3.19.75 \(\int \frac {e^{e^{\frac {-256+32 x^3-x^6+x^4 \log (5)}{x^3}}+x} (-x^4+e^{\frac {-256+32 x^3-x^6+x^4 \log (5)}{x^3}} (-768+3 x^6-x^4 \log (5)))}{x^4} \, dx\)

Optimal. Leaf size=28 \[ 1-e^{e^{x \left (-\left (\frac {16}{x^2}-x\right )^2+\log (5)\right )}+x} \]

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Rubi [A]  time = 0.83, antiderivative size = 27, normalized size of antiderivative = 0.96, number of steps used = 1, number of rules used = 1, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {6706} \begin {gather*} -e^{5^x e^{-\frac {x^6-32 x^3+256}{x^3}}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(E^(E^((-256 + 32*x^3 - x^6 + x^4*Log[5])/x^3) + x)*(-x^4 + E^((-256 + 32*x^3 - x^6 + x^4*Log[5])/x^3)*(-7
68 + 3*x^6 - x^4*Log[5])))/x^4,x]

[Out]

-E^(5^x/E^((256 - 32*x^3 + x^6)/x^3) + x)

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-e^{5^x e^{-\frac {256-32 x^3+x^6}{x^3}}+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.14, size = 24, normalized size = 0.86 \begin {gather*} -e^{5^x e^{32-\frac {256}{x^3}-x^3}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^((-256 + 32*x^3 - x^6 + x^4*Log[5])/x^3) + x)*(-x^4 + E^((-256 + 32*x^3 - x^6 + x^4*Log[5])/x^
3)*(-768 + 3*x^6 - x^4*Log[5])))/x^4,x]

[Out]

-E^(5^x*E^(32 - 256/x^3 - x^3) + x)

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fricas [A]  time = 0.87, size = 28, normalized size = 1.00 \begin {gather*} -e^{\left (x + e^{\left (-\frac {x^{6} - x^{4} \log \relax (5) - 32 \, x^{3} + 256}{x^{3}}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4*log(5)+3*x^6-768)*exp((x^4*log(5)-x^6+32*x^3-256)/x^3)-x^4)*exp(exp((x^4*log(5)-x^6+32*x^3-25
6)/x^3)+x)/x^4,x, algorithm="fricas")

[Out]

-e^(x + e^(-(x^6 - x^4*log(5) - 32*x^3 + 256)/x^3))

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giac [A]  time = 0.24, size = 22, normalized size = 0.79 \begin {gather*} -e^{\left (x + e^{\left (-x^{3} + x \log \relax (5) - \frac {256}{x^{3}} + 32\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4*log(5)+3*x^6-768)*exp((x^4*log(5)-x^6+32*x^3-256)/x^3)-x^4)*exp(exp((x^4*log(5)-x^6+32*x^3-25
6)/x^3)+x)/x^4,x, algorithm="giac")

[Out]

-e^(x + e^(-x^3 + x*log(5) - 256/x^3 + 32))

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maple [A]  time = 0.12, size = 23, normalized size = 0.82

method	result	size
risch	\(-{\mathrm e}^{5^{x} {\mathrm e}^{-\frac {\left (x^{3}-16\right )^{2}}{x^{3}}}+x}\)	\(23\)
norman	\(-{\mathrm e}^{{\mathrm e}^{\frac {x^{4} \ln \relax (5)-x^{6}+32 x^{3}-256}{x^{3}}}+x}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^4*ln(5)+3*x^6-768)*exp((x^4*ln(5)-x^6+32*x^3-256)/x^3)-x^4)*exp(exp((x^4*ln(5)-x^6+32*x^3-256)/x^3)+x
)/x^4,x,method=_RETURNVERBOSE)

[Out]

-exp(5^x*exp(-(x^3-16)^2/x^3)+x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {{\left (x^{4} - {\left (3 \, x^{6} - x^{4} \log \relax (5) - 768\right )} e^{\left (-\frac {x^{6} - x^{4} \log \relax (5) - 32 \, x^{3} + 256}{x^{3}}\right )}\right )} e^{\left (x + e^{\left (-\frac {x^{6} - x^{4} \log \relax (5) - 32 \, x^{3} + 256}{x^{3}}\right )}\right )}}{x^{4}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^4*log(5)+3*x^6-768)*exp((x^4*log(5)-x^6+32*x^3-256)/x^3)-x^4)*exp(exp((x^4*log(5)-x^6+32*x^3-25
6)/x^3)+x)/x^4,x, algorithm="maxima")

[Out]

-integrate((x^4 - (3*x^6 - x^4*log(5) - 768)*e^(-(x^6 - x^4*log(5) - 32*x^3 + 256)/x^3))*e^(x + e^(-(x^6 - x^4
*log(5) - 32*x^3 + 256)/x^3))/x^4, x)

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mupad [B]  time = 1.29, size = 23, normalized size = 0.82 \begin {gather*} -{\mathrm {e}}^{5^x\,{\mathrm {e}}^{32}\,{\mathrm {e}}^{-x^3}\,{\mathrm {e}}^{-\frac {256}{x^3}}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x + exp((x^4*log(5) + 32*x^3 - x^6 - 256)/x^3))*(exp((x^4*log(5) + 32*x^3 - x^6 - 256)/x^3)*(x^4*log
(5) - 3*x^6 + 768) + x^4))/x^4,x)

[Out]

-exp(5^x*exp(32)*exp(-x^3)*exp(-256/x^3))*exp(x)

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sympy [A]  time = 0.52, size = 26, normalized size = 0.93 \begin {gather*} - e^{x + e^{\frac {- x^{6} + x^{4} \log {\relax (5 )} + 32 x^{3} - 256}{x^{3}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**4*ln(5)+3*x**6-768)*exp((x**4*ln(5)-x**6+32*x**3-256)/x**3)-x**4)*exp(exp((x**4*ln(5)-x**6+32*
x**3-256)/x**3)+x)/x**4,x)

[Out]

-exp(x + exp((-x**6 + x**4*log(5) + 32*x**3 - 256)/x**3))

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