right ) \int \frac {-1+5 x}{x^6 \left (e^2-e^{\frac {1}{x}} x\right )} \, dx-\frac {1}{9} \left (64 e^4 (1+\log (2))^2\right ) \int \frac {-1+2 x}{x^6 \left (e^2-e^{\frac {1}{x}} x\right )^2} \, dx+\frac {1}{9} \left (32 e^6 (1+\log (2))^2\right ) \int \frac {-1+x}{x^6 \left (e^2-e^{\frac {1}{x}} x\right )^3} \, dx\\ &=\frac {1}{9} \int \left (27-\frac {64 (1+\log (2))^2}{x^5}\right ) \, dx+\frac {1}{9} \left (32 e^2 (1+\log (2))^2\right ) \int \left (\frac {1}{x^6 \left (-e^2+e^{\frac {1}{x}} x\right )}-\frac {5}{x^5 \left (-e^2+e^{\frac {1}{x}} x\right )}\right ) \, dx-\frac {1}{9} \left (64 e^4 (1+\log (2))^2\right ) \int \left (-\frac {1}{x^6 \left (-e^2+e^{\frac {1}{x}} x\right )^2}+\frac {2}{x^5 \left (-e^2+e^{\frac {1}{x}} x\right )^2}\right ) \, dx+\frac {1}{9} \left (32 e^6 (1+\log (2))^2\right ) \int \left (\frac {1}{x^6 \left (-e^2+e^{\frac {1}{x}} x\right )^3}-\frac {1}{x^5 \left (-e^2+e^{\frac {1}{x}} x\right )^3}\right ) \, dx\\ &=3 x+\frac {16 (1+\log (2))^2}{9 x^4}+\frac {1}{9} \left (32 e^2 (1+\log (2))^2\right ) \int \frac {1}{x^6 \left (-e^2+e^{\frac {1}{x}} x\right )} \, dx-\frac {1}{9} \left (160 e^2 (1+\log (2))^2\right ) \int \frac {1}{x^5 \left (-e^2+e^{\frac {1}{x}} x\right )} \, dx+\frac {1}{9} \left (64 e^4 (1+\log (2))^2\right ) \int \frac {1}{x^6 \left (-e^2+e^{\frac {1}{x}} x\right )^2} \, dx-\frac {1}{9} \left (128 e^4 (1+\log (2))^2\right ) \int \frac {1}{x^5 \left (-e^2+e^{\frac {1}{x}} x\right )^2} \, dx+\frac {1}{9} \left (32 e^6 (1+\log (2))^2\right ) \int \frac {1}{x^6 \left (-e^2+e^{\frac {1}{x}} x\right )^3} \, dx-\frac {1}{9} \left (32 e^6 (1+\log (2))^2\right ) \int \frac {1}{x^5 \left (-e^2+e^{\frac {1}{x}} x\right )^3} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 64, normalized size = 1.78 \begin {gather*} \frac {27 e^4 x^3-54 e^{2+\frac {1}{x}} x^4+e^{2/x} \left (27 x^5+16 (1+\log (2))^2\right )}{9 x^2 \left (e^2-e^{\frac {1}{x}} x\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(64*x^2 + 27*E^((3*(-1 + 2*x))/x)*x^4 - 81*E^((2*(-1 + 2*x))/x)*x^5 - 27*x^7 + 128*x^2*Log[2] + 64*x
^2*Log[2]^2 + E^((-1 + 2*x)/x)*(-32 - 32*x + 81*x^6 + (-64 - 64*x)*Log[2] + (-32 - 32*x)*Log[2]^2))/(9*E^((3*(
-1 + 2*x))/x)*x^4 - 27*E^((2*(-1 + 2*x))/x)*x^5 + 27*E^((-1 + 2*x)/x)*x^6 - 9*x^7),x]

[Out]

(27*E^4*x^3 - 54*E^(2 + x^(-1))*x^4 + E^(2/x)*(27*x^5 + 16*(1 + Log[2])^2))/(9*x^2*(E^2 - E^x^(-1)*x)^2)

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fricas [B]  time = 1.13, size = 86, normalized size = 2.39 \begin {gather*} \frac {27 \, x^{5} - 54 \, x^{4} e^{\left (\frac {2 \, x - 1}{x}\right )} + 27 \, x^{3} e^{\left (\frac {2 \, {\left (2 \, x - 1\right )}}{x}\right )} + 16 \, \log \relax (2)^{2} + 32 \, \log \relax (2) + 16}{9 \, {\left (x^{4} - 2 \, x^{3} e^{\left (\frac {2 \, x - 1}{x}\right )} + x^{2} e^{\left (\frac {2 \, {\left (2 \, x - 1\right )}}{x}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*x^4*exp((2*x-1)/x)^3-81*x^5*exp((2*x-1)/x)^2+((-32*x-32)*log(2)^2+(-64*x-64)*log(2)+81*x^6-32*x-
32)*exp((2*x-1)/x)+64*x^2*log(2)^2+128*x^2*log(2)-27*x^7+64*x^2)/(9*x^4*exp((2*x-1)/x)^3-27*x^5*exp((2*x-1)/x)
^2+27*x^6*exp((2*x-1)/x)-9*x^7),x, algorithm="fricas")

[Out]

1/9*(27*x^5 - 54*x^4*e^((2*x - 1)/x) + 27*x^3*e^(2*(2*x - 1)/x) + 16*log(2)^2 + 32*log(2) + 16)/(x^4 - 2*x^3*e
^((2*x - 1)/x) + x^2*e^(2*(2*x - 1)/x))

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giac [B]  time = 0.28, size = 82, normalized size = 2.28 \begin {gather*} -\frac {\frac {54 \, e^{\left (-\frac {1}{x} + 2\right )}}{x} - \frac {27 \, e^{\left (-\frac {2}{x} + 4\right )}}{x^{2}} - \frac {16 \, \log \relax (2)^{2}}{x^{5}} - \frac {32 \, \log \relax (2)}{x^{5}} - \frac {16}{x^{5}} - 27}{9 \, {\left (\frac {1}{x} - \frac {2 \, e^{\left (-\frac {1}{x} + 2\right )}}{x^{2}} + \frac {e^{\left (-\frac {2}{x} + 4\right )}}{x^{3}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*x^4*exp((2*x-1)/x)^3-81*x^5*exp((2*x-1)/x)^2+((-32*x-32)*log(2)^2+(-64*x-64)*log(2)+81*x^6-32*x-
32)*exp((2*x-1)/x)+64*x^2*log(2)^2+128*x^2*log(2)-27*x^7+64*x^2)/(9*x^4*exp((2*x-1)/x)^3-27*x^5*exp((2*x-1)/x)
^2+27*x^6*exp((2*x-1)/x)-9*x^7),x, algorithm="giac")

[Out]

-1/9*(54*e^(-1/x + 2)/x - 27*e^(-2/x + 4)/x^2 - 16*log(2)^2/x^5 - 32*log(2)/x^5 - 16/x^5 - 27)/(1/x - 2*e^(-1/
x + 2)/x^2 + e^(-2/x + 4)/x^3)

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maple [A]  time = 0.31, size = 36, normalized size = 1.00

method	result	size
risch	\(3 x +\frac {\frac {16 \ln \relax (2)^{2}}{9}+\frac {32 \ln \relax (2)}{9}+\frac {16}{9}}{x^{2} \left (x -{\mathrm e}^{\frac {2 x -1}{x}}\right )^{2}}\)	\(36\)
norman	\(\frac {\left (\frac {16 \ln \relax (2)^{2}}{9}+\frac {32 \ln \relax (2)}{9}+\frac {16}{9}\right ) x +3 x^{6}+3 x^{4} {\mathrm e}^{\frac {4 x -2}{x}}-6 x^{5} {\mathrm e}^{\frac {2 x -1}{x}}}{x^{3} \left (x -{\mathrm e}^{\frac {2 x -1}{x}}\right )^{2}}\)	\(73\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((27*x^4*exp((2*x-1)/x)^3-81*x^5*exp((2*x-1)/x)^2+((-32*x-32)*ln(2)^2+(-64*x-64)*ln(2)+81*x^6-32*x-32)*exp(
(2*x-1)/x)+64*x^2*ln(2)^2+128*x^2*ln(2)-27*x^7+64*x^2)/(9*x^4*exp((2*x-1)/x)^3-27*x^5*exp((2*x-1)/x)^2+27*x^6*
exp((2*x-1)/x)-9*x^7),x,method=_RETURNVERBOSE)

[Out]

3*x+16/9*(ln(2)^2+2*ln(2)+1)/x^2/(x-exp((2*x-1)/x))^2

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maxima [B]  time = 0.49, size = 76, normalized size = 2.11 \begin {gather*} -\frac {54 \, x^{4} e^{\left (\frac {1}{x} + 2\right )} - 27 \, x^{3} e^{4} - {\left (27 \, x^{5} + 16 \, \log \relax (2)^{2} + 32 \, \log \relax (2) + 16\right )} e^{\frac {2}{x}}}{9 \, {\left (x^{4} e^{\frac {2}{x}} - 2 \, x^{3} e^{\left (\frac {1}{x} + 2\right )} + x^{2} e^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*x^4*exp((2*x-1)/x)^3-81*x^5*exp((2*x-1)/x)^2+((-32*x-32)*log(2)^2+(-64*x-64)*log(2)+81*x^6-32*x-
32)*exp((2*x-1)/x)+64*x^2*log(2)^2+128*x^2*log(2)-27*x^7+64*x^2)/(9*x^4*exp((2*x-1)/x)^3-27*x^5*exp((2*x-1)/x)
^2+27*x^6*exp((2*x-1)/x)-9*x^7),x, algorithm="maxima")

[Out]

-1/9*(54*x^4*e^(1/x + 2) - 27*x^3*e^4 - (27*x^5 + 16*log(2)^2 + 32*log(2) + 16)*e^(2/x))/(x^4*e^(2/x) - 2*x^3*
e^(1/x + 2) + x^2*e^4)

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mupad [B]  time = 3.40, size = 34, normalized size = 0.94 \begin {gather*} 3\,x+\frac {\frac {32\,\ln \relax (2)}{9}+\frac {16\,{\ln \relax (2)}^2}{9}+\frac {16}{9}}{x^2\,{\left (x-{\mathrm {e}}^{2-\frac {1}{x}}\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((64*x^2*log(2)^2 - exp((2*x - 1)/x)*(32*x + log(2)*(64*x + 64) + log(2)^2*(32*x + 32) - 81*x^6 + 32) + 27*
x^4*exp((3*(2*x - 1))/x) - 81*x^5*exp((2*(2*x - 1))/x) + 128*x^2*log(2) + 64*x^2 - 27*x^7)/(27*x^6*exp((2*x -
1)/x) + 9*x^4*exp((3*(2*x - 1))/x) - 27*x^5*exp((2*(2*x - 1))/x) - 9*x^7),x)

[Out]

3*x + ((32*log(2))/9 + (16*log(2)^2)/9 + 16/9)/(x^2*(x - exp(2 - 1/x))^2)

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sympy [A]  time = 0.21, size = 48, normalized size = 1.33 \begin {gather*} 3 x + \frac {16 \log {\relax (2 )}^{2} + 16 + 32 \log {\relax (2 )}}{9 x^{4} - 18 x^{3} e^{\frac {2 x - 1}{x}} + 9 x^{2} e^{\frac {2 \left (2 x - 1\right )}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((27*x**4*exp((2*x-1)/x)**3-81*x**5*exp((2*x-1)/x)**2+((-32*x-32)*ln(2)**2+(-64*x-64)*ln(2)+81*x**6-3
2*x-32)*exp((2*x-1)/x)+64*x**2*ln(2)**2+128*x**2*ln(2)-27*x**7+64*x**2)/(9*x**4*exp((2*x-1)/x)**3-27*x**5*exp(
(2*x-1)/x)**2+27*x**6*exp((2*x-1)/x)-9*x**7),x)

[Out]

3*x + (16*log(2)**2 + 16 + 32*log(2))/(9*x**4 - 18*x**3*exp((2*x - 1)/x) + 9*x**2*exp(2*(2*x - 1)/x))

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