3.60.4 \(\int (-16+e^{2 x} (32-2 \log (\frac {6561}{625}))+\log (\frac {6561}{625})) \, dx\)

Optimal. Leaf size=18 \[ \left (e^{2 x}-x\right ) \left (16-\log \left (\frac {6561}{625}\right )\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.44, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2194} \begin {gather*} e^{2 x} \left (16-\log \left (\frac {6561}{625}\right )\right )-x \left (16-\log \left (\frac {6561}{625}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-16 + E^(2*x)*(32 - 2*Log[6561/625]) + Log[6561/625],x]

[Out]

E^(2*x)*(16 - Log[6561/625]) - x*(16 - Log[6561/625])

Rule 2194

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]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-x \left (16-\log \left (\frac {6561}{625}\right )\right )+\left (2 \left (16-\log \left (\frac {6561}{625}\right )\right )\right ) \int e^{2 x} \, dx\\ &=e^{2 x} \left (16-\log \left (\frac {6561}{625}\right )\right )-x \left (16-\log \left (\frac {6561}{625}\right )\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 0.94 \begin {gather*} -\left (\left (e^{2 x}-x\right ) \left (-16+\log \left (\frac {6561}{625}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-16 + E^(2*x)*(32 - 2*Log[6561/625]) + Log[6561/625],x]

[Out]

-((E^(2*x) - x)*(-16 + Log[6561/625]))

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fricas [A]  time = 0.65, size = 18, normalized size = 1.00 \begin {gather*} {\left (\log \left (\frac {625}{6561}\right ) + 16\right )} e^{\left (2 \, x\right )} - x \log \left (\frac {625}{6561}\right ) - 16 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(625/6561)+32)*exp(x)^2-log(625/6561)-16,x, algorithm="fricas")

[Out]

(log(625/6561) + 16)*e^(2*x) - x*log(625/6561) - 16*x

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giac [A]  time = 0.12, size = 18, normalized size = 1.00 \begin {gather*} {\left (\log \left (\frac {625}{6561}\right ) + 16\right )} e^{\left (2 \, x\right )} - x \log \left (\frac {625}{6561}\right ) - 16 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(625/6561)+32)*exp(x)^2-log(625/6561)-16,x, algorithm="giac")

[Out]

(log(625/6561) + 16)*e^(2*x) - x*log(625/6561) - 16*x

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maple [A]  time = 0.04, size = 16, normalized size = 0.89

method	result	size
derivativedivides	\(\left (16+\ln \left (\frac {625}{6561}\right )\right ) \left ({\mathrm e}^{2 x}-\ln \left ({\mathrm e}^{x}\right )\right )\)	\(16\)
default	\(-16 x +{\mathrm e}^{2 x} \ln \left (\frac {625}{6561}\right )+16 \,{\mathrm e}^{2 x}-\ln \left (\frac {625}{6561}\right ) x\)	\(23\)
norman	\(\left (-4 \ln \relax (5)+8 \ln \relax (3)-16\right ) x +\left (4 \ln \relax (5)-8 \ln \relax (3)+16\right ) {\mathrm e}^{2 x}\)	\(29\)
risch	\(4 \,{\mathrm e}^{2 x} \ln \relax (5)-8 \,{\mathrm e}^{2 x} \ln \relax (3)+16 \,{\mathrm e}^{2 x}-4 x \ln \relax (5)+8 x \ln \relax (3)-16 x\)	\(37\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*ln(625/6561)+32)*exp(x)^2-ln(625/6561)-16,x,method=_RETURNVERBOSE)

[Out]

(16+ln(625/6561))*(exp(x)^2-ln(exp(x)))

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maxima [A]  time = 0.36, size = 18, normalized size = 1.00 \begin {gather*} {\left (\log \left (\frac {625}{6561}\right ) + 16\right )} e^{\left (2 \, x\right )} - x \log \left (\frac {625}{6561}\right ) - 16 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(625/6561)+32)*exp(x)^2-log(625/6561)-16,x, algorithm="maxima")

[Out]

(log(625/6561) + 16)*e^(2*x) - x*log(625/6561) - 16*x

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mupad [B]  time = 4.57, size = 16, normalized size = 0.89 \begin {gather*} {\mathrm {e}}^{2\,x}\,\left (\ln \left (\frac {625}{6561}\right )+16\right )+x\,\left (\ln \left (\frac {6561}{625}\right )-16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2*x)*(2*log(625/6561) + 32) - log(625/6561) - 16,x)

[Out]

exp(2*x)*(log(625/6561) + 16) + x*(log(6561/625) - 16)

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sympy [B]  time = 0.11, size = 29, normalized size = 1.61 \begin {gather*} x \left (-16 - 4 \log {\relax (5 )} + 8 \log {\relax (3 )}\right ) + \left (- 8 \log {\relax (3 )} + 4 \log {\relax (5 )} + 16\right ) e^{2 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*ln(625/6561)+32)*exp(x)**2-ln(625/6561)-16,x)

[Out]

x*(-16 - 4*log(5) + 8*log(3)) + (-8*log(3) + 4*log(5) + 16)*exp(2*x)

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