3.20.75 \(\int (-2 x-8 x^7+e^{5/x} (5 x^6-8 x^7)-\log (\log (25))) \, dx\) [1975]

Optimal. Leaf size=26 \[ x \left (-x+\left (-1-e^{5/x}\right ) x^7-\log (\log (25))\right ) \]

[Out]

(x^7*(-1-exp(5/x))-ln(2*ln(5))-x)*x

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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.05, antiderivative size = 35, normalized size of antiderivative = 1.35, number of steps used = 6, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1607, 2258, 2250} \begin {gather*} -3125000 \text {Gamma}\left (-8,-\frac {5}{x}\right )-390625 \text {Gamma}\left (-7,-\frac {5}{x}\right )-x^8-x^2-x \log (\log (25)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-2*x - 8*x^7 + E^(5/x)*(5*x^6 - 8*x^7) - Log[Log[25]],x]

[Out]

-x^2 - x^8 - 3125000*Gamma[-8, -5/x] - 390625*Gamma[-7, -5/x] - x*Log[Log[25]]

Rule 1607

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 2258

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-x^2-x^8-x \log (\log (25))+\int e^{5/x} \left (5 x^6-8 x^7\right ) \, dx\\ &=-x^2-x^8-x \log (\log (25))+\int e^{5/x} (5-8 x) x^6 \, dx\\ &=-x^2-x^8-x \log (\log (25))+\int \left (5 e^{5/x} x^6-8 e^{5/x} x^7\right ) \, dx\\ &=-x^2-x^8-x \log (\log (25))+5 \int e^{5/x} x^6 \, dx-8 \int e^{5/x} x^7 \, dx\\ &=-x^2-x^8-3125000 \Gamma \left (-8,-\frac {5}{x}\right )-390625 \Gamma \left (-7,-\frac {5}{x}\right )-x \log (\log (25))\\ \end {aligned} \end {gather*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.02, size = 35, normalized size = 1.35 \begin {gather*} -x^2-x^8-3125000 \Gamma \left (-8,-\frac {5}{x}\right )-390625 \Gamma \left (-7,-\frac {5}{x}\right )-x \log (\log (25)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-2*x - 8*x^7 + E^(5/x)*(5*x^6 - 8*x^7) - Log[Log[25]],x]

[Out]

-x^2 - x^8 - 3125000*Gamma[-8, -5/x] - 390625*Gamma[-7, -5/x] - x*Log[Log[25]]

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Maple [A]
time = 0.42, size = 31, normalized size = 1.19

method	result	size
default	\(-x^{8} {\mathrm e}^{\frac {5}{x}}-x^{2}-x^{8}-x \ln \left (2 \ln \left (5\right )\right )\)	\(31\)
derivativedivides	\(-x^{8}-x^{2}-x^{8} {\mathrm e}^{\frac {5}{x}}-x \ln \left (2\right )-x \ln \left (\ln \left (5\right )\right )\)	\(34\)
risch	\(-x^{8}-x^{2}-x^{8} {\mathrm e}^{\frac {5}{x}}-x \ln \left (2\right )-x \ln \left (\ln \left (5\right )\right )\)	\(34\)
norman	\(\left (-\ln \left (2\right )-\ln \left (\ln \left (5\right )\right )\right ) x -x^{2}-x^{8}-x^{8} {\mathrm e}^{\frac {5}{x}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-ln(2*ln(5))+(-8*x^7+5*x^6)*exp(5/x)-8*x^7-2*x,x,method=_RETURNVERBOSE)

[Out]

-x^8*exp(5/x)-x^2-x^8-x*ln(2*ln(5))

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Maxima [A]
time = 0.27, size = 30, normalized size = 1.15 \begin {gather*} -x^{8} e^{\frac {5}{x}} - x^{8} - x^{2} - x \log \left (2 \, \log \left (5\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-log(2*log(5))+(-8*x^7+5*x^6)*exp(5/x)-8*x^7-2*x,x, algorithm="maxima")

[Out]

-x^8*e^(5/x) - x^8 - x^2 - x*log(2*log(5))

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Fricas [A]
time = 0.38, size = 30, normalized size = 1.15 \begin {gather*} -x^{8} e^{\frac {5}{x}} - x^{8} - x^{2} - x \log \left (2 \, \log \left (5\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-log(2*log(5))+(-8*x^7+5*x^6)*exp(5/x)-8*x^7-2*x,x, algorithm="fricas")

[Out]

-x^8*e^(5/x) - x^8 - x^2 - x*log(2*log(5))

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Sympy [A]
time = 0.04, size = 26, normalized size = 1.00 \begin {gather*} - x^{8} e^{\frac {5}{x}} - x^{8} - x^{2} + x \left (- \log {\left (2 \right )} - \log {\left (\log {\left (5 \right )} \right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-ln(2*ln(5))+(-8*x**7+5*x**6)*exp(5/x)-8*x**7-2*x,x)

[Out]

-x**8*exp(5/x) - x**8 - x**2 + x*(-log(2) - log(log(5)))

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Giac [A]
time = 0.39, size = 30, normalized size = 1.15 \begin {gather*} -x^{8} e^{\frac {5}{x}} - x^{8} - x^{2} - x \log \left (2 \, \log \left (5\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-log(2*log(5))+(-8*x^7+5*x^6)*exp(5/x)-8*x^7-2*x,x, algorithm="giac")

[Out]

-x^8*e^(5/x) - x^8 - x^2 - x*log(2*log(5))

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Mupad [B]
time = 1.15, size = 28, normalized size = 1.08 \begin {gather*} -x\,\ln \left (\ln \left (25\right )\right )-x^8\,{\mathrm {e}}^{5/x}-x^2-x^8 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(5/x)*(5*x^6 - 8*x^7) - log(2*log(5)) - 2*x - 8*x^7,x)

[Out]

- x*log(log(25)) - x^8*exp(5/x) - x^2 - x^8

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