3.42.96 \(\int \frac {e^{-\frac {75}{x^2+x^5-x \log (2)}} (150 x+375 x^4-75 \log (2))}{x^4+x^{10}-2 x^3 \log (2)+x^2 \log ^2(2)+x^4 (2 x^3-2 x^2 \log (2))} \, dx\)

Optimal. Leaf size=18 \[ e^{-\frac {75}{x \left (x+x^4-\log (2)\right )}} \]

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Rubi [A]  time = 0.73, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 73, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.041, Rules used = {6688, 12, 6706} \begin {gather*} e^{-\frac {75}{x \left (x^4+x-\log (2)\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(150*x + 375*x^4 - 75*Log[2])/(E^(75/(x^2 + x^5 - x*Log[2]))*(x^4 + x^10 - 2*x^3*Log[2] + x^2*Log[2]^2 + x
^4*(2*x^3 - 2*x^2*Log[2]))),x]

[Out]

E^(-75/(x*(x + x^4 - Log[2])))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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} &=\int \frac {75 e^{-\frac {75}{x \left (x+x^4-\log (2)\right )}} \left (2 x+5 x^4-\log (2)\right )}{x^2 \left (x+x^4-\log (2)\right )^2} \, dx\\ &=75 \int \frac {e^{-\frac {75}{x \left (x+x^4-\log (2)\right )}} \left (2 x+5 x^4-\log (2)\right )}{x^2 \left (x+x^4-\log (2)\right )^2} \, dx\\ &=e^{-\frac {75}{x \left (x+x^4-\log (2)\right )}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 18, normalized size = 1.00 \begin {gather*} e^{-\frac {75}{x^2+x^5-x \log (2)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(150*x + 375*x^4 - 75*Log[2])/(E^(75/(x^2 + x^5 - x*Log[2]))*(x^4 + x^10 - 2*x^3*Log[2] + x^2*Log[2]
^2 + x^4*(2*x^3 - 2*x^2*Log[2]))),x]

[Out]

E^(-75/(x^2 + x^5 - x*Log[2]))

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fricas [A]  time = 0.56, size = 17, normalized size = 0.94 \begin {gather*} e^{\left (-\frac {75}{x^{5} + x^{2} - x \log \relax (2)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((375*x^4-75*log(2)+150*x)*exp(-75/(x^5-x*log(2)+x^2))/(x^10+(-2*x^2*log(2)+2*x^3)*x^4+x^2*log(2)^2-2
*x^3*log(2)+x^4),x, algorithm="fricas")

[Out]

e^(-75/(x^5 + x^2 - x*log(2)))

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giac [A]  time = 0.12, size = 17, normalized size = 0.94 \begin {gather*} e^{\left (-\frac {75}{x^{5} + x^{2} - x \log \relax (2)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((375*x^4-75*log(2)+150*x)*exp(-75/(x^5-x*log(2)+x^2))/(x^10+(-2*x^2*log(2)+2*x^3)*x^4+x^2*log(2)^2-2
*x^3*log(2)+x^4),x, algorithm="giac")

[Out]

e^(-75/(x^5 + x^2 - x*log(2)))

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maple [A]  time = 0.21, size = 20, normalized size = 1.11

method	result	size
gosper	\({\mathrm e}^{\frac {75}{x \left (-x^{4}+\ln \relax (2)-x \right )}}\)	\(20\)
risch	\({\mathrm e}^{\frac {75}{x \left (-x^{4}+\ln \relax (2)-x \right )}}\)	\(20\)
norman	\(\frac {x \ln \relax (2) {\mathrm e}^{-\frac {75}{x^{5}-x \ln \relax (2)+x^{2}}}-x^{2} {\mathrm e}^{-\frac {75}{x^{5}-x \ln \relax (2)+x^{2}}}-x^{5} {\mathrm e}^{-\frac {75}{x^{5}-x \ln \relax (2)+x^{2}}}}{x \left (-x^{4}+\ln \relax (2)-x \right )}\)	\(84\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((375*x^4-75*ln(2)+150*x)*exp(-75/(x^5-x*ln(2)+x^2))/(x^10+(-2*x^2*ln(2)+2*x^3)*x^4+x^2*ln(2)^2-2*x^3*ln(2)
+x^4),x,method=_RETURNVERBOSE)

[Out]

exp(75/x/(-x^4+ln(2)-x))

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maxima [B]  time = 0.55, size = 56, normalized size = 3.11 \begin {gather*} e^{\left (-\frac {75 \, x^{3}}{x^{4} \log \relax (2) + x \log \relax (2) - \log \relax (2)^{2}} - \frac {75}{x^{4} \log \relax (2) + x \log \relax (2) - \log \relax (2)^{2}} + \frac {75}{x \log \relax (2)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((375*x^4-75*log(2)+150*x)*exp(-75/(x^5-x*log(2)+x^2))/(x^10+(-2*x^2*log(2)+2*x^3)*x^4+x^2*log(2)^2-2
*x^3*log(2)+x^4),x, algorithm="maxima")

[Out]

e^(-75*x^3/(x^4*log(2) + x*log(2) - log(2)^2) - 75/(x^4*log(2) + x*log(2) - log(2)^2) + 75/(x*log(2)))

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mupad [B]  time = 3.56, size = 17, normalized size = 0.94 \begin {gather*} {\mathrm {e}}^{-\frac {75}{x^5+x^2-\ln \relax (2)\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-75/(x^2 - x*log(2) + x^5))*(150*x - 75*log(2) + 375*x^4))/(x^2*log(2)^2 - x^4*(2*x^2*log(2) - 2*x^3)
- 2*x^3*log(2) + x^4 + x^10),x)

[Out]

exp(-75/(x^2 - x*log(2) + x^5))

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sympy [A]  time = 0.84, size = 15, normalized size = 0.83 \begin {gather*} e^{- \frac {75}{x^{5} + x^{2} - x \log {\relax (2 )}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((375*x**4-75*ln(2)+150*x)*exp(-75/(x**5-x*ln(2)+x**2))/(x**10+(-2*x**2*ln(2)+2*x**3)*x**4+x**2*ln(2)
**2-2*x**3*ln(2)+x**4),x)

[Out]

exp(-75/(x**5 + x**2 - x*log(2)))

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