3.40.69 \(\int \frac {-216 x+216 x^3-54 x^5+e^{4/x} (-216+54 x+108 x^2+27 x^3)+(-54 x-27 x^3) \log (2)}{16 x^3+e^{8/x} x^3-16 x^5+4 x^7+(8 x^3-4 x^5) \log (2)+x^3 \log ^2(2)+e^{4/x} (-8 x^3+4 x^5-2 x^3 \log (2))} \, dx\)

Optimal. Leaf size=32 \[ \frac {27}{2 x-\frac {x \left (e^{4/x}-\log (2)\right )}{2-x^2}} \]

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Rubi [F]  time = 2.98, 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 {-216 x+216 x^3-54 x^5+e^{4/x} \left (-216+54 x+108 x^2+27 x^3\right )+\left (-54 x-27 x^3\right ) \log (2)}{16 x^3+e^{8/x} x^3-16 x^5+4 x^7+\left (8 x^3-4 x^5\right ) \log (2)+x^3 \log ^2(2)+e^{4/x} \left (-8 x^3+4 x^5-2 x^3 \log (2)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-216*x + 216*x^3 - 54*x^5 + E^(4/x)*(-216 + 54*x + 108*x^2 + 27*x^3) + (-54*x - 27*x^3)*Log[2])/(16*x^3 +
E^(8/x)*x^3 - 16*x^5 + 4*x^7 + (8*x^3 - 4*x^5)*Log[2] + x^3*Log[2]^2 + E^(4/x)*(-8*x^3 + 4*x^5 - 2*x^3*Log[2]
)),x]

[Out]

216*Defer[Int][(E^(4/x) + 2*x^2 - 4*(1 + Log[2]/4))^(-2), x] - 216*(4 + Log[2])*Defer[Int][1/(x^3*(E^(4/x) + 2
*x^2 - 4*(1 + Log[2]/4))^2), x] + 108*(8 + Log[2])*Defer[Int][1/(x*(E^(4/x) + 2*x^2 - 4*(1 + Log[2]/4))^2), x]
- 216*Defer[Int][x/(E^(4/x) + 2*x^2 - 4*(1 + Log[2]/4))^2, x] - 108*Defer[Int][x^2/(E^(4/x) + 2*x^2 - 4*(1 +
Log[2]/4))^2, x] + 27*Defer[Int][(E^(4/x) + 2*x^2 - 4*(1 + Log[2]/4))^(-1), x] + 54*Defer[Int][1/(x^2*(E^(4/x)
+ 2*x^2 - 4*(1 + Log[2]/4))), x] + 108*Defer[Int][1/(x*(E^(4/x) + 2*x^2 - 4*(1 + Log[2]/4))), x] + 216*Defer[
Int][1/(x^3*(-E^(4/x) - 2*x^2 + 4*(1 + Log[2]/4))), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-216 x+216 x^3-54 x^5+e^{4/x} \left (-216+54 x+108 x^2+27 x^3\right )+\left (-54 x-27 x^3\right ) \log (2)}{e^{8/x} x^3-16 x^5+4 x^7+\left (8 x^3-4 x^5\right ) \log (2)+e^{4/x} \left (-8 x^3+4 x^5-2 x^3 \log (2)\right )+x^3 \left (16+\log ^2(2)\right )} \, dx\\ &=\int \frac {27 \left (e^{4/x} \left (-8+2 x+4 x^2+x^3\right )-x \left (8+2 x^4+x^2 (-8+\log (2))+\log (4)\right )\right )}{x^3 \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2} \, dx\\ &=27 \int \frac {e^{4/x} \left (-8+2 x+4 x^2+x^3\right )-x \left (8+2 x^4+x^2 (-8+\log (2))+\log (4)\right )}{x^3 \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2} \, dx\\ &=27 \int \left (\frac {8 x^3-8 x^4-4 x^5+32 x^2 \left (1+\frac {\log (2)}{8}\right )-32 \left (1+\frac {\log (2)}{4}\right )}{x^3 \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2}+\frac {-8+2 x+4 x^2+x^3}{x^3 \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )}\right ) \, dx\\ &=27 \int \frac {8 x^3-8 x^4-4 x^5+32 x^2 \left (1+\frac {\log (2)}{8}\right )-32 \left (1+\frac {\log (2)}{4}\right )}{x^3 \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2} \, dx+27 \int \frac {-8+2 x+4 x^2+x^3}{x^3 \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )} \, dx\\ &=27 \int \left (\frac {1}{e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )}+\frac {2}{x^2 \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )}+\frac {4}{x \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )}+\frac {8}{x^3 \left (-e^{4/x}-2 x^2+4 \left (1+\frac {\log (2)}{4}\right )\right )}\right ) \, dx+27 \int \frac {4 \left (2-x^2\right ) \left (-4+2 x^2+x^3-\log (2)\right )}{x^3 \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2} \, dx\\ &=27 \int \frac {1}{e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )} \, dx+54 \int \frac {1}{x^2 \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )} \, dx+108 \int \frac {1}{x \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )} \, dx+108 \int \frac {\left (2-x^2\right ) \left (-4+2 x^2+x^3-\log (2)\right )}{x^3 \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2} \, dx+216 \int \frac {1}{x^3 \left (-e^{4/x}-2 x^2+4 \left (1+\frac {\log (2)}{4}\right )\right )} \, dx\\ &=27 \int \frac {1}{e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )} \, dx+54 \int \frac {1}{x^2 \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )} \, dx+108 \int \frac {1}{x \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )} \, dx+108 \int \left (\frac {2}{\left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2}-\frac {2 x}{\left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2}-\frac {x^2}{\left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2}+\frac {2 (-4-\log (2))}{x^3 \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2}+\frac {8+\log (2)}{x \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2}\right ) \, dx+216 \int \frac {1}{x^3 \left (-e^{4/x}-2 x^2+4 \left (1+\frac {\log (2)}{4}\right )\right )} \, dx\\ &=27 \int \frac {1}{e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )} \, dx+54 \int \frac {1}{x^2 \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )} \, dx-108 \int \frac {x^2}{\left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2} \, dx+108 \int \frac {1}{x \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )} \, dx+216 \int \frac {1}{\left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2} \, dx-216 \int \frac {x}{\left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2} \, dx+216 \int \frac {1}{x^3 \left (-e^{4/x}-2 x^2+4 \left (1+\frac {\log (2)}{4}\right )\right )} \, dx-(216 (4+\log (2))) \int \frac {1}{x^3 \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2} \, dx+(108 (8+\log (2))) \int \frac {1}{x \left (e^{4/x}+2 x^2-4 \left (1+\frac {\log (2)}{4}\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.72, size = 71, normalized size = 2.22 \begin {gather*} \frac {27 \left (32-8 x^3+8 x^4+4 x^5-4 x^2 (8+\log (2))+\log (256)\right )}{4 x \left (-4+e^{4/x}+2 x^2-\log (2)\right ) \left (-4+2 x^2+x^3-\log (2)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-216*x + 216*x^3 - 54*x^5 + E^(4/x)*(-216 + 54*x + 108*x^2 + 27*x^3) + (-54*x - 27*x^3)*Log[2])/(16
*x^3 + E^(8/x)*x^3 - 16*x^5 + 4*x^7 + (8*x^3 - 4*x^5)*Log[2] + x^3*Log[2]^2 + E^(4/x)*(-8*x^3 + 4*x^5 - 2*x^3*
Log[2])),x]

[Out]

(27*(32 - 8*x^3 + 8*x^4 + 4*x^5 - 4*x^2*(8 + Log[2]) + Log[256]))/(4*x*(-4 + E^(4/x) + 2*x^2 - Log[2])*(-4 + 2
*x^2 + x^3 - Log[2]))

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fricas [A]  time = 0.93, size = 31, normalized size = 0.97 \begin {gather*} \frac {27 \, {\left (x^{2} - 2\right )}}{2 \, x^{3} + x e^{\frac {4}{x}} - x \log \relax (2) - 4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^3+108*x^2+54*x-216)*exp(4/x)+(-27*x^3-54*x)*log(2)-54*x^5+216*x^3-216*x)/(x^3*exp(4/x)^2+(-2*
x^3*log(2)+4*x^5-8*x^3)*exp(4/x)+x^3*log(2)^2+(-4*x^5+8*x^3)*log(2)+4*x^7-16*x^5+16*x^3),x, algorithm="fricas"
)

[Out]

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

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^3+108*x^2+54*x-216)*exp(4/x)+(-27*x^3-54*x)*log(2)-54*x^5+216*x^3-216*x)/(x^3*exp(4/x)^2+(-2*
x^3*log(2)+4*x^5-8*x^3)*exp(4/x)+x^3*log(2)^2+(-4*x^5+8*x^3)*log(2)+4*x^7-16*x^5+16*x^3),x, algorithm="giac")

[Out]

Timed out

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maple [A]  time = 0.53, size = 30, normalized size = 0.94

method	result	size
risch	\(-\frac {27 \left (x^{2}-2\right )}{x \left (-2 x^{2}+\ln \relax (2)-{\mathrm e}^{\frac {4}{x}}+4\right )}\)	\(30\)
norman	\(\frac {-27 x^{3}+54 x}{x^{2} \left (-2 x^{2}+\ln \relax (2)-{\mathrm e}^{\frac {4}{x}}+4\right )}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((27*x^3+108*x^2+54*x-216)*exp(4/x)+(-27*x^3-54*x)*ln(2)-54*x^5+216*x^3-216*x)/(x^3*exp(4/x)^2+(-2*x^3*ln(
2)+4*x^5-8*x^3)*exp(4/x)+x^3*ln(2)^2+(-4*x^5+8*x^3)*ln(2)+4*x^7-16*x^5+16*x^3),x,method=_RETURNVERBOSE)

[Out]

-27*(x^2-2)/x/(-2*x^2+ln(2)-exp(4/x)+4)

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maxima [A]  time = 0.55, size = 30, normalized size = 0.94 \begin {gather*} \frac {27 \, {\left (x^{2} - 2\right )}}{2 \, x^{3} - x {\left (\log \relax (2) + 4\right )} + x e^{\frac {4}{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^3+108*x^2+54*x-216)*exp(4/x)+(-27*x^3-54*x)*log(2)-54*x^5+216*x^3-216*x)/(x^3*exp(4/x)^2+(-2*
x^3*log(2)+4*x^5-8*x^3)*exp(4/x)+x^3*log(2)^2+(-4*x^5+8*x^3)*log(2)+4*x^7-16*x^5+16*x^3),x, algorithm="maxima"
)

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {216\,x+\ln \relax (2)\,\left (27\,x^3+54\,x\right )-{\mathrm {e}}^{4/x}\,\left (27\,x^3+108\,x^2+54\,x-216\right )-216\,x^3+54\,x^5}{x^3\,{\ln \relax (2)}^2-{\mathrm {e}}^{4/x}\,\left (2\,x^3\,\ln \relax (2)+8\,x^3-4\,x^5\right )+\ln \relax (2)\,\left (8\,x^3-4\,x^5\right )+x^3\,{\mathrm {e}}^{8/x}+16\,x^3-16\,x^5+4\,x^7} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(216*x + log(2)*(54*x + 27*x^3) - exp(4/x)*(54*x + 108*x^2 + 27*x^3 - 216) - 216*x^3 + 54*x^5)/(x^3*log(2
)^2 - exp(4/x)*(2*x^3*log(2) + 8*x^3 - 4*x^5) + log(2)*(8*x^3 - 4*x^5) + x^3*exp(8/x) + 16*x^3 - 16*x^5 + 4*x^
7),x)

[Out]

int(-(216*x + log(2)*(54*x + 27*x^3) - exp(4/x)*(54*x + 108*x^2 + 27*x^3 - 216) - 216*x^3 + 54*x^5)/(x^3*log(2
)^2 - exp(4/x)*(2*x^3*log(2) + 8*x^3 - 4*x^5) + log(2)*(8*x^3 - 4*x^5) + x^3*exp(8/x) + 16*x^3 - 16*x^5 + 4*x^
7), x)

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sympy [A]  time = 0.17, size = 26, normalized size = 0.81 \begin {gather*} \frac {27 x^{2} - 54}{2 x^{3} + x e^{\frac {4}{x}} - 4 x - x \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x**3+108*x**2+54*x-216)*exp(4/x)+(-27*x**3-54*x)*ln(2)-54*x**5+216*x**3-216*x)/(x**3*exp(4/x)**
2+(-2*x**3*ln(2)+4*x**5-8*x**3)*exp(4/x)+x**3*ln(2)**2+(-4*x**5+8*x**3)*ln(2)+4*x**7-16*x**5+16*x**3),x)

[Out]

(27*x**2 - 54)/(2*x**3 + x*exp(4/x) - 4*x - x*log(2))

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