3.20.43 \(\int (8 e^{5+8 x}-2 \log (\frac {3}{2 x})+(2-2 \log (\frac {3}{2 x})) \log (x)) \, dx\) [1943]

Optimal. Leaf size=21 \[ e^{5+8 x}-2 x \log \left (\frac {3}{2 x}\right ) \log (x) \]

[Out]

exp(8*x+5)-2*x*ln(3/2/x)*ln(x)

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Rubi [A]
time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {2225, 2332, 2408, 12} \begin {gather*} e^{8 x+5}-2 x \log \left (\frac {3}{2 x}\right ) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[8*E^(5 + 8*x) - 2*Log[3/(2*x)] + (2 - 2*Log[3/(2*x)])*Log[x],x]

[Out]

E^(5 + 8*x) - 2*x*Log[3/(2*x)]*Log[x]

Rule 12

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

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2408

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.)), x_Symbol] :> With[{u =
IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[SimplifyIntegrand[u/x, x], x],
x]] /; FreeQ[{a, b, c, d, e, f, n, p, r}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-\left (2 \int \log \left (\frac {3}{2 x}\right ) \, dx\right )+8 \int e^{5+8 x} \, dx+\int \left (2-2 \log \left (\frac {3}{2 x}\right )\right ) \log (x) \, dx\\ &=e^{5+8 x}-2 x-2 x \log \left (\frac {3}{2 x}\right )-2 x \log \left (\frac {3}{2 x}\right ) \log (x)-\int -2 \log \left (\frac {3}{2 x}\right ) \, dx\\ &=e^{5+8 x}-2 x-2 x \log \left (\frac {3}{2 x}\right )-2 x \log \left (\frac {3}{2 x}\right ) \log (x)+2 \int \log \left (\frac {3}{2 x}\right ) \, dx\\ &=e^{5+8 x}-2 x \log \left (\frac {3}{2 x}\right ) \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.02, size = 21, normalized size = 1.00 \begin {gather*} e^{5+8 x}-2 x \log \left (\frac {3}{2 x}\right ) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[8*E^(5 + 8*x) - 2*Log[3/(2*x)] + (2 - 2*Log[3/(2*x)])*Log[x],x]

[Out]

E^(5 + 8*x) - 2*x*Log[3/(2*x)]*Log[x]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs. \(2(18)=36\).
time = 0.37, size = 86, normalized size = 4.10

method	result	size
norman	\({\mathrm e}^{8 x +5}-2 x \ln \left (\frac {3}{2 x}\right ) \ln \left (x \right )\)	\(19\)
risch	\(2 x \ln \left (x \right )^{2}+\left (-2+2 \ln \left (2\right )-2 \ln \left (3\right )\right ) x \ln \left (x \right )-2 x \ln \left (2\right )+2 x \ln \left (3\right )-2 x \ln \left (\frac {3}{2 x}\right )+{\mathrm e}^{8 x +5}\)	\(48\)
default	\(2 x \ln \left (2\right ) \ln \left (x \right )-2 x \ln \left (2\right )-2 x \ln \left (3\right ) \ln \left (x \right )+2 x \ln \left (3\right )-2 \ln \left (\frac {1}{x}\right ) x \left (\ln \left (\frac {1}{x}\right )+\ln \left (x \right )\right )-2 \left (\ln \left (\frac {1}{x}\right )+\ln \left (x \right )\right ) x +2 x \ln \left (\frac {1}{x}\right )^{2}+4 x \ln \left (\frac {1}{x}\right )+2 x \ln \left (x \right )+{\mathrm e}^{8 x +5}-2 x \ln \left (\frac {3}{2 x}\right )\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*ln(3/2/x)+2)*ln(x)-2*ln(3/2/x)+8*exp(8*x+5),x,method=_RETURNVERBOSE)

[Out]

2*x*ln(2)*ln(x)-2*x*ln(2)-2*x*ln(3)*ln(x)+2*x*ln(3)-2*ln(1/x)*x*(ln(1/x)+ln(x))-2*(ln(1/x)+ln(x))*x+2*x*ln(1/x
)^2+4*x*ln(1/x)+2*x*ln(x)+exp(8*x+5)-2*x*ln(3/2/x)

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Maxima [A]
time = 0.27, size = 18, normalized size = 0.86 \begin {gather*} -2 \, x \log \left (x\right ) \log \left (\frac {3}{2 \, x}\right ) + e^{\left (8 \, x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(3/2/x)+2)*log(x)-2*log(3/2/x)+8*exp(8*x+5),x, algorithm="maxima")

[Out]

-2*x*log(x)*log(3/2/x) + e^(8*x + 5)

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Fricas [A]
time = 0.37, size = 29, normalized size = 1.38 \begin {gather*} -2 \, x \log \left (\frac {3}{2}\right ) \log \left (\frac {3}{2 \, x}\right ) + 2 \, x \log \left (\frac {3}{2 \, x}\right )^{2} + e^{\left (8 \, x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(3/2/x)+2)*log(x)-2*log(3/2/x)+8*exp(8*x+5),x, algorithm="fricas")

[Out]

-2*x*log(3/2)*log(3/2/x) + 2*x*log(3/2/x)^2 + e^(8*x + 5)

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Sympy [A]
time = 0.09, size = 31, normalized size = 1.48 \begin {gather*} 2 x \log {\left (x \right )}^{2} + \left (- 2 x \log {\left (3 \right )} + 2 x \log {\left (2 \right )}\right ) \log {\left (x \right )} + e^{8 x + 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*ln(3/2/x)+2)*ln(x)-2*ln(3/2/x)+8*exp(8*x+5),x)

[Out]

2*x*log(x)**2 + (-2*x*log(3) + 2*x*log(2))*log(x) + exp(8*x + 5)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 52 vs. \(2 (18) = 36\).
time = 0.38, size = 52, normalized size = 2.48 \begin {gather*} -2 \, x \log \left (3\right ) \log \left (x\right ) + 2 \, x \log \left (2\right ) \log \left (x\right ) + 2 \, x \log \left (x\right )^{2} + 2 \, x \log \left (3\right ) - 2 \, x \log \left (2\right ) - 2 \, x \log \left (x\right ) - 2 \, x \log \left (\frac {3}{2 \, x}\right ) + e^{\left (8 \, x + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(3/2/x)+2)*log(x)-2*log(3/2/x)+8*exp(8*x+5),x, algorithm="giac")

[Out]

-2*x*log(3)*log(x) + 2*x*log(2)*log(x) + 2*x*log(x)^2 + 2*x*log(3) - 2*x*log(2) - 2*x*log(x) - 2*x*log(3/2/x)
+ e^(8*x + 5)

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Mupad [B]
time = 1.18, size = 30, normalized size = 1.43 \begin {gather*} {\mathrm {e}}^{8\,x+5}+2\,x\,\ln \left (2\right )\,\ln \left (x\right )-2\,x\,\ln \left (3\right )\,\ln \left (x\right )-2\,x\,\ln \left (\frac {1}{x}\right )\,\ln \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(8*exp(8*x + 5) - 2*log(3/(2*x)) - log(x)*(2*log(3/(2*x)) - 2),x)

[Out]

exp(8*x + 5) + 2*x*log(2)*log(x) - 2*x*log(3)*log(x) - 2*x*log(1/x)*log(x)

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